# Introduction and Background (Tuesday, January 11)

::: {.remark}
References: [@jacobson_2009].
:::

::: {.remark}
Idea: study representation by studying associated geometric objects, and use homological methods to bridge the two. The representation theory side will mostly be rings/modules, and the geometric side will involve algebraic geometry and commutative algebra. Throughout the course, all rings will be unital and all actions on the left.
:::

::: {.example title="of categories of modules"}
Recall the definition of a left \( R{\hbox{-}} \)module. Some examples:

-   \( k\in \mathsf{Field}\implies {\mathsf{k}{\hbox{-}}\mathsf{Mod}} = { \mathsf{Vect} }_k \)
-   \( R={\mathbb{Z}}\implies {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} = {\mathsf{Ab}}{\mathsf{Grp}} \).
-   \( A\in{\mathsf{Alg}}_{/ {k}}  \), which is a ring \( (A, +, \cdot) \) where \( (A, +, .) \) (using scalar multiplication) is a vector space.
    -   E.g. \( \operatorname{Mat}(n\times n, {\mathbb{C}}) \).
    -   E.g. for \( G \) a finite group, the group algebra \( kG \) for \( k\in \mathsf{Field} \).
    -   E.g. \( U({\mathfrak{g}}) \) for \( {\mathfrak{g}}\in \mathsf{Lie}{\mathsf{Alg}} \) or a super algebra.
:::

::: {.remark}
Connecting this to representation theory: for \( A\in {\mathsf{Alg}}_{/ {k}}  \) and \( M\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}} \), a representation of \( A \) is a morphism of algebras \( A \xrightarrow{\rho} {\mathfrak{gl}}_n(k) \), the algebra of all \( n\times n \) matrices (not necessarily invertible). Note that for groups, one instead asks for maps \( kG\to \operatorname{GL}_n \), the invertible matrices. There is a correspondence between \( {\mathsf{A}{\hbox{-}}\mathsf{Mod}} \rightleftharpoons{\mathsf{Rep}}(A) \): given \( M \), one can define the action as
\[
\rho: A &\to \mathop{\mathrm{End}}_k(M) \\
\rho(a)(m) &= a.m
.\]
:::

::: {.remark}
Recall the definitions of:

-   Morphisms of \( R{\hbox{-}} \)modules: \( f(r.m_1 + m_2) = r.f(m_1) + f(m_2) \)
-   Submodules: \( N\leq M \iff r.n \in N \) and \( N \) is closed under \( + \).
-   Quotient modules: \( M/N = \left\{{m + N}\right\} \).
-   The fundamental homomorphism theorem: for \( M \xrightarrow{f} N \), there is an induced \( \psi: M/\ker f\to N \) where \( M/\ker f\cong \operatorname{im}f \).

```{=tex}
\begin{tikzcd}
    M && N & \textcolor{rgb,255:red,92;green,92;blue,214}{f(m)} \\
    \\
    {M/\ker f} \\
    \textcolor{rgb,255:red,92;green,92;blue,214}{m + \ker f}
    \arrow["\eta"', from=1-1, to=3-1]
    \arrow["f", from=1-1, to=1-3]
    \arrow["{\exists \psi}"', dashed, from=3-1, to=1-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, maps to, from=4-1, to=1-4]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJNIl0sWzIsMCwiTiJdLFswLDIsIk0vXFxrZXIgZiJdLFswLDMsIm0gKyBcXGtlciBmIixbMjQwLDYwLDYwLDFdXSxbMywwLCJmKG0pIixbMjQwLDYwLDYwLDFdXSxbMCwyLCJcXGV0YSIsMl0sWzAsMSwiZiJdLFsyLDEsIlxcZXhpc3RzIFxccHNpIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzMsNCwiIiwyLHsiY29sb3VyIjpbMjQwLDYwLDYwXSwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==)

-   The fundamental SES
    \[
    0\to \ker f \xhookrightarrow{g} M \xrightarrow{f} \operatorname{im}f \to 0
    ,\]
    where one generally needs \( \operatorname{im}g = \ker f \) for exactness.
    -   More generally, need monomorphisms, epimorphisms.
:::

::: {.example title="?"}
Some examples:

-   \( f:{\mathbb{Z}}\to {\mathbb{Z}} \) where \( f(m) \coloneqq 4m \) yields \( 0\to {\mathbb{Z}}\xrightarrow{f} {\mathbb{Z}}\to {\mathbb{Z}}/4\to 0 \) in \( {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \).
-   In \( {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}} \), one can take \( 0\to {\mathbb{C}}\xrightarrow{\Delta: x\mapsto (x, x)} {\mathbb{C}}{ {}^{ \scriptscriptstyle\times^{2} }  } \to {\mathbb{C}}\to 0 \).
:::

::: {.remark}
Direct sums, products, and indecomposables. Let \( I \) be an index set and \( \left\{{M_k}\right\}_{k\in I} \) \( R{\hbox{-}} \)modules to define the **direct product** \( \prod_{k\in I} M_k \coloneqq\left\{{(m_k)_{k\in I} {~\mathrel{\Big\vert}~}m_k\in M_k }\right\} \), the set of all ordered sequences of elements from the \( M_k \), with addition defined pointwise. For the **direct sum** \( \bigoplus _{k\in I} M_k \) to be those sequences with only finitely many nonzero components. For internal direct sums, if \( M = M_1 + M_2 \) then \( M \cong M_1 \oplus M_2 \) iff \( M \cap M_2 = 0 \). An **irreducible representation** is a simple \( R{\hbox{-}} \)module, and an **indecomposable representation** is an indecomposable \( R{\hbox{-}} \)module. An \( R{\hbox{-}} \)module is **simple** iff its only submodules are \( 0, M \), and **indecomposable** iff \( M \not\cong M_1 \oplus M_2 \) for any \( M_i\not\cong M \). Note that simple \( \implies \) indecomposable.

> Note: is it possible for \( M \cong M \oplus M \)?
:::

::: {.example title="?"}
Some examples:

-   Simple objects in \( {\mathsf{k}{\hbox{-}}\mathsf{Mod}} \) are isomorphic to \( k \), and indecomposables are also isomorphic to \( k \) if we restrict to finite dimensional modules.

-   Simple objects in \( {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \) are cyclic groups of prime order, \( C_p \). Indecomposables are \( {\mathbb{Z}}, C_{p^k} \), using the classification theorem to rule out composites.

-   For \( A\in {\mathsf{Alg}}^{\mathrm{fd}}_{/ {k}}  \), the simple objects in \( {\mathsf{A}{\hbox{-}}\mathsf{Mod}} \) are hard to determine in general. The same goes for indecomposables, and is undecidable in many cases (equivalent to the word problem in finite groups).

    > See **finite**, **tame**, and **wild** representation types.
:::

::: {.remark}
Toward homological algebra: free and projective modules. An \( R{\hbox{-}} \)module \( M \) is **free** iff \( M\cong \bigoplus_{i\in I} R_i \) for some indexing set where \( R_i \cong R \) as a left \( R{\hbox{-}} \)module. Equivalently, \( M \) has a linearly independent spanning set, or there exists an \( X \) and a unique \( \phi \) such that the following diagram commutes:

```{=tex}
\begin{tikzcd}
    M \\
    \\
    X && N
    \arrow["{\mathsf{Set}}", from=3-1, to=3-3]
    \arrow["{\iota\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}}", hook', from=3-1, to=1-1]
    \arrow["{\exists ! \phi \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}}", dashed, from=1-1, to=3-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJYIl0sWzAsMCwiTSJdLFsyLDIsIk4iXSxbMCwyLCJcXFNldCJdLFswLDEsIlxcaW90YVxcaW4gXFxtb2Rze1J9IiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJib3R0b20ifX19XSxbMSwyLCJcXGV4aXN0cyAhIFxccGhpIFxcaW4gXFxtb2Rze1J9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)

Every \( M\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) is the image of a free \( R{\hbox{-}} \)module: let \( X\coloneqq\left\{{m_i}\right\}_{i\in I} \) generate \( M \), so \( X\hookrightarrow M \) by inclusion. Define \( X \to \bigoplus \bigoplus_{i\in I} R_i \) sending \( m_i \to (0,\cdots, 1, \cdots, 0) \) with a 1 in the \( i \)th position, then since \( X \) is a generating set this will lift to a surjection \( \bigoplus _i R_i\to M \). We can use this to define a free resolution:

```{=tex}
\begin{tikzcd}
    {\ker \delta_1} \\
    \cdots & \textcolor{rgb,255:red,92;green,92;blue,214}{\exists F_1} && {F_0} && M && 0 \\
    && \textcolor{rgb,255:red,214;green,92;blue,92}{\ker \delta_0} \\
    & \textcolor{rgb,255:red,214;green,92;blue,92}{0} && \textcolor{rgb,255:red,92;green,92;blue,214}{0}
    \arrow[from=4-2, to=3-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, no head, from=3-3, to=4-2]
    \arrow["{\delta_0}", from=2-4, to=2-6]
    \arrow[from=2-6, to=2-8]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, hook, from=3-3, to=2-4]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, two heads, from=2-2, to=3-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-3, to=4-4]
    \arrow["{\exists\delta_1}", color={rgb,255:red,92;green,92;blue,214}, dashed, from=2-2, to=2-4]
    \arrow[hook, from=1-1, to=2-2]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)
:::

::: {.remark}
Let \( A\in {\mathsf{Alg}}^{\mathrm{fd}}_{/ {k}}  \) and \( F \cong \bigoplus A \) be free, and suppose \( e\in A \) is idempotent, so \( e^2 = e \) -- these are useful because they can split algebras up. There is a *Pierce decomposition* of \( 1 \) given by \( 1 = e + (1-e) \). Noting that \( 1-e \) is also idempotent, there is a decomposition \( A \cong Ae \oplus A(1-e) \). Since \( Ae \) is direct summand of \( A \) which is free, this yields a way to construct projective modules.
:::

# Thursday, January 13

::: {.remark}
Last time:

-   \( R{\hbox{-}} \)modules and their morphisms
-   Free resolutions \( F \twoheadrightarrow R \).

Today: projective modules and their resolutions.

> See Krull-Schmidt theorem.
:::

::: {.remark}
Recall the definition of projective modules \( P \) and injective modules \( I \):

```{=tex}
\begin{tikzcd}
    &&&&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{P} \\
    \\
    {\forall \xi:} & 0 && A && B && C && 0 \\
    \\
    &&& \textcolor{rgb,255:red,92;green,92;blue,214}{I}
    \arrow[from=3-2, to=3-4]
    \arrow[from=3-4, to=3-6]
    \arrow[from=3-6, to=3-8]
    \arrow[from=3-8, to=3-10]
    \arrow[from=1-8, to=3-8]
    \arrow["\exists", dashed, from=1-8, to=3-6]
    \arrow["\exists", dashed, from=3-6, to=5-4]
    \arrow[from=3-4, to=5-4]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMSwyLCIwIl0sWzMsMiwiQSJdLFs1LDIsIkIiXSxbNywyLCJDIl0sWzksMiwiMCJdLFs3LDAsIlAiLFsyNDAsNjAsNjAsMV1dLFszLDQsIkkiLFsyNDAsNjAsNjAsMV1dLFswLDIsIlxcZm9yYWxsOiJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdLFs1LDNdLFs1LDIsIlxcZXhpc3QiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMiw2LCJcXGV4aXN0IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsNl1d)
:::

::: {.exercise title="?"}
Show that free implies projective using the universal properties, and conclude that every \( R{\hbox{-}} \)module has a projective cover.
:::

::: {.remark}
Forming projective resolutions: take the minimal \( P_0 \xrightarrow[]{\delta_0} { \mathrel{\mkern-16mu}\rightarrow }\,  M\to 0 \) such that \( \Omega^1 \coloneqq\ker \delta_0 \) has no projective summands. Continue in such a minimal way:

```{=tex}
\begin{tikzcd}
    & 0 && 0 \\
    & {\Omega^2} && {\Omega^1} \\
    \\
    \cdots && {P_1} && {P_0} && M && 0
    \arrow[two heads, from=4-5, to=4-7]
    \arrow[from=4-7, to=4-9]
    \arrow[hook, two heads, from=2-4, to=4-5]
    \arrow[two heads, from=4-3, to=2-4]
    \arrow["\exists", dashed, from=4-3, to=4-5]
    \arrow[dashed, from=4-1, to=4-3]
    \arrow[hook, no head, from=2-2, to=4-3]
    \arrow[from=1-2, to=2-2]
    \arrow[from=1-4, to=2-4]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)
:::

::: {.remark}
For modules \( M \) over an algebra \( A \), if \( \dim_k(M) \) is finite, then each \( P_i \) can be chosen to be finite dimensional. Otherwise, define a **complexity** or **rate of growth** \( s c_A(M) \geq 0 \) such that \( \dim P_n \leq C n^{s-1} \) for some constant \( C \). A theorem we'll prove is that \( s \) is finite when \( A = k G \) for every finite dimensional \( G{\hbox{-}} \)module. When \( A = kG \), this is a numerical invariant but has a nice geometric interpretation in terms of support varieties \( V_A(M) \), an affine algebraic variety where \( \dim V_A(M) = c_A(M) \).
:::

::: {.exercise title="?"}
Recall the definition of a SES \( \xi: 0\to A \xrightarrow{d_1} B \xrightarrow{d_2} C\to 0 \) and show that TFAE:

-   \( \xi \) splits
-   \( \xi \) admits a right section \( s_r: C\to B \)
-   \( \xi \) admits a left section \( s_\ell B\to A \)

> Hint: for the right section, show that \( s_r \) is injective. Get that \( \operatorname{im}f + \operatorname{im}h \subseteq M_2 \), use exactness to write \( \operatorname{im}d_1 = \ker d_2 \) and show that \( \ker d_2 \cap\operatorname{im}s_r = \emptyset \).
:::

::: {.warnings}
It's not necessarily true that if \( B \cong A \oplus C \) that \( \xi \) splits: consider

```{=tex}
\begin{tikzcd}
    0 && {C_2} && {C_4} && {C_2} && 0
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiQ18yIl0sWzQsMCwiQ180Il0sWzYsMCwiQ18yIl0sWzgsMCwiMCJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdXQ==)
:::

::: {.exercise title="?"}
Show that for \( P \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \), TFAE:

-   \( P \) is projective.
-   Every SES \( \xi: 0\to A\to B\to P \to 0 \) splits.
-   There exists a free module \( F \) such that \( F = P \oplus K \).
:::

::: {.exercise title="?"}
Show that \( \bigoplus_{i\in I} P_i \) is projective iff each \( P_i \) is projective.
:::

::: {.example title="?"}
-   If \( R=k\in \mathsf{Field} \), then every \( M\in {\mathsf{k}{\hbox{-}}\mathsf{Mod}} \) is free and thus projective since \( M \cong \bigoplus_{i\in I} k \) with \( k \) free in \( {\mathsf{k}{\hbox{-}}\mathsf{Mod}} \).

-   If \( R={\mathbb{Z}} \), let \( P\in {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \) be projective and \( F \) free and consider \( 0\to K\to F\to P\to 0 \). Since \( F\cong P \oplus K \), \( P \) is a submodule of \( F \), making \( P \) free since \( {\mathbb{Z}} \) is a PID. So projective implies free.

-   Not every \( M\in {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \) is projective: take \( C_6\in {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \), then \( C_6 \cong C_2 \oplus C_3 \) so \( C_2, C_3 \) are projective in \( {\mathsf{C_6}{\hbox{-}}\mathsf{Mod}} \) but not free here.
:::

::: {.exercise title="?"}
Let \( Q\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) and show TFAE:

-   \( Q \) is injective
-   Every SES \( \xi: 0\to Q\to B\to C\to 0 \) splits.
:::

::: {.exercise title="?"}
Show that \( \prod_{i\in I}Q_i \) is injective iff each \( Q_i \) is injective. Note that one needs to use direct products instead of direct sums here.
:::

::: {.theorem title="?"}
The category \( {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) has enough injectives, i.e. for every \( M\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) there is an injective \( Q \) and a SES \( 0\to M\hookrightarrow Q \).
:::

::: {.proof title="Sketch"}
See Hungerford or Weibel. Prove it first for \( \mathsf{C} = {\mathsf{Z}{\hbox{-}}\mathsf{Mod}} \). The idea now is to apply
\[
F({-}) \coloneqq\mathop{\mathrm{Hom}}_{\mathbb{Z}}(R,{-}): ({{\mathbb{Z}}}, {{\mathbb{Z}}}){\hbox{-}}\mathsf{biMod} &\to ({R}, {{\mathbb{Z}}}){\hbox{-}}\mathsf{biMod}
,\]
the left-exact contravariant hom. Using that \( R\in ({R}, {R}){\hbox{-}}\mathsf{biMod}\hookrightarrow({{\mathbb{Z}}}, {R}){\hbox{-}}\mathsf{biMod} \), one can use the right action \( R \) on itself to define a left action on \( \mathop{\mathrm{Hom}}_{\mathbb{Z}}(R, M) \). Then check that

-   \( f \) is left exact
-   \( f \) sends injectives to injectives.
-   If \( R\in{\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \) has an \( R{\hbox{-}} \)module structure, then \( F(R) \) is again an \( R{\hbox{-}} \)module.
:::

::: {.exercise title="?"}
Show that for \( M\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) that \( \mathop{\mathrm{Hom}}_{\mathbb{Z}}(R, M) \cong M \).

> Hint: try \( f\mapsto f(1) \).
:::

::: {.remark}
Next week:

-   Tensor products
-   Categories
-   Tensor and Hom
:::

# Tensor Products (Tuesday, January 18)

::: {.remark}
Setup: \( R\in \mathsf{Ring}, M_R \in \mathsf{Mod}{\hbox{-}}\mathsf{R} \), and \( {}_R N \in \mathsf{R}{\hbox{-}}\mathsf{Mod} \). Note that \( R \) is not necessarily commutative. The goal is to define \( M\otimes_R N \) as an abelian group.
:::

::: {.definition title="The Tensor Product"}
The **balanced product** of \( M \) and \( N \) is a \( P \in {\mathsf{Ab}}{\mathsf{Grp}} \) with a map \( f: M\times N\to P \) such that

-   \( f(x+x', y) = f(x, y) + f(x', y) \)
-   \( f(x, y+y') = f(x,y) + f(x, y') \)
-   \( f(ax, y) = f(x, ay) \).

The **tensor product** \( (M\otimes_R N, \otimes) \) of \( M \) and \( N \) is the initial balanced product, i.e. if \( P \) is a balanced product with \( M\times N \xrightarrow{f} P \) then there is a unique map \( \psi: M\otimes_R N\to P \):

```{=tex}
\begin{tikzcd}
    & {M\otimes_R N} \\
    \\
    {M\times N} && P
    \arrow["\otimes", from=3-1, to=1-2]
    \arrow["{\exists !\psi}", dashed, from=1-2, to=3-3]
    \arrow["f"', from=3-1, to=3-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJNXFx0aW1lcyBOIl0sWzIsMiwiUCJdLFsxLDAsIk1cXHRlbnNvcl9SIE4iXSxbMCwyLCJcXHRlbnNvciJdLFsyLDEsIlxcZXhpc3RzICFcXHBzaSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDEsImYiLDJdXQ==)

Uniqueness follows from the standard argument on universal properties:

```{=tex}
\begin{tikzcd}
    &&& {(M\otimes N)_1} \\
    \\
    {M\times N} &&& {(M\otimes N)_2} \\
    \\
    &&& {(M\otimes N)_1}
    \arrow["{\otimes_2}"{description}, from=3-1, to=3-4]
    \arrow["{\otimes_1}"{description}, from=3-1, to=5-4]
    \arrow["{\exists \psi_{12}}"{description}, curve={height=-18pt}, dashed, from=5-4, to=3-4]
    \arrow["{\exists \psi_{21}}"{description}, curve={height=-18pt}, dashed, from=3-4, to=1-4]
    \arrow["\operatorname{id}"', curve={height=30pt}, from=5-4, to=1-4]
    \arrow["{\otimes_1}"{description}, from=3-1, to=1-4]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJNXFx0aW1lcyBOIl0sWzMsNCwiKE1cXHRlbnNvciBOKV8xIl0sWzMsMiwiKE1cXHRlbnNvciBOKV8yIl0sWzMsMCwiKE1cXHRlbnNvciBOKV8xIl0sWzAsMiwiXFx0ZW5zb3JfMiIsMV0sWzAsMSwiXFx0ZW5zb3JfMSIsMV0sWzEsMiwiXFxleGlzdHMgXFxwc2lfezEyfSIsMSx7ImN1cnZlIjotMywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsMywiXFxleGlzdHMgXFxwc2lfezIxfSIsMSx7ImN1cnZlIjotMywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMywiXFxpZCIsMix7ImN1cnZlIjo1fV0sWzAsMywiXFx0ZW5zb3JfMSIsMV1d)

Existence: let \( \mathsf{Free}({-}): {\mathsf{Set}}\to {\mathsf{Ab}}{\mathsf{Grp}} \) and \( F\coloneqq\mathsf{Free}(M\times N) \), then set \( M\otimes_R N \coloneqq F/G \) where \( G \) is generated by

-   \( (x+x', y) - \qty{ (x, y) + (x', y) } \)
-   \( (x, y+y') - \qty{ (x, y) + (x, y') } \)
-   \( (ax, y) - (x, ay) \).

Then define the map as
\[
\otimes: M\times N\to F \\
(x, y) &\mapsto x\otimes y \coloneqq(x, y) + G
.\]

Why it satisfies the universal property: use the universal property of free groups to get a map to \( F \) and check that the following diagram commutes:

```{=tex}
\begin{tikzcd}
    {M\times N} && F && {M\otimes_R N \coloneqq F/G} \\
    \\
    && P
    \arrow[from=1-1, to=3-3]
    \arrow["\otimes"', from=1-1, to=1-3]
    \arrow["{({-})/G}"', from=1-3, to=1-5]
    \arrow["{\exists }"', dashed, from=1-3, to=3-3]
    \arrow["{\exists  \psi}", from=1-5, to=3-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJNXFx0aW1lcyBOIl0sWzIsMCwiRiJdLFs0LDAsIk1cXHRlbnNvcl9SIE4gXFxkYSBGL0ciXSxbMiwyLCJQIl0sWzAsM10sWzAsMSwiXFx0ZW5zb3IiLDJdLFsxLDIsIihcXHdhaXQpL0ciLDJdLFsxLDMsIlxcZXhpc3RzICIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsyLDMsIlxcZXhpc3RzICBcXHBzaSJdXQ==)

Morphisms: for \( f:M\to M' \) and \( g: N\to N' \), form
\[
f\otimes g: M\otimes N &\to M'\otimes N' \\
x\otimes y &\mapsto f(x) \otimes g(y)
.\]
:::

::: {.warnings}
Note every \( z\in M\otimes_R N \) is a simple tensor of the form \( z=x\otimes y \)!
:::

::: {.example title="?"}
-   For \( R=k\in \mathsf{Field} \), \( M\otimes_k N \in ({k}, {k}){\hbox{-}}\mathsf{biMod} \). If \( M = \left\langle{m_i}\right\rangle \) and \( N = \left\langle{n_j}\right\rangle \), then \( M\otimes_k n = \left\langle{m_i\otimes n_j}\right\rangle \) and \( \dim_k M\otimes_k N = \dim_k M \cdot \dim_k N \).

-   For \( A\in {\mathsf{Ab}}{\mathsf{Grp}} \), \( A\otimes_{\mathbb{Z}}{\mathbb{Z}}\cong A \) since \( x\otimes y = xy\otimes 1 \).

-   \( M\coloneqq C_p\otimes_{\mathbb{Z}}{\mathbb{Q}}= 0 \). It suffices to check on simple tensors:
    \[
    x\otimes y
    &= x\otimes{p\over p} y \\
    &= x\otimes p\qty{1\over p} y \\
    &= px\otimes\qty{1\over p} y \\
    &= 0\otimes{1\over p}y \\
    &= 0
    .\]

-   More generally, if \( A\in {\mathsf{Ab}}{\mathsf{Grp}} \) is torsion then \( A\otimes_{\mathbb{Z}}{\mathbb{Q}}= 0 \).
:::

::: {.definition title="Categories"}
A category \( \mathsf{C} \) is a class of objects \( A\in \mathsf{C} \) and for any pair \( (A, B) \), a set of morphism \( \mathop{\mathrm{Hom}}_{\mathsf{C}}(A, B) \) such that

1.  \( (A, B) \neq (C, D)\implies \mathop{\mathrm{Hom}}(A, B) \) and \( \mathop{\mathrm{Hom}}(C, D) \) are disjoint.
2.  Associativity of composition: \( (h\circ g)\circ f = h\circ(g\circ f) \)
3.  Identities: \( \exists ! \operatorname{id}_A \in \mathop{\mathrm{Hom}}_{\mathsf{C}}(A, A) \) for all \( A\in \mathsf{C} \).

A **subcategory** \( \mathsf{D} \leq \mathsf{C} \) is a subclass of objects and morphisms, and is **full** if \( \mathop{\mathrm{Hom}}_{\mathsf{D}}(A, B) = \mathop{\mathrm{Hom}}_{\mathsf{C}}(A, B) \) for all objects in \( \mathsf{D} \).
:::

::: {.example title="?"}
Examples of categories:

-   \( \mathsf{C} = {\mathsf{Set}} \),
-   \( \mathsf{C} = {\mathsf{Grp}} \),
-   \( \mathsf{C} = {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \),
-   \( \mathsf{C} = {\mathsf{Top}} \) with continuous maps.
:::

::: {.example title="?"}
Examples of fullness:

-   \( {\mathsf{Grp}}\leq {\mathsf{Set}} \) is not a full subcategory, since not all set morphisms are group morphisms.
-   \( {\mathsf{Ab}}{\mathsf{Grp}}\leq {\mathsf{Grp}} \) is a full subcategory.
:::

::: {.remark}
Recall the definition of covariant and contravariant functors, which requires that \( F(\operatorname{id}_A) = \operatorname{id}_{F(A)} \).
:::

# Thursday, January 20

::: {.remark}
RIP Brian Parshall and Fred Cohen... 😔
:::

::: {.remark}
Recall the definition of a covariant functor. Some examples:

-   \( F(R) = U(R) = R^{\times}= {\mathbb{G}}_m(R) \), the group of units of \( R \).
-   The forgetful functor \( {\mathsf{Grp}}\to {\mathsf{Set}} \).
-   \( \mathop{\mathrm{Hom}}_{\mathbb{Z}}(R, {-}) \) for \( R\in ({{\mathbb{Z}}}, {R}){\hbox{-}}\mathsf{biMod} \) is a functor \( \mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}\to \mathsf{R}{\hbox{-}}\mathsf{Mod} \).
:::

::: {.exercise title="?"}
Formulate \( \mathop{\mathrm{Hom}}_{\mathbb{Z}}({-}, {-}) \) in terms of functors between bimodule categories. How does this "use up an action" in the way \( {-}\otimes_{\mathbb{Z}}{-} \) does?
:::

::: {.remark}
Recall that contravariant functors reverse arrows. Functors with the same variance can be composed.
:::

::: {.definition title="Full and Faithful Functors"}
Let \( F: \mathsf{C}\to \mathsf{D} \) and consider the set map
\[
F_{AB}: \mathop{\mathrm{Hom}}(A, B) &\to \mathop{\mathrm{Hom}}(FA, FB) \\
f &\mapsto F(f)
.\]
We say \( F \) is **full** if \( F_{AB} \) is injective for all \( A, B\in \mathsf{C} \), and **faithful** if \( F_{AB} \) is surjective for all \( A, B \).
:::

::: {.definition title="Natural Transformations"}
A morphism of functors \( \eta: F\to G \) for \( F,G:\mathsf{C}\to \mathsf{D} \) is a **natural transformation**: a family of maps \( \eta_A\in \mathop{\mathrm{Hom}}_{\mathsf{D}}(FA, GA) \) satisfying the following naturality condition:

```{=tex}
\begin{tikzcd}
    A &&& FA && GA \\
    &&&&&& {\in \mathsf{D}} \\
    B &&& FB && GB
    \arrow["{\eta_A}", from=1-4, to=1-6]
    \arrow["{G(f)}", from=1-6, to=3-6]
    \arrow[""{name=0, anchor=center, inner sep=0}, "{F(f)}", from=1-4, to=3-4]
    \arrow["{\eta_B}"', from=3-4, to=3-6]
    \arrow[""{name=1, anchor=center, inner sep=0}, "{f \in \mathsf{C}}"', from=1-1, to=3-1]
    \arrow[shorten <=19pt, shorten >=19pt, Rightarrow, from=1, to=0]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMywwLCJGQSJdLFszLDIsIkZCIl0sWzUsMCwiR0EiXSxbNSwyLCJHQiJdLFswLDAsIkEiXSxbMCwyLCJCIl0sWzYsMSwiXFxpbiBcXGNhdCBEIl0sWzAsMiwiXFxldGFfQSJdLFsyLDMsIkcoZikiXSxbMCwxLCJGKGYpIl0sWzEsMywiXFxldGFfQiIsMl0sWzQsNSwiZiBcXGluIFxcY2F0e0N9IiwyXSxbMTEsOSwiIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

If \( \eta_A \) is an isomorphism for all \( A\in \mathsf{C} \), then \( \eta \) is a **natural isomorphism**.
:::

::: {.exercise title="?"}
For \( \mathsf{C}, \mathsf{D} = { \mathsf{Vect} }^{{\mathrm{fd}}}_{/ {k}}  \) finite-dimensional vector spaces, take \( F = \operatorname{id} \) and \( G({-}) = ({-}) {}^{ \vee } {}^{ \vee } \). Note that \( \mathop{\mathrm{Hom}}(FV, GV) \cong \mathop{\mathrm{Hom}}(V, V {}^{ \vee } {}^{ \vee }) \cong \mathop{\mathrm{Hom}}(V, V) \), so set \( \eta_V \) to be the image of \( \operatorname{id}_V \) under this chain of isomorphisms. Show that \( \left\{{\eta_V }\right\}_{V\in \mathsf{C}} \) assemble to a natural transformation \( F\to G \).
:::

::: {.definition title="Isomorphisms and Equivalences of categories"}
Two categories \( \mathsf{C}, \mathsf{D} \) are **isomorphic** if there are functors \( F, G \) with \( F\circ G = \operatorname{id}_{\mathsf{D}}, G\circ F = \operatorname{id}_{\mathsf{C}} \) *equal* to the identities. They are **equivalent** if \( F\circ G, G\circ F \) are instead *naturally isomorphic* to the identity.
:::

::: {.example title="?"}
Some examples:

-   \( \mathsf{C} = {\mathsf{Ab}}{\mathsf{Grp}} \) and \( \mathsf{D} = {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \) by taking \( G:\mathsf{D}\to \mathsf{C} \) the forgetful functor, and for \( F \), using the same underlying set and defining the \( {\mathbb{Z}}{\hbox{-}} \)module structure by \( n\cdot m \coloneqq m + m + \cdots + m \).

-   \( \mathsf{C}={\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) and \( \mathsf{D} = {\mathsf{\operatorname{Mat}_{n\times n}(R)}{\hbox{-}}\mathsf{Mod}} \). For \( {\mathsf{k}{\hbox{-}}\mathsf{Mod}} \), the simple objects are \( k \), but for \( {\mathsf{\operatorname{Mat}_{n\times n}(R)}{\hbox{-}}\mathsf{Mod}} \), the simple objects are \( k^n \), so these categories are not isomorphic. However, it turns out that they are equivalent.

Producing inverse functors can be difficult, so we have the following:
:::

::: {.proposition title="A useful criterion for equivalence of categories"}
Let \( F:\mathsf{C}\to \mathsf{D} \), then there exists an inverse inducing an *equivalence* iff

-   \( F \) is fully faithful,
-   Surjectivity on objects: for every \( A'\in \mathsf{D} \), there exists an \( A\in \mathsf{C} \) such that \( F(A) \cong A' \).
:::

::: {.proof title="?"}
\( \implies \): Suppose \( F, G \) induce an equivalence \( \mathsf{C} \simeq\mathsf{D} \), so \( F\circ G\simeq\operatorname{id}_{\mathsf{D}} \) and \( G\circ F \simeq\operatorname{id}_{\mathsf{C}} \). To show \( f\to F(f) \) is injective, check that
\[
F(f) &= F(g) \\
\implies GF(f) &= GF(g) \\
\operatorname{id}(f) &= \operatorname{id}(g) \\
\implies f= g
.\]
:::

::: {.exercise title="?"}
Show surjectivity.

A hint:

Let \( A'\in \mathsf{D} \) with \( FG \simeq\operatorname{id}_{\mathsf{D}} \) and \( \eta_{A'} \in \mathop{\mathrm{Hom}}_{\mathsf{D}}(A', FGA') \) is an iso. Set \( A \coloneqq GA'\in \mathsf{C} \) and use that
\[
\mathop{\mathrm{Hom}}_{\mathsf{D}}(A', FGA') \coloneqq\mathop{\mathrm{Hom}}(A', FA)
,\]
So if there is an isomorphism in \( \mathop{\mathrm{Hom}}(A', FA) \), there exists an isomorphism in \( \mathop{\mathrm{Hom}}(FA, A') \) and thus \( FA \cong A' \).

> \#todo Missed a bit here so this doesn't make sense as-is!
:::

::: {.proposition title="?"}
Let \( R\in \mathsf{Ring} \) and set \( S\coloneqq\operatorname{Mat}_{n\times n}(R) \), then \( {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \simeq{\mathsf{S}{\hbox{-}}\mathsf{Mod}} \).
:::

# Tuesday, January 25

::: {.remark}
Recall isomorphisms \( \mathsf{C} \cong \mathsf{D} \) of categories, so \( F\circ G = \operatorname{id} \), vs equivalences of categories \( \mathsf{C} \simeq\mathsf{D} \) so \( F\circ G \cong \operatorname{id} \).
:::

::: {.theorem title="?"}
For \( F:\mathsf{C} \to \mathsf{D} \) and \( G:\mathsf{D}\to \mathsf{C} \) and write \( \psi_F: \mathop{\mathrm{Hom}}_{\mathsf{C}}(A, B) \to \mathop{\mathrm{Hom}}_{\mathsf{D}}(F(A), F(B)) \). This pair induces an equivalence iff

1.  \( F \) is faithful, i.e. \( \psi_F \) is injective,
2.  \( F \) is full, i.e. \( \psi_F \) is surjective,
3.  For any \( D\in \mathsf{D} \), there exists a \( C\in \mathsf{C} \) with \( F(C) \cong D \).
:::

::: {.proposition title="?"}
Let \( R\in \mathsf{Ring} \) and \( S=\operatorname{Mat}_{n\times n}(R) \), then \( {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \simeq{\mathsf{S}{\hbox{-}}\mathsf{Mod}} \).
:::

::: {.proof title="?"}
Define a functor \( F:{\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to {\mathsf{S}{\hbox{-}}\mathsf{Mod}} \) by \( F(M) \coloneqq\prod_{k\leq n} M \), regarding this as a column vector and letting \( S \) act by matrix multiplication. On morphisms, define \( F(f)(\mathbf{x}) = {\left[ {f(x_1), \cdots, f(x_n)} \right]} \) for \( \mathbf{x} \in \prod M \). Then \( F(\operatorname{id}) = \operatorname{id} \), and (exercise) \( F(f) \) is a morphism of \( S{\hbox{-}} \)modules and composes correctly:
\[
F(g\circ f)(\mathbf{x}) = {\left[ {gf(x_1), \cdots, gf(x_n)} \right]} = F(g){\left[ {f(x_1), \cdots, f(x_n) } \right]} = \qty{ F(g)\circ F(f) } \mathbf{x}
.\]
So this defines a functor.

::: {.claim}
\( F \) is fully faithful.
:::

-   Faithfulness: if \( F(f_1) = F(f_2) \), then \( f_1(x_j) = f_2(x_j) \) for all \( j \), making \( f_1=f_2 \).

-   Fullness: let \( g\in \mathop{\mathrm{Hom}}_S(M^n, N^n) \) for \( M, N \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) and \( e_{ij} \) be the elementary matrix with a 1 only in the \( i, j \) position. Check that \( e_{11} M^n = \left\{{{\left[ {x,0,\cdots} \right]} {~\mathrel{\Big\vert}~}x\in M}\right\} \), \( e_{11} N^n = \left\{{{\left[ {y,0,\cdots} \right]}{~\mathrel{\Big\vert}~}y\in N}\right\} \), and \( \operatorname{diag}(x) \) be a matrix with only copies of \( x \) on the diagonal. Then \( g(e_{11} M^n) \subseteq e_{11} g(M^n) \subseteq e_{11}N^n \) and \( g{\left[ {x, 0, \cdots} \right]} = {\left[ {y, 0, \cdots} \right]} \). Define \( f:M\to N \) by \( f(x) = y \), then on one hand,
    \[
    g(\operatorname{diag}(a) {\left[ {x, 0,\cdots} \right]}) = g{\left[ {ax, 0, \cdots} \right]} = {\left[ {f(ax), 0, \cdots} \right]}
    ,\]
    but since \( g \) is a morphism of \( S{\hbox{-}} \)modules, this also equals \( \operatorname{diag}(a)\cdot g{\left[ {x,0,\cdots} \right]} = {\left[ {ay,0,\cdots} \right]} \). Then \( f(ax) = ay = af(x) \), so \( f \) is a morphism of \( R{\hbox{-}} \)modules.

    Note that \( e_{j1} \mathbf{x} = {\left[ {0, \cdots, x,\cdots 0} \right]} \) with \( x \) in the \( j \)th position. Check that \( g(e_{j1}\mathbf{x}) = g{\left[ {0, \cdots, x, \cdots, 0} \right]} \). The LHS is
    \[
    e_{j1} g(\mathbf{x}) = e_{j1}{\left[ {f(x), 0, \cdots} \right]} = {\left[ { 0,\cdots, f(x), \cdots, 0} \right]}
    \]
    with \( f(x) \) in the \( j \)th position. Hence \( g(\mathbf{x}) = {\left[ {f(x_1), \cdots, f(x_n)} \right]} \), making \( F \) full.

> See also Jacobson *Basic Algebra Part II* p.31.
:::

::: {.exercise title="Tensors commute with direct sums"}
Show that
\[
\qty{ \bigoplus _{\alpha \in I} M_\alpha } \otimes_R N 
&\cong \bigoplus _{\alpha\in I} \qty{M_\alpha \otimes_R N}
,\]
and similarly for \( M\otimes(\oplus N_\alpha) \).
:::

::: {.remark}
Define functors \( F,G{\mathsf{R}{\hbox{-}}\mathsf{Mod}} \to{\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \) by \( F({-}) \coloneqq M\otimes_R ({-}) \) and \( G({-}) \coloneqq({-})\otimes_R N \) on objects, and on morphisms \( f:N\to N' \), set \( F(f) \coloneqq\operatorname{id}\otimes f \) and similarly for \( G \). Recall the definition of exactness, left-exactness, and right-exactness.
:::

::: {.example title="Tensoring may not be left exact"}
Consider
\[
\xi: 0\to p{\mathbb{Z}}\xrightarrow{f}  {\mathbb{Z}}\xrightarrow{g}  {\mathbb{Z}}/p{\mathbb{Z}}\to 0
\]
and apply \( ({-})\otimes_{\mathbb{Z}}{\mathbb{Z}}/p{\mathbb{Z}} \). Use that \( p{\mathbb{Z}}\cong {\mathbb{Z}} \) in \( {\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}} \) to get
\[
F(\xi): C_p \xrightarrow{f\otimes\operatorname{id}} C_p \xrightarrow{g\otimes\operatorname{id}}  C_p
,\]
and
\[
(f\otimes\operatorname{id})(px\otimes y) = px\otimes y = x\otimes py = 0
,\]
using that \( f \) is the inclusion.
:::

::: {.exercise title="?"}
Show that \( M\otimes_R({-}) \) and \( ({-})\otimes_R N \) are right exact for any \( M, N \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \).
:::

::: {.solution}
Let \( 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 \) which maps to \( M\otimes A \xrightarrow{\operatorname{id}\otimes f} M\otimes B \xrightarrow{\operatorname{id}\otimes g} C \).

-   Show \( \operatorname{id}\otimes g \) is surjective: write \( m\in M\otimes C \) as \( m=\sum x_i\otimes y_j \), pull back the \( y_j \) via \( g \) to get \( z_j \) with \( g(z_j) = y_j \). Then
    \[
    (\operatorname{id}\otimes g)(\sum x_i \otimes z_J) = \sum x_i\otimes g(z_j) = \sum x_i \otimes y_j
    .\]

-   Exactness, \( \operatorname{im}(\operatorname{id}\otimes f) = \ker (\operatorname{id}\otimes g) \): Use that \( gf=0 \) by exactness of the original sequence, and \( (\operatorname{id}\otimes g)\circ (\operatorname{id}\otimes f) = \operatorname{id}\otimes(g\circ f) = 0 \), so \( \operatorname{im}(\operatorname{id}\otimes f) \subseteq \ker(\operatorname{id}\otimes g) \).

    -   For the reverse containment, use that \( \operatorname{id}\otimes g: M\otimes B\to M\otimes C \) and define a map
        \[
        \Gamma: {M\otimes B \over \operatorname{im}(\operatorname{id}\otimes f)} \to M\otimes C \\
        m\otimes n + \operatorname{im}(\operatorname{id}\otimes f)&\mapsto m\otimes g(n)
        .\]
        Then \( \phi \) is an isomorphism iff \( \operatorname{im}(\operatorname{id}\otimes f) = \ker (\operatorname{id}\otimes g) \). Define
        \[
        \Psi: M\times C &\to {M\otimes B\over \operatorname{im}(\operatorname{id}\otimes f)} \\
        (x, y) &\mapsto x \otimes z + \operatorname{im}(\operatorname{id}\otimes f)
        ,\]
        where \( g(z) = y \), so \( z \) is a lift of \( y \).

    Why is this well-defined? Check \( g(z_1) = y = g(z_2) \) implies \( z_1 -z_2\in \ker g = \operatorname{im}f \), so write \( f(y) = z_1-z_2 \) for some \( y \). Then \( x\otimes z_1 + \operatorname{im}f = x\otimes z_2 + \operatorname{im}f \).

    Why does this factor through the tensor product? Check that \( \Psi \) is a balanced product, this yields \( \mkern 1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu: M\otimes C\to {M\otimes B\over \operatorname{im}(\operatorname{id}\otimes f)} \). Now check that \( \mkern 1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu, \Gamma \) are mutually inverse:
    \[
    \Gamma\Psi(x\otimes y) 
    &= \Gamma(x\otimes z + \operatorname{im}(\operatorname{id}\otimes f)) = x\otimes g(z) = x\otimes y \\
    \Psi\Gamma(x\otimes z + \operatorname{im}(\operatorname{id}\otimes f)) 
    &= (x\otimes g(z) ) = x\otimes z + \operatorname{im}f
    .\]
:::

::: {.question}
When is \( M\otimes_R ({-}) \) exact?
:::

# Thursday, January 27

::: {.remark}
Recall that \( M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) is flat iff for every \( N, N' \) and \( f\in \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(N, N') \), the induced map
\[
\operatorname{id}_M\otimes f: M\otimes_R N \to M\otimes_R N'
\]
is a monomorphism. Equivalently, \( M\otimes_R ({-}) \) is left exact and thus exact.
:::

::: {.proposition title="?"}
\( M \coloneqq\bigoplus _{\alpha\in I} M_\alpha \) is flat iff \( M_\alpha \) is flat for all \( \alpha\in I \).
:::

::: {.proof title="?"}
\[
M\otimes_R({-}) \coloneqq(\bigoplus M_\alpha)\otimes_R ({-}) \cong \bigoplus (M_\alpha \otimes_R ({-}) )
.\]
:::

::: {.exercise title="?"}
Show that projective \( \implies \) flat.
:::

::: {.exercise title="?"}
Prove that the hom functors \( \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, {-}), \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}({-}, M): {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\to {\mathbb{Z}{\hbox{-}}\mathsf{Mod}} \) are left exact.
:::

::: {.exercise title="?"}
Show that

-   \( P \) is projective iff \( \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(P, {-}) \) is exact
-   \( I \) is projective iff \( \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}({-}, I) \) is exact
:::

::: {.remark}
An object \( Z\in \mathsf{C} \) is a zero object iff \( \mathop{\mathrm{Hom}}_{\mathsf{C}}(A, Z), \mathop{\mathrm{Hom}}_{\mathsf{C}}(Z, A) \) are singletons for all \( A\in \mathsf{C} \). Write this as \( 0_A \in \mathop{\mathrm{Hom}}_{\mathsf{C}}(A, Z) \). If \( \mathsf{C} \) has a zero object, define the zero morphism as \( 0_{AB} \coloneqq 0_{B} \circ 0_A \in \mathop{\mathrm{Hom}}_{\mathsf{C}}(A, B) \).
:::

# Tuesday, February 01

::: {.definition title="Additive categories"}
A category \( \mathsf{C} \) is **additive** iff

-   \( \mathsf{C} \) has zero object
-   There exists a binary operation \( +: \mathop{\mathrm{Hom}}(A, B){ {}^{ \scriptscriptstyle\times^{2} }  }\to \mathop{\mathrm{Hom}}(A, B) \) for all \( A, B\in \mathsf{C} \) making \( \mathop{\mathrm{Hom}}(A ,B) \) an abelian group.
-   Distributivity with respect to composition: \( (g_1 + g_2)f = g_1f + g_2 f \)
-   For any collection \( \left\{{A_1,\cdots, A_n}\right\} \), there exists an object \( A \), projections \( p_j: A\to A_j \) with sections \( i_k: A_k\to A \) with \( p_j i_j = \operatorname{id}_A \), \( p_j i_k = 0 \) for \( j\neq k \), and \( \sum i_j p_j = \operatorname{id}_A \).
:::

::: {.definition title="Monomorphisms and epimorphisms"}
A morphism: \( k:K\to A \) is **monic** iff whenever \( g_1, g_2: L\to K \), \( kg_1 = kg_2 \implies g_1 = g_2 \):

```{=tex}
\begin{tikzcd}
    L && K && A
    \arrow["k", from=1-3, to=1-5]
    \arrow["{g_1}", shift left=3, from=1-1, to=1-3]
    \arrow["{g_2}", shift right=3, from=1-1, to=1-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJMIl0sWzIsMCwiSyJdLFs0LDAsIkEiXSxbMSwyLCJrIl0sWzAsMSwiZ18xIiwwLHsib2Zmc2V0IjotM31dLFswLDEsImdfMiIsMCx7Im9mZnNldCI6M31dXQ==)

Define \( k \) to be **epic** by reversing the arrows.
:::

::: {.definition title="Kernel"}
Assume \( \mathsf{C} \) has a zero object. Then for \( f:A\to B \), the *morphism* \( k: K\to A \) is the **kernel** of \( f \) iff

-   \( k \) is monic
-   \( fk=0 \)
-   For any \( g:G\to A \) with \( fg=0 \), there exists a \( g' \) with \( g=kg' \).
:::

::: {.example title="?"}
For \( f\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(A, B) \), take \( k: \ker f\hookrightarrow A \). If \( g\in \mathsf{C}(G, A) \) with \( f(g(x)) = 0 \) for all \( x\in G \), then \( \operatorname{im}g \subseteq \ker f \) and we can factor \( g \) as \( G \xrightarrow{g'} \ker f \xhookrightarrow{k} A \).
:::

::: {.definition title="Cokernel"}
For \( f: A\to B \), a morphism \( c: B\to C \) is a **cokernel of \( f \)** iff

-   \( c \) is epic,
-   \( cf=0 \)
-   For any \( h\in \mathsf{C}(B, H) \) with \( hf=0 \), there is a lift \( h': C\to G \) with \( h=h'c \).
:::

::: {.example title="?"}
For \( \mathsf{C} = {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) and \( f\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(A, B) \), set \( c: B\to B/\operatorname{im}f \).
:::

::: {.exercise title="?"}
Show that kernels are unique. Sketch:

-   Set \( k:K\to A \), \( k': K'\to A \).
-   Factor \( k=k' u_1 \) and \( k' = ku_2 \).
-   Then \( k\operatorname{id}= k(u_2 u_1) \implies \operatorname{id}= u_2 u_1 \), similarly \( u_1u_2=\operatorname{id} \).
:::

::: {.definition title="Abelian categories"}
\( \mathsf{C} \) is **abelian** iff \( \mathsf{C} \) is additive and

-   A5: Every morphism admits kernels and cokernels.
-   A6: Every monic is the kernel of its cokernel, and every epic is the cokernel of its kernel.
-   A7: Every morphism \( f \) factors as \( f=me \) with \( m \) monic and \( e \) epic.
:::

::: {.example title="?"}
For \( f\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(A, B) \),

-   A5: Take \( k: \ker f\hookrightarrow A \) and \( c: B\twoheadrightarrow B/\operatorname{im}f \)
-   A6: For \( m: A\hookrightarrow B \) monic, consider the composition \( A\hookrightarrow B \xrightarrow{\operatorname{coker}m} B/A \) and check \( A\cong \ker(\operatorname{coker}m) \).
-   A7: Use the 1st isomorphism theorem:

```{=tex}
\begin{tikzcd}
    A &&&& B \\
    \\
    & {A/\ker f} && {\operatorname{im}f}
    \arrow["f", from=1-1, to=1-5]
    \arrow["i"', two heads, from=1-1, to=3-2]
    \arrow["{\text{1st iso}}"', from=3-2, to=3-4]
    \arrow["m"', hook, from=3-4, to=1-5]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJBIl0sWzQsMCwiQiJdLFszLDIsIlxcaW0gZiJdLFsxLDIsIkEvXFxrZXIgZiJdLFswLDEsImYiXSxbMCwzLCJpIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMiwiXFx0ZXh0ezFzdCBpc299IiwyXSxbMiwxLCJtIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=)
:::

::: {.remark}
Some notes:

-   Recall the definition the category of chain complexes \( \mathsf{Ch}(\mathsf{C}) \) over an abelian category: \( d_i d_{i+1} = 0 \), so \( \operatorname{im}d_i \subseteq \ker d_{i+1} \).
-   Every exact sequence is an acyclic complex.
-   \( \mathsf{C}\hookrightarrow\mathsf{Ch}(\mathsf{C}) \) by \( M\mapsto \cdots \to 0 \to M \to 0 \to \cdots \). Note that this isn't an acyclic complex.
-   Morphisms between complexes: chain maps, just levelwise maps forming commutative squares, i.e. maps commuting with the differentials.
-   \( \mathsf{Ch}(\mathsf{C}) \) is additive: given \( \alpha_\bullet, \beta_\bullet\in \mathsf{Ch}\mathsf{C}( (A, d), (B, \delta) ) \), check that \( (\alpha_{i-1} + \beta_{i-1})d_i = \delta_i (\alpha_i + \beta_i) \).
-   There are direct sums: \( (A \oplus B)_i \coloneqq A_i \oplus B_i \) with \( d \coloneqq d_A + d_B \).
-   Define cycles as \( Z_i \coloneqq\ker\qty{ C_i \xrightarrow{d_i} C_{i-1}} \) for \( C_\bullet \in \mathsf{Ch}(\mathsf{C}) \), and boundaries \( B_i \coloneqq\operatorname{im}\qty{C_{i+1} \xrightarrow{d_{i+1}} C_i} \subseteq \ker d_i \).
-   Define \( H_i(C_\bullet )\coloneqq Z_i/B_i \).
-   Show that chain morphisms induce morphisms on homology:
    -   Let \( \alpha\in \mathsf{Ch}(\mathsf{C})(C, C') \), then \( \alpha_i(Z_i) \subseteq Z_i' \).
    -   Check \( d_2(a_i(Z_i)) = a_{i-1} d_i(Z_i) = 0 \).
    -   Factor \( Z_i \xrightarrow{\alpha_i} Z_i' \twoheadrightarrow Z_i'/B_i' \).
    -   Show that \( x\in B_i \) maps lands in \( B_i' \) to get well-defined map on \( H_i \).
    -   Use \( \alpha(B_i) \subseteq Z_i' \), so pull back \( x\in B_i \) to \( y\in C_{i+1} \).
    -   Check \( d_{i+1}(y) = x \), so \( \alpha(d_{i+1}(y)) = \alpha(x) \).
    -   The LHS is \( d_{i+1}'(\alpha_{i+1}(y)) \), so \( \alpha_i(x) in \operatorname{im}d_{i+1}' = B_{i+1}' \)
-   Chain homotopies: for \( \alpha, \beta\in \mathsf{Ch}(\mathsf{C})(C, C') \), write \( \alpha \simeq\beta \) iff there exists \( \left\{{s_i: C_i \to C_{i+1}' }\right\} \) with \( \alpha_i - \beta_i = d_{i+1}' s_i + s_{i-1} d_i \).

```{=tex}
\begin{tikzcd}
    \cdots && {C_{i+1}} && {C_{i}} && {C_{i-1}} && \cdots \\
    \\
    \cdots && {C_{i+1}'} && {C_{i}'} && {C_{i-1}'} && \cdots
    \arrow[from=1-1, to=1-3]
    \arrow["{d_{i+1}}", from=1-3, to=1-5]
    \arrow["{d_{i+1}}", color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow[from=3-1, to=3-3]
    \arrow["{d_{i}'}", from=3-5, to=3-7]
    \arrow["{d_{i+1}'}", color={rgb,255:red,214;green,92;blue,92}, from=3-3, to=3-5]
    \arrow[from=3-7, to=3-9]
    \arrow[from=1-3, to=3-3]
    \arrow["{\alpha_i-\beta_i}"{description}, color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=3-5]
    \arrow[from=1-7, to=3-7]
    \arrow["{s_i}"{description}, color={rgb,255:red,214;green,92;blue,92}, from=1-5, to=3-3]
    \arrow["{s_{i-1}}"{description}, color={rgb,255:red,214;green,153;blue,92}, from=1-7, to=3-5]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)
:::

# Thursday, February 03

## Projective Resolutions and Chain Maps

::: {.remark}
Also check that \( \simeq \) is an equivalence relation, i.e. it is symmetric, transitive, and reflexive. For transitivity: given
\[
\alpha_i - \beta_i &= d_{i+1}' s_i +s_{i-1} d_i \\
\beta_i - \gamma_i &= d_{i+1}' t_{i} + t_{i-1} d_i
,\]
one can write
\[
\alpha_i - \gamma_i &= d_{i+1}'(s_i + t_i) + (s_{i-1} + t_{i-1} ) d_i
.\]
:::

::: {.theorem title="?"}
Let \( \alpha, \beta \in \mathsf{Ch}\mathsf{C}(A, B) \) with induced maps \( \widehat{\alpha}, \widehat{\beta }\in \mathsf{Ch}\mathsf{C}(H^* A, H^* B) \) on homology. If \( \alpha \simeq\beta \), then \( \widehat{\alpha }= \widehat{\beta} \).
:::

::: {.proof title="?"}
A computation:
\[
\widehat{\alpha}_{1}(&\left.z_{1}+B_{i}\right)=\alpha_{1}\left(z_{i}\right)+B_{i}^{\prime} \\
&=\beta_{i}\left(z_{i}\right)+\delta_{i+1}^{\prime} s_{1}\left(z_{i}\right)+s_{i-1}^{\prime \prime} \delta_{i}\left(z_{i}\right) + B_i'\\
&=\beta_{i}\left(z_{i}\right)+B_{i}^{\prime} \\
&=\widehat{\beta}_{i}\left(z_{i}+B_{i}\right)
\]
:::

::: {.remark}
Roadmap:

-   Homological algebra
-   Commutative rings
-   Support theory
-   Tensor triangular geometry
:::

::: {.definition title="?"}
Let \( M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \). A **projective complex** for \( M \) is a chain complex \( (C_i, d_i)_{i\in {\mathbb{Z}}} \), indexed homologically:
\[
\cdots \to C_2 \xrightarrow{d_2} C_1 \xrightarrow{d_1} C_0 \xrightarrow{d_0\coloneqq{\varepsilon}} 0
.\]

In particular, \( d^2 = 0 \), but this complex need not be exact. A **projective resolution** of \( M \) is an *exact* projective complex in the following sense:

-   \( H_{k\geq 1}({ {C}_{\scriptscriptstyle \bullet}} ) = 0 \)
-   \( H_0({ {C}_{\scriptscriptstyle \bullet}} ) = C_0/d(C_1) = C_0/\ker {\varepsilon}\cong M \).
:::

::: {.example title="?"}
Some projective resolutions:

-   For \( M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \), projective resolutions exist since we can find covers by free modules:

```{=tex}
\begin{tikzcd}
    \cdots & {F_2} & {F_1} & {F_0} & M & 0 \\
    && {\ker d_1} & {\ker {\varepsilon}}
    \arrow[from=1-4, to=2-4]
    \arrow[from=2-4, to=1-5]
    \arrow[from=1-3, to=2-3]
    \arrow[from=2-3, to=1-4]
    \arrow[from=1-2, to=1-3]
    \arrow["{d_1}", from=1-3, to=1-4]
    \arrow["{\varepsilon}", two heads, from=1-4, to=1-5]
    \arrow[from=1-5, to=1-6]
    \arrow[from=1-1, to=1-2]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJcXGNkb3RzIl0sWzEsMCwiRl8yIl0sWzIsMCwiRl8xIl0sWzMsMCwiRl8wIl0sWzQsMCwiTSJdLFs1LDAsIjAiXSxbMywxLCJcXGtlciBcXGVwcyJdLFsyLDEsIlxca2VyIGRfMSJdLFszLDZdLFs2LDRdLFsyLDddLFs3LDNdLFsxLDJdLFsyLDMsImRfMSJdLFszLDQsIlxcZXBzIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzQsNV0sWzAsMV1d)

-   For \( M\in {\mathsf{Z}{\hbox{-}}\mathsf{Mod}} \), every module has a 2-stage resolution:

```{=tex}
\begin{tikzcd}
    0 & {\ker {\varepsilon}\cong {\mathbb{Z}}{ {}^{ \scriptscriptstyle\oplus^{m} }  }} & {{\mathbb{Z}}{ {}^{ \scriptscriptstyle\oplus^{n} }  }} & M & 0
    \arrow[from=1-4, to=1-5]
    \arrow["{\varepsilon}", two heads, from=1-3, to=1-4]
    \arrow[from=1-2, to=1-3]
    \arrow[from=1-1, to=1-2]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbNCwwLCIwIl0sWzMsMCwiTSJdLFsyLDAsIlxcWlpcXHN1bXBvd2Vye259Il0sWzEsMCwiXFxrZXIgXFxlcHMgXFxjb25nIFxcWlpcXHN1bXBvd2Vye219Il0sWzAsMCwiMCJdLFsxLDBdLFsyLDEsIlxcZXBzIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMl0sWzQsM11d)
:::

::: {.theorem title="?"}
For \( \mu \in \mathsf{C}(M, M') \) and \( C \coloneqq({ {C}_{\scriptscriptstyle \bullet}} , d) \twoheadrightarrow M, C' \coloneqq({ {C}_{\scriptscriptstyle \bullet}} ', d')\twoheadrightarrow M' \), there is an induced chain map \( \alpha \in \mathsf{Ch}\mathsf{C}(C, C') \). Moreover, any other chain map \( \beta \) is chain homotopic to \( \alpha \).

> Note that \( C \) can in fact be any projective complex over \( M \), not necessarily a resolution.
:::

::: {.proof title="?"}
Using that \( C_0 \) is projective, there is a lift of the following form:

```{=tex}
\begin{tikzcd}
    {C_0} && M \\
    \\
    {C_0'} && {M'}
    \arrow["\mu", from=1-3, to=3-3]
    \arrow["{\varepsilon}"', two heads, from=3-1, to=3-3]
    \arrow["{\varepsilon}", two heads, from=1-1, to=1-3]
    \arrow["{\exists \alpha_0}"', dashed, from=1-1, to=3-1]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJDXzAnIl0sWzIsMiwiTSciXSxbMiwwLCJNIl0sWzAsMCwiQ18wIl0sWzIsMSwiXFxtdSJdLFswLDEsIlxcZXBzIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMiwiXFxlcHMiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywwLCJcXGV4aXN0cyBcXGFscGhhXzAiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

Now inductively, we want to construct the following lift:

```{=tex}
\begin{tikzcd}
    {C_n} && {C_{n-1}} && {C_{n-2}} \\
    \\
    {C_{n}'} && {C_{n-1}'} && {C_{n-2}'} \\
    & {\operatorname{im}d_n' = \ker d_{n-1}'}
    \arrow["{d_n}", from=1-1, to=1-3]
    \arrow["{d_{n-1}}", from=1-3, to=1-5]
    \arrow["{\alpha_{n-2}}", from=1-5, to=3-5]
    \arrow["{d_{n-1}'}"', from=3-3, to=3-5]
    \arrow["{d_n'}"', from=3-1, to=3-3]
    \arrow["{\alpha_{n-1}}", from=1-3, to=3-3]
    \arrow["\exists"', dashed, from=1-1, to=3-1]
    \arrow[two heads, from=3-1, to=4-2]
    \arrow[hook, from=4-2, to=3-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)

STS \( \operatorname{im}\alpha_{n-1} d_n \subseteq \ker d_{n-1}' \), which follows from
\[
d_{n-1}' \alpha_{n-1} d_n(x) = \alpha_{n-1} d_{n-1} d_n(x)
.\]

So there is a map \( C_n \to \operatorname{im}d_n' \), and using projectivity produces the desired lift by the same argument as in the case case:

```{=tex}
\begin{tikzcd}
    {C_n} && {C_{n-1}} && {C_{n-2}} \\
    \\
    {C_{n}'} && {C_{n-1}'} && {C_{n-2}'} \\
    & {\operatorname{im}d_n' = \ker d_{n-1}'}
    \arrow["{d_n}", from=1-1, to=1-3]
    \arrow["{d_{n-1}}", from=1-3, to=1-5]
    \arrow["{\alpha_{n-2}}", from=1-5, to=3-5]
    \arrow["{d_{n-1}'}"', from=3-3, to=3-5]
    \arrow["{d_n'}"'{pos=0.4}, from=3-1, to=3-3]
    \arrow["{\alpha_{n-1}}", from=1-3, to=3-3]
    \arrow[""{name=0, anchor=center, inner sep=0}, "{\exists \text{ by projectivity}}"', dashed, from=1-1, to=3-1]
    \arrow[two heads, from=3-1, to=4-2]
    \arrow[hook, from=4-2, to=3-3]
    \arrow[""{name=1, anchor=center, inner sep=0}, "\exists"{description}, curve={height=-18pt}, dashed, from=1-1, to=4-2]
    \arrow[shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)

To see that any two such maps are chain homotopic, set \( \gamma \coloneqq\alpha - \beta \), then
\[
{\varepsilon}'( \gamma_0) = {\varepsilon}'( \alpha_i - \beta_i) = \mu{\varepsilon}- \mu {\varepsilon}=0
,\]
and
\[
d_n'(\gamma_n) 
&- d_n'( \alpha_n - \beta_n) \\
&= d_n' \alpha_n - d_n' \beta_n \\
&= \alpha_{n-1} d_n - \beta_{n-1} d_n \\
&= \gamma_{n-1} d_n
,\]
so \( \gamma \) yields a well-defined chain map.

We'll now construct the chain homotopy inductively. There is a lift \( s_0 \) of the following form:

```{=tex}
\begin{tikzcd}
    && {C_0} \\
    \\
    {C_1'} && {C'_0} & {M'} & 0 \\
    & {\operatorname{im}d_1'}
    \arrow["{d_1'}"', two heads, from=3-1, to=4-2]
    \arrow[hook, from=4-2, to=3-3]
    \arrow["{\gamma_0}"', from=1-3, to=3-3]
    \arrow["{\exists s_0}"', dashed, from=1-3, to=3-1]
    \arrow[from=3-1, to=3-3]
    \arrow["{{\varepsilon}'}", two heads, from=3-3, to=3-4]
    \arrow[from=3-4, to=3-5]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMywyLCJNJyJdLFs0LDIsIjAiXSxbMiwyLCJDJ18wIl0sWzEsMywiXFxpbSBkXzEnIl0sWzIsMCwiQ18wIl0sWzAsMiwiQ18xJyJdLFs1LDMsImRfMSciLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywyLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFs0LDIsIlxcZ2FtbWFfMCIsMl0sWzQsNSwiXFxleGlzdHMgc18wIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzUsMl0sWzIsMCwiXFxlcHMnIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzAsMV1d)

This follows because \( \operatorname{im}d_1' = \ker {\varepsilon}' \) and \( {\varepsilon}' \gamma_0 = 0 \) by the previous calculation.

Assuming all \( s_{i\leq n-1} \) are constructed, set \( \gamma_i = d_{i+1}' s_i + s_{i-1} d_i \). Setting \( \gamma_n - s_{n-1}d_n: C_n \to C_n' \), then
\[
d_n'( \gamma_n - s_{n-1} d_n)
&= d_n' \gamma_n - d_n' s_{n-1} d_n \\
&= \gamma_{n-1} d_n - d_n' s_{n-1} d_n \\
&= (\gamma_{n-1} - d_n' s_{n-1})d_n \\
&= s_{n-2} d_{n-1} d_n \\
&= 0
,\]
using \( d^2 = 0 \). Now there is a lift \( s_n \) of the following form:

```{=tex}
\begin{tikzcd}
    && {C_n} \\
    \\
    {C_{n+1}'} && {C_n'} && {C_{n-1}} \\
    & {\operatorname{im}d_{n+1} = \ker d_n'}
    \arrow["{\gamma_n - s_{n-1} d_N}", from=1-3, to=3-3]
    \arrow["{d_n'}", from=3-3, to=3-5]
    \arrow[from=3-1, to=3-3]
    \arrow["{d_{n+1}}"', from=3-1, to=4-2]
    \arrow[from=4-2, to=3-3]
    \arrow["{s_n}"', dashed, from=1-3, to=3-1]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJDX24iXSxbMiwyLCJDX24nIl0sWzQsMiwiQ197bi0xfSJdLFswLDIsIkNfe24rMX0nIl0sWzEsMywiXFxpbSBkX3tuKzF9ID0gXFxrZXIgZF9uJyJdLFswLDEsIlxcZ2FtbWFfbiAtIHNfe24tMX0gZF9OIl0sWzEsMiwiZF9uJyJdLFszLDFdLFszLDQsImRfe24rMX0iLDJdLFs0LDFdLFswLDMsInNfbiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

Thus follows from the fact that \( \operatorname{im}\gamma_n - s_{n-1} d_n \subseteq \ker d_n' \) and projectivity of \( C_n \).
:::

::: {.remark}
Dually one can construct injective resolutions \( 0 \to M \xhookrightarrow{\eta} { {D}_{\scriptscriptstyle \bullet}}  \)
:::

## Derived Functors

::: {.remark}
Setup: \( F: {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\to {\mathbb{Z}{\hbox{-}}\mathsf{Mod}} \) is an additive covariant functor, e.g. \( ({-}) \otimes_R N \) or \( M\otimes_R({-}) \), and \( { {C}_{\scriptscriptstyle \bullet}}  \xrightarrow[]{{\varepsilon}} { \mathrel{\mkern-16mu}\rightarrow }\,  M \) a complex over \( M \). We define the left-derived functors as \( (L_n F)(M) \coloneqq H_n(F({ {C}_{\scriptscriptstyle \bullet}} )) \).
:::

# Tuesday, February 08

::: {.remark}
Defining derived functors: for \( F \) an additive functor and \( M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \), take a projective resolution and apply \( F \):
\[
\cdots \to C_2 \xrightarrow{d_2} C_1 \xrightarrow{d_1} C_0 \xrightarrow{{\varepsilon}= d_0} M \to 0 \leadsto F(C_2) \xrightarrow{Fd_2} F(C_1) \xrightarrow{Fd_1} \cdots
,\]
so \( { {C}_{\scriptscriptstyle \bullet}}  \rightrightarrows F \).

Define the left-derived functor
\[
{\mathbb{L}}F M \coloneqq H_n F{ {C}_{\scriptscriptstyle \bullet}} 
.\]
:::

::: {.remark}
Any \( \mu \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(M, M') \) induces a chain map \( \widehat{\alpha }\in \mathsf{Ch}{\mathsf{R}{\hbox{-}}\mathsf{Mod}}(H_* F{ {C}_{\scriptscriptstyle \bullet}} , H_* F{ {C}_{\scriptscriptstyle \bullet}} ' ) \), where \( \alpha \) is any lift of \( \mu \) to their resolutions.

```{=tex}
\begin{tikzcd}
    {{ {C}_{\scriptscriptstyle \bullet}} } && M \\
    \\
    {{ {C}_{\scriptscriptstyle \bullet}} '} && {M'}
    \arrow["{\varepsilon}", Rightarrow, from=1-1, to=1-3]
    \arrow["\mu", from=1-3, to=3-3]
    \arrow["{{\varepsilon}'}"', Rightarrow, from=3-1, to=3-3]
    \arrow["\alpha"', from=1-1, to=3-1]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMiwwLCJNIl0sWzIsMiwiTSciXSxbMCwyLCJcXGNvbXBsZXh7Q30nIl0sWzAsMCwiXFxjb21wbGV4e0N9Il0sWzMsMCwiXFxlcHMiLDAseyJsZXZlbCI6Mn1dLFswLDEsIlxcbXUiXSxbMiwxLCJcXGVwcyciLDIseyJsZXZlbCI6Mn1dLFszLDIsIlxcYWxwaGEiLDJdXQ==)
:::

::: {.exercise title="?"}
Show that any two lifts \( \alpha, \alpha' \) induce the same map on homology.
:::

::: {.remark}
Similarly, \( {\mathbb{L}}F(M) \) does not depend on the choice of resolution:

```{=tex}
\begin{tikzcd}
    {{ {C}_{\scriptscriptstyle \bullet}} } && M &&&& {F{ {C}_{\scriptscriptstyle \bullet}} } && {F(M)} \\
    \\
    {{ {C}_{\scriptscriptstyle \bullet}} '} && M & \leadsto &&& {F{ {C}_{\scriptscriptstyle \bullet}} '} && {F(M)} \\
    \\
    {{ {C}_{\scriptscriptstyle \bullet}} } && M &&&& {F{ {C}_{\scriptscriptstyle \bullet}} } && {F(M)}
    \arrow["{\operatorname{id}_M}", from=1-3, to=3-3]
    \arrow["{\operatorname{id}_M}", from=3-3, to=5-3]
    \arrow["\alpha", from=1-1, to=3-1]
    \arrow["\beta", from=3-1, to=5-1]
    \arrow["{\varepsilon}", from=5-1, to=5-3]
    \arrow["{\varepsilon}", from=3-1, to=3-3]
    \arrow["{\varepsilon}", from=1-1, to=1-3]
    \arrow[from=1-9, to=3-9]
    \arrow[from=3-9, to=5-9]
    \arrow[from=5-7, to=5-9]
    \arrow[from=3-7, to=3-9]
    \arrow[from=1-7, to=1-9]
    \arrow["{F(\alpha)}", from=1-7, to=3-7]
    \arrow["{F(\beta)}"', from=3-7, to=5-7]
    \arrow["{\operatorname{id}_{{ {C}_{\scriptscriptstyle \bullet}} }}"', curve={height=30pt}, from=1-1, to=5-1]
    \arrow["{\therefore \operatorname{id}_{F { {C}_{\scriptscriptstyle \bullet}} }}"', curve={height=30pt}, dashed, from=1-7, to=5-7]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)
:::

::: {.definition title="Projective resolution of a SES"}
For \( 0\to M' \to M\to M'' \to 0 \) in \( \mathsf{C} \), a **projective resolution** is a collection of chain maps forming projective resolutions of each of the constituent modules:

```{=tex}
\begin{tikzcd}
    0 && {{ {C}_{\scriptscriptstyle \bullet}} '} && {{ {C}_{\scriptscriptstyle \bullet}} } && {{ {C}_{\scriptscriptstyle \bullet}} ''} && 0 \\
    \\
    0 && {M'} && M && {M''} && 0
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=3-7, to=3-9]
    \arrow[from=1-7, to=1-9]
    \arrow[from=1-5, to=1-7]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-1, to=1-3]
    \arrow[Rightarrow, from=1-3, to=3-3]
    \arrow[Rightarrow, from=1-5, to=3-5]
    \arrow[Rightarrow, from=1-7, to=3-7]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMiwiMCJdLFsyLDIsIk0nIl0sWzQsMiwiTSJdLFs2LDIsIk0nJyJdLFs4LDIsIjAiXSxbMCwwLCIwIl0sWzIsMCwiXFxjb21wbGV4e0N9JyJdLFs0LDAsIlxcY29tcGxleHtDfSJdLFs2LDAsIlxcY29tcGxleHtDfScnIl0sWzgsMCwiMCJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdLFs4LDldLFs3LDhdLFs2LDddLFs1LDZdLFs2LDEsIiIsMSx7ImxldmVsIjoyfV0sWzcsMiwiIiwxLHsibGV2ZWwiOjJ9XSxbOCwzLCIiLDEseyJsZXZlbCI6Mn1dXQ==)
:::

::: {.exercise title="?"}
Show that such resolutions exist. This involves constructing \( {\varepsilon}: C_0 \to M \):

```{=tex}
\begin{tikzcd}
    0 && {C_0'} && {C \cong C_0' \oplus C_0''} && {C_0''} && 0 \\
    \\
    0 && {M'} && M && {M''} && 0
    \arrow[from=3-1, to=3-3]
    \arrow["\gamma", hook, from=3-3, to=3-5]
    \arrow["\sigma", two heads, from=3-5, to=3-7]
    \arrow[from=3-7, to=3-9]
    \arrow[from=1-7, to=1-9]
    \arrow["{p_0}", two heads, from=1-5, to=1-7]
    \arrow["{\iota_0}", hook, from=1-3, to=1-5]
    \arrow[from=1-1, to=1-3]
    \arrow["{\varepsilon}"', from=1-3, to=3-3]
    \arrow["{\therefore \exists {\varepsilon}}"', dashed, from=1-5, to=3-5]
    \arrow["{{\varepsilon}''}"', two heads, from=1-7, to=3-7]
    \arrow["{\exists {\varepsilon}^*}"', dashed, from=1-7, to=3-5]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)

The claim is that \( {\varepsilon}(x, x'') \coloneqq\gamma {\varepsilon}'(x') + {\varepsilon}^*(x'') \) works. To prove surjectivity, use the following:
:::

::: {.proposition title="Short Five Lemma"}
Given a commutative diagram of the following form

```{=tex}
\begin{tikzcd}
    0 && A && B && C && 0 \\
    \\
    0 && {A'} && {B'} && {C'} && 0
    \arrow[from=3-1, to=3-3]
    \arrow["s", from=3-3, to=3-5]
    \arrow["t", from=3-5, to=3-7]
    \arrow[from=3-7, to=3-9]
    \arrow[from=1-1, to=1-3]
    \arrow["p", from=1-3, to=1-5]
    \arrow["q", from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow["h"', from=1-7, to=3-7]
    \arrow["g"', from=1-5, to=3-5]
    \arrow["f"', from=1-3, to=3-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMCwiMCJdLFsyLDAsIkEiXSxbNCwwLCJCIl0sWzYsMCwiQyJdLFs4LDAsIjAiXSxbMCwyLCIwIl0sWzIsMiwiQSciXSxbNCwyLCJCJyJdLFs2LDIsIkMnIl0sWzgsMiwiMCJdLFs1LDZdLFs2LDcsInMiXSxbNyw4LCJ0Il0sWzgsOV0sWzAsMV0sWzEsMiwicCJdLFsyLDMsInEiXSxbMyw0XSxbMyw4LCJoIiwyXSxbMiw3LCJnIiwyXSxbMSw2LCJmIiwyXV0=)

If \( g,h \) are mono (resp. epi, resp. iso) then \( f \) is mono (resp. epi, resp. iso).
:::

::: {.proof title="of surjectivity, alternative by diagram chase"}
```{=tex}
\envlist
```
-   Let \( x\in M \)

-   Set \( y=\sigma(x) \)

-   Find \( z\in C_0 \) such that \( {\varepsilon}'' p_0 (z) = y \).

-   Consider \( {\varepsilon}(z) - x \) and apply \( \sigma \):
    \[
    \sigma({\varepsilon}(z) - x) 
    &= \sigma {\varepsilon}(x) - \sigma(x) \\
    &= {\varepsilon}'' p_0(x) - \sigma(x) \\
    &= y-y \\
    &= 0
    .\]

-   So \( {\varepsilon}(z) - x\in \ker \sigma = \operatorname{im}\gamma \)

-   Pull back to \( w\in C_0' \) such that \( \gamma {\varepsilon}'(w) = {\varepsilon}(z) - x \)

-   Check \( {\varepsilon}i_0 (w) = \gamma {\varepsilon}'(w) = {\varepsilon}(z) - x \), so \( {\varepsilon}(i_0(w) - z) = -x \).
:::

::: {.proof title="of existence"}
The setup:

```{=tex}
\begin{tikzcd}
    && 0 && 0 && 0 \\
    \\
    0 && {\ker {\varepsilon}'} && {\ker {\varepsilon}} && {\ker {\varepsilon}''} && 0 \\
    \\
    0 && {C_0'} && {C_0} && {C_0''} && 0 \\
    \\
    0 && {M'} && M && {M''} && 0
    \arrow[from=7-1, to=7-3]
    \arrow["\gamma", hook, from=7-3, to=7-5]
    \arrow["\sigma", two heads, from=7-5, to=7-7]
    \arrow[from=7-7, to=7-9]
    \arrow[from=5-7, to=5-9]
    \arrow["{p_0}", two heads, from=5-5, to=5-7]
    \arrow["{\iota_0}", hook, from=5-3, to=5-5]
    \arrow[from=5-1, to=5-3]
    \arrow["{{\varepsilon}'}"', from=5-3, to=7-3]
    \arrow["{\therefore \exists {\varepsilon}}"', dashed, from=5-5, to=7-5]
    \arrow["{{\varepsilon}''}"', two heads, from=5-7, to=7-7]
    \arrow["{\exists {\varepsilon}^*}"', dashed, from=5-7, to=7-5]
    \arrow[from=3-1, to=3-3]
    \arrow["f", hook, from=3-3, to=3-5]
    \arrow["g", from=3-5, to=3-7]
    \arrow[from=3-7, to=3-9]
    \arrow[hook, from=3-7, to=5-7]
    \arrow[hook, from=3-5, to=5-5]
    \arrow[hook, from=3-3, to=5-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=1-5, to=3-5]
    \arrow[from=1-7, to=3-7]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)

This is exact and commutative by a diagram chase:

-   \( f = i \circ \downarrow_{\ker {\varepsilon}'} \) shows \( g(\ker {\varepsilon}) \subseteq \ker {\varepsilon}'' \)
-   \( g = p \circ \downarrow_{\ker {\varepsilon}} \) shows \( f(\ker {\varepsilon}') \subseteq \ker {\varepsilon} \).

To show exactness along the top line:

-   \( f \) is injective, since it's the restriction of an injective map.
-   \( g \) is surjective:
    -   Let \( x\in \ker {\varepsilon}'' \), so \( {\varepsilon}''(x) = 0 \).
    -   \( \exists y\in C_0 \) with \( p_0(y) = x \) by surjectivity of \( p_0 \).
    -   Check \( {\varepsilon}''(p_0(y)) = {\varepsilon}(x) = 0 \) in \( M'' \), so \( \sigma{\varepsilon}(y) = 0 \)
    -   Thus \( {\varepsilon}(y)\in \ker \sigma = \operatorname{im}\gamma \)
    -   By surjectivity there exists \( w \in C_0' \) such that \( \gamma( {\varepsilon}'(w)) = {\varepsilon}(y) \).
    -   Use commutativity to verify
        \[
        {\varepsilon}(i_0(w) - y) 
        &= {\varepsilon}(i_0(w)) - {\varepsilon}(y) \\
        &= \gamma{\varepsilon}'(w) - {\varepsilon}(y) \\
        &= {\varepsilon}(y) - {\varepsilon}(y) \\
        &= 0
        .\]
    -   Then
        \[
        g(i_0(w) - y) 
        &= p_0(i_0 (w)) - g(y) \\
        &= -g(y) \\
        &= -p_0(y) \\
        &= -x
        .\]
-   Exactness at the middle, i.e. \( \operatorname{im}f = \ker g \):
    -   \( \operatorname{im}f \subseteq \ker g \) by exactness of the second row, so it STS \( \ker g \subseteq \operatorname{im}f \).
    -   Let \( y\in \ker g \), then by commutativity \( y\in \ker p_0 = \operatorname{im}i_0 \). Note that \( y\in \ker {\varepsilon} \) by definition.
    -   Write \( y = i_0(x) \) for some \( x\in C_0' \)
    -   Note \( \gamma {\varepsilon}' (x) = {\varepsilon}i_0(x) = {\varepsilon}(y) = 0 \) since \( y\in \ker {\varepsilon} \).
    -   Since \( \gamma' \) is mono, \( {\varepsilon}'(x) = 0 \), so \( y = i_0(x) = f(x) \).
:::

::: {.proposition title="?"}
For \( F: {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\to{\mathbb{Z}{\hbox{-}}\mathsf{Mod}} \) additive and a SES
\[
\xi: 0\to M' \xrightarrow{f} M \xrightarrow{g} M'' \to 0
,\]

note that there are morphisms
\[
{\mathbb{L}}F M'' \to {\mathbb{L}}F M\to {\mathbb{L}}FM'
.\]

There is a connecting morphism
\[
\Delta: {\mathbb{L}}F M'' \to \Sigma^{-1} {\mathbb{L}}F M'
,\]
which in components looks like

```{=tex}
\begin{tikzcd}
    0 && {{\mathbb{L}}_0 F(M'')} && {{\mathbb{L}}_0 F(M)} && {{\mathbb{L}}_0 F(M')} \\
    \\
    && {{\mathbb{L}}_1 F(M'')} && {{\mathbb{L}}_1 F(M)} && {{\mathbb{L}}_1 F(M')} \\
    \\
    && {{\mathbb{L}}_2 F(M'')} && \cdots
    \arrow[from=1-3, to=1-1]
    \arrow[from=1-5, to=1-3]
    \arrow[from=1-7, to=1-5]
    \arrow[from=3-3, to=1-7]
    \arrow[from=3-5, to=3-3]
    \arrow[from=3-7, to=3-5]
    \arrow[from=5-3, to=3-7]
    \arrow[from=5-5, to=5-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMCwwLCIwIl0sWzIsMCwiXFxMTF8wIEYoTScnKSJdLFs0LDAsIlxcTExfMCBGKE0pIl0sWzYsMCwiXFxMTF8wIEYoTScpIl0sWzIsMiwiXFxMTF8xIEYoTScnKSJdLFs0LDIsIlxcTExfMSBGKE0pIl0sWzYsMiwiXFxMTF8xIEYoTScpIl0sWzIsNCwiXFxMTF8yIEYoTScnKSJdLFs0LDQsIlxcY2RvdHMiXSxbMSwwXSxbMiwxXSxbMywyXSxbNCwzXSxbNSw0XSxbNiw1XSxbNyw2XSxbOCw3XV0=)
:::

# Thursday, February 10

Missed! Please send me notes. :)

# Tuesday, February 15

# Tuesday, February 22

## Prime Ideals

::: {.remark}
Plan: commutative ring theory, aiming toward tensor triangular geometry.
:::

::: {.remark}
```{=tex}
\envlist
```
-   Recall the definition of prime ideals.
-   Show \( {\mathfrak{p}}\in \operatorname{Spec}R \iff R/{\mathfrak{p}} \) is an integral domain.
-   Recall \( {\mathfrak{m}}\in \operatorname{mSpec}R \iff R/{\mathfrak{m}} \) is a field.
-   Recall the definition of a monoid
-   Note that \( R\setminus{\mathfrak{p}}\ni 1 \) and \( R\setminus{\mathfrak{p}} \) is a submonoid of \( (R, \cdot) \).
-   Examples of primes:
    -   \( \left\langle{p}\right\rangle \in \operatorname{Spec}R \) and if \( p\neq 0 \) then \( \left\langle{p}\right\rangle \in \operatorname{mSpec}R \).
    -   \( R = k[x] \) is a PID and \( \left\langle{f}\right\rangle \in \operatorname{Spec}R \iff f \) is irreducible.
-   Recall the set of nilpotent elements and the nilradical \( {\sqrt{0_{R}} } \).
    -   Show \( {\sqrt{0_{R}} } \in \operatorname{Id}(R) \).
    -   Show that \( R_{ \text{red} }\coloneqq R/{\sqrt{0_{R}} } \) is reduced (no nonzero nilpotents).
:::

::: {.lemma title="Prime Avoidance"}
Let \( A, I_j \in \operatorname{Id}(R) \) where at most two of the \( I_j \) are not prime and \( A \subseteq \displaystyle\bigcup_j I_j \). Then \( A \subseteq I_j \) for some \( j \).
:::

::: {.proof title="of lemma"}
The case \( n=1 \) is clear. For \( n>1 \), if \( A \subseteq \tilde I_k \coloneqq I_1 \cup I_2 \cup\cdots \widehat{I}_k \cup\cdots \cup I_n \) then the result holds by the IH. So suppose \( A \not\subseteq \tilde I_k \) and pick some \( a_k \not\in \tilde I_k \). Since \( A \subseteq \displaystyle\bigcup I_j \), we must have \( a_k\in I_k \).

Case 1: \( n=2 \). If \( a_1 + a_2\in A \) with \( a_1 \in I_1 \setminus I_2 \) and \( a_2\in I_2\setminus I_1 \), then \( a_1 + a_2\not\in I_1 \cup I_2 \) -- otherwise \( a_1 + a_2 \in I_1 \implies a_2\in I_1 \), and similarly if \( a_1 + a_2\in I_2 \). So \( A \subseteq I_1 \cup I_2 \).

Case 2: \( n>2 \). At least one \( I_j \) is prime, without loss of generality \( I_1 \). However, \( a_1 + a_2a_3\cdots a_n\in A \setminus\displaystyle\bigcup_{j\geq 1} I_j \). Since \( a_j\in I_j \), we have \( a_2\cdots a_n \in I_j \), contradicting \( a_1\not\in I_j \) for \( j\neq 1 \).
:::

::: {.proposition title="?"}
Let \( S\leq (R, \cdot) \) be a submonoid and \( P\in \operatorname{Id}(R) \) proper with \( P \cap S = \emptyset \) and \( P \) is maximal with respect to this property, so if \( P' \supseteq P \) and \( P' \cap S = \emptyset \) then \( P' = P \). Then \( P\in \operatorname{Spec}R \) is prime.
:::

::: {.proof title="?"}
By contrapositive, we'll show \( a,b\not\in P \implies ab\not\in P \). If \( a,b\not\in P \), then \( P \subsetneq aR + P, bR + P \) is a proper subset. By maximality, \( (aR + P) \cap S \neq \emptyset \) and \( (bR + P) \cap S \neq \emptyset \). Pick \( s_1, s_2\in S \) with \( s_1 = x_1 a + p_1, s_2 = x_2 b + p_2 \). Then \( s_1 s_2\in S \) and thus
\[
s_1 s_2 = x_1x_2 ab + x_1 ap_2 + x_2 b p_1 + p_1 p_2\in x_1x_2 ab + P + P + P
,\]
hence \( ab\not\in P \) -- otherwise \( S \cap P \neq \emptyset \). \( \contradiction \)
:::

::: {.proposition title="?"}
Let \( S \leq R \) be a monoid and let \( I \in \operatorname{Id}(R) \) with \( I \cap S = \emptyset \). Then there exists some \( p\in \operatorname{Spec}R \) such that

-   \( I \subseteq p \)
-   \( p \cap S = \emptyset \)
:::

::: {.proof title="?"}
Set \( B = \left\{{I' \supseteq I {~\mathrel{\Big\vert}~}I' \cap S = \emptyset}\right\} \), then \( B \neq \emptyset \). Apply Zorn's lemma to get a maximal element \( p \), which is prime by the previous proposition.
:::

::: {.theorem title="Krull"}
\[
{\sqrt{0_{R}} } = \cap_{p\in \operatorname{Spec}R} p
.\]
:::

::: {.exercise title="?"}
Prove this!
:::

## Localization

::: {.remark}
Recall the definition of \( {\mathbb{Q}} \) as \( {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{S} } \right] } \) where \( S = {\mathbb{Z}}\setminus\left\{{0}\right\} \) using the arithmetic of fractions. More generally, for \( D \) an integral domain, there is a field of fractions \( F \) with \( D \hookrightarrow F \) satisfying a universal property and thus uniqueness. Recall the definition of localization and the universal property: if \( \eta: R\to R' \) with \( \eta(S) \subseteq (R')^{\times} \) then \( \exists \tilde\eta: R \left[ { \scriptstyle { {S}^{-1}}  } \right] \to R' \).
:::

::: {.remark}
Next time:

-   Existence of \( R \left[ { \scriptstyle { {S}^{-1}}  } \right] \)
-   Localization for \( {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \).
-   Localization using tensor products.
:::

# Tuesday, March 01

::: {.remark}
Recall the definition of the localization of an \( R\in \mathsf{CRing}^{\operatorname{unital}} \) at a submonoid \( S \leq (M, \cdot) \), written \( R \left[ { \scriptstyle { {S}^{-1}}  } \right] \). Similarly for \( M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \), one can form \( M \left[ { \scriptstyle { {S}^{-1}}  } \right] \), and \( ({-}) \left[ { \scriptstyle { {S}^{-1}}  } \right] \) is a functor where the induced map on \( M \xrightarrow{f} N \) is \( f_S(m/s) \coloneqq f(m)/s \).
:::

::: {.proposition title="?"}
For \( I\in \operatorname{Id}(R) \), let \( j(I) \coloneqq\left\{{a\in R{~\mathrel{\Big\vert}~}a/s\in I \text{ for some } s\in S}\right\} \) which is again an ideal in \( R \). Then

1.  \( j(I)_S = I \),
2.  \( I_S = R_S \iff I \) contains an element of \( S \).
:::

::: {.proof title="of 2"}
\( \impliedby \): \( I_S \subseteq R_S \) is clear. Let \( x/t\in R_S \) and \( s\in I \cap S \), then \( {sx\over st} = {x\over t}\in I_S \).

\( \implies \): Write \( 1=i/s \) to produce \( t\in s \) with \( t(s-i) = 0 \). Then \( z=ts \in S \) and \( z=it\in I \) so \( z \in I \cap S \).
:::

::: {.proposition title="?"}
Let \( P\in \operatorname{Spec}R \) with \( S \cap p = \emptyset \), then \( j(P_S) = P \).
:::

::: {.proof title="?"}
\( \supseteq \): Clear.

\( \subseteq \): Let \( a\in j(P_S) \), so \( a/s=p/t \) for \( s,t\in S, p\in P \) and \( \exists u\in S \) such that \( u(at-sp)=0\in P \), so \( uat - usp\in P \) where \( usp\in P \). Thus \( uat\in P \implies a(ut)\in P\implies a\in P \), since \( ut\in S \) and \( ut\not\in P \).
:::

::: {.proposition title="?"}
There is an order-preserving correspondence
\[
\left\{{p\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}p \cap S = \emptyset}\right\} &\rightleftharpoons\operatorname{Spec}R \left[ { \scriptstyle { {S}^{-1}}  } \right] \\
P &\mapsto P \left[ { \scriptstyle { {S}^{-1}}  } \right] \\
j(P') &\mapsfrom P'
.\]
:::

::: {.proof title="?"}
We need to show

1.  \( P \left[ { \scriptstyle { {S}^{-1}}  } \right] \in \operatorname{Spec}R \left[ { \scriptstyle { {S}^{-1}}  } \right] \) is actually prime.
2.  If \( P'\in \operatorname{Spec}R \left[ { \scriptstyle { {S}^{-1}}  } \right] \) then \( j(P')\in \operatorname{Spec}R \) with \( j(P') \cap S = \emptyset \).

For one:
\[
{x\over t}, {y\over t} \in P_S 
&\implies {xy\over st} \in P_S \\
&\implies xy \in j(P_S) = P \\
&\implies x\in P \text{ or } y\in P \\
&\implies x/s\in P \text{ or } y/s\in P
.\]

For two:
\[
xy\in j(P') 
&\implies {xy\over s}\in P' \\
&\implies {x\over 1}{y\over s}\in P' \\
&\implies {x\over 1}\in P' \text{ or } {y\over s}\in P' \\
&\implies {x}\in P' \text{ or } {y}\in P' \\
.\]

If \( x\in j(P') \cap S \) then \( {x\over t}\in P' \) so \( {t\over x}{x\over t}\in P' \). \( \contradiction \)

One can then check that these two maps compose to the identity.
:::

::: {.exercise title="?"}
Show that if \( p\in \operatorname{Spec}R \) then \( R_p \in \mathsf{Loc}\mathsf{Ring} \) is local. Use that the image of \( p \) in \( R_p \) is \( P_p = R_p\setminus R_p^{\times} \), making it maximal and unique.
:::

::: {.exercise title="?"}
Show that

1.  \( M=0 \iff M_S = 0 \) for all \( S \),
2.  \( M=0 \iff M_p = 0\, \forall p\in \operatorname{mSpec}R \),
3.  \( M=0 \iff M_p = 0\, \forall p\in \operatorname{Spec}R \), noting that this is a stronger condition than maximal.

For (2), use that \( \operatorname{Ann}_R(x) \) is a proper ideal and thus contained in a maximal, and show by contradiction that \( x/1\neq 0\in M_p \).
:::

::: {.exercise title="?"}
Show that if \( f\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(M, N) \) then

-   \( f \) injective (resp. surjective) \( \implies f_S \) injective (resp. surjective)
-   If \( f_p \) is injective for all \( p\in \operatorname{Spec}R \), then \( f \) is injective (resp. surjective)
-   If \( M \) is flat then \( M_S \) is flat
-   If \( M_p \) is flat for all \( p \) then \( M \) is flat.
:::

::: {.remark}
Recall that for \( A \subseteq R \), \( V(A) \coloneqq\left\{{p\in \operatorname{Spec}R{~\mathrel{\Big\vert}~}p\supseteq A}\right\} \). Letting \( I(A) \) be the ideal generated by \( A \), then check that \( V(I(A)) = V(A) \) and \( V(I) = V(\sqrt I) \).
:::

::: {.exercise title="?"}
Check that defining closed sets as \( \left\{{V(A) {~\mathrel{\Big\vert}~}A \subseteq R}\right\} \) forms the basis for a topology on \( \operatorname{Spec}R \), and \( V(p) \cap V(q) = V(pq) \).
:::

::: {.remark}
Next time: generic points, idempotents, irreducible sets.
:::

# Tuesday, March 15

> See <https://www.math.ucla.edu/~balmer/Pubfile/TTG.pdf>

::: {.remark}
Recall that \( V(B) \coloneqq\left\{{p\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}p\supseteq B}\right\} \) are the closed sets for the Zariski topology, and \( V(B) = V(\left\langle{B}\right\rangle) \). Write \( I(A) = \displaystyle\bigcap_{p\in A} p \) for the vanishing ideal of \( A \), and note \( V(I(A)) = { \operatorname{cl}} _{\operatorname{Spec}R} A \). Recall \( \sqrt{J} = \displaystyle\bigcap_{p\supseteq J} = \left\{{x\in R {~\mathrel{\Big\vert}~}\exists n\, \text{ such that } x^n \in J}\right\} \), so \( \sqrt{0} \) is the nilradical, i.e. all nilpotent elements. An ideal \( J \) is radical iff \( \sqrt J = J \).
:::

::: {.theorem title="?"}
For \( X=\operatorname{Spec}R \), \( I(V(J)) = \sqrt{J} \), and there is a bijection between closed subsets of \( X \) and radical ideals in \( R \).
:::

::: {.proof title="?"}
\[
I(V(J)) = \displaystyle\bigcap_{p\in V(J)} p = \displaystyle\bigcap_{p\supseteq J} p = \sqrt{J}
,\]
and
\[
J \xrightarrow{V} V(J) \xrightarrow{I} I(V(J)) = \sqrt{J} = J
.\]
:::

::: {.remark}
Recall that \( X \) is **reducible** iff \( X= X_1 \cup X_2 \) with \( X_i \) nonempty proper and closed.
:::

::: {.theorem title="?"}
For \( R\in \mathsf{CRing} \), a closed subset \( A \subseteq X \) is irreducible iff \( I(A) \) is a prime ideal.
:::

::: {.proof title="?"}
\( \implies \): Suppose \( A \) is irreducible, let \( fg\in I(A) = \displaystyle\bigcap_{p\in A} p \). Then \( fg\in p\implies f\in [ \) without loss of generality for all \( p\in A \), and \( A = (A \cap V(f)) \cup(A \cap V(g)) \) so \( A \subseteq V(f) \) or \( A \subseteq V(g) \). Thus \( f\in \sqrt{\left\langle{f}\right\rangle} = I(V(f)) \subseteq I(A) \) (similarly for \( g \)).

\( \impliedby \): Suppose \( I(A) \) is a prime ideal and \( A = A_1 \cup A_2 \) with \( A_j \) closed, so \( I(A) \subseteq I(A_j) \). Then
\[
I(A) = I(A_1 \cup A_2) = I(A_1) \cap I(A_2)
.\]
If \( I(A_j) \subsetneq I(A) \) are proper containments, then one reaches a contradiction: if \( x\in I(A_1) \) and \( y\in I(A_2) \), use that \( xy\in I(A) \) to conclude \( x\in I(A) \) or \( y\in I(A) \).
:::

::: {.theorem title="?"}
Let \( X\in {\mathsf{Top}} \); TFAE:

1.  \( X \) is irreducible.
2.  Any two open nonempty sets intersect.
3.  Any nonempty open is dense in \( X \).
:::

::: {.proposition title="?"}
```{=tex}
\envlist
```
1.  Any irreducible subset of \( X \) is entirely contained in a single irreducible component.

2.  Any space is a union of its irreducible components.
:::

::: {.remark}
-   A space is Noetherian iff any descending chain of closed sets stabilizes, and if \( R \) is a Noetherian ring then \( X=\operatorname{Spec}R \) is a Noetherian space. Note that the converse may not hold in general!
-   A Noetherian space has a unique decomposition into irreducibles.
-   Any irreducible component is the closure of a point.
-   Any nonempty irreducible closed subset \( A \subseteq \operatorname{Spec}R \) contains a unique generic point \( p = I(A) \).
:::

::: {.remark}
Coming up:

-   Group cohomology, the Hopf algebra structure on \( kG \)
-   Cohomology using minimal resolutions
-   \( R = H^0(G; k) = \operatorname{Ext} _{kG}^0(k, k) \) which is a Noetherian ring
-   Use minimal resolutions to define \( c_{kG}(M) \), the rate of growth of a minimal projective resolution of \( M \) (1977)
-   Support varieties: \( R\coloneqq\operatorname{Ext} ^i_{kG}(k,k)\curvearrowright\tilde M\coloneqq\operatorname{Ext} ^0_{kG}(M, M) \), let \( J = \operatorname{Ann}_R(\tilde M) \) and \( V_G(M) = \operatorname{Spec}(R/J) \).
-   An equality of numerical invariants: \( c_{kG}(M) = \dim V_G(M) \).
-   Paul Balmer's tensor triangular geometry.
:::

# Tuesday, March 22

## Hilbert-Serre

::: {.remark}
Setup: \( V\in {\mathsf{gr}\,}_{\mathbb{Z}}{\mathsf{k}{\hbox{-}}\mathsf{Mod}} \) a graded vector space, so \( V = \bigoplus _{r\geq 0} V_r \) with \( \dim_k V_r < \infty \). Define the **Poincare series**
\[
p(V, t) = \sum_{r\geq 0} \dim V_r t^r
.\]
:::

::: {.theorem title="Hilbert-Serre"}
Let \( R\in {\mathsf{gr}\,}_{\mathbb{Z}}\mathsf{CRing} \) be of finite type over \( A_0 \) for \( A\in {{k}{\hbox{-}}\mathsf{Alg}} \) and suppose \( R \) is finitely generated over \( A_0 \) by homogeneous elements of degrees \( k_1,\cdots, k_s \). Supposing \( V\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}} \),
\[
p(V, t) = {f(t) \over \prod_{1\leq j\leq s} 1-t^{k_j} }, \qquad f(t) \in {\mathbb{Z}}[t]
.\]
:::

::: {.proposition title="?"}
Suppose that
\[
p(V, t) = {f(t) \over \prod_{1\leq j\leq s} 1-t^{k_j} } = \sum_{r\geq 0} a_r t^r, \qquad f(t) \in {\mathbb{Z}}[t], a_r\in {\mathbb{Z}}_{\geq 0}
.\]
Let \( \gamma \) be the order of the pole of \( p(t) \) at \( t=1 \). Then

1.  There exists \( K > 0 \) such that \( a_n \leq K n^{\gamma-1} \) for \( n\geq 0 \)

2.  There does *not* exist \( k > 0 \) such that \( a_n \leq k n^{\gamma - 2} \).
:::

::: {.definition title="?"}
Let \( V \) be a graded vector space of finite type over \( k \). The **rate of growth** \( \gamma(V) \) of \( V \) is the smallest \( \gamma \) such that \( \dim V_n \leq C n^{\gamma-1} \) for all \( n\geq 0 \) for some constant \( C \).
:::

::: {.remark}
Compare this to the complexity \( C_G(M) = \gamma(P_0) \) where \( P^0 \rightrightarrows M \) is a minimal projective resolution.
:::

## Finite Generation of Cohomology

::: {.remark}
Fix \( G \in {\mathsf{Fin}}{\mathsf{Grp}} \). Recall that \(  { {H}^{\scriptscriptstyle \bullet}} (G; k)  { {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{G}(k, k) \) has an algebra structure given by concatenation of LESs:

```{=tex}
\begin{tikzcd}
    {\xi_M:} & 0 & k & {M_1} & \cdots & {M_n} & k & 0 & {\in \operatorname{Ext} ^n_G(k, k)} \\
    \\
    {\xi_N:} & 0 & k & {N_1} & \cdots & {N_m} & k & 0 & {\in \operatorname{Ext} ^m_G(k, k)}
    \arrow[from=1-2, to=1-3]
    \arrow[from=1-3, to=1-4]
    \arrow[from=1-4, to=1-5]
    \arrow[from=1-5, to=1-6]
    \arrow[from=1-6, to=1-7]
    \arrow[from=1-7, to=1-8]
    \arrow["{\xi_M \cdot \xi_N}"{description}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-7, to=3-3]
    \arrow[from=3-3, to=3-4]
    \arrow[from=3-4, to=3-5]
    \arrow[from=3-5, to=3-6]
    \arrow[from=3-6, to=3-7]
    \arrow[from=3-7, to=3-8]
    \arrow[from=3-2, to=3-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=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)

Recall that \( \operatorname{Ext} ^n_{G}(k, k) = \mathop{\mathrm{Hom}}_{kG}(P_n, k) \), providing the additive structure. Moreover, \( \operatorname{Ext} _{kG}(M, M) \) is a ring, and if \( N\in {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \), then \( \operatorname{Ext} _{kG}{N, M} \in {\mathsf{\operatorname{Ext} _{kG}(M, M)}{\hbox{-}}\mathsf{Mod}} \). Similarly \( \operatorname{Ext} ^0_{kG}(N, M) \in {\mathsf{ { {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} (k, k)}{\hbox{-}}\mathsf{Mod}} \) by tensoring LESs.
:::

::: {.remark}
There is a coproduct
\[
kG &\xrightarrow{\Delta} kG \otimes_k kG \\
g &\mapsto g\otimes g
.\]
There is a cup product:

```{=tex}
\begin{tikzcd}
    {\bigoplus _{s+t=m} \operatorname{Ext} _{kG}^s(k, N) \otimes_k \operatorname{Ext} ^t_{kG}(k, M)} && {\operatorname{Ext} ^{m}_{kG{ {}^{ \scriptstyle\otimes_{k}^{2} }  }}(k\otimes_k N, k\otimes_k M) } \\
    \\
    && {\operatorname{Ext} _{kG}^m(N, M)}
    \arrow["\cong", tail reversed, from=1-1, to=1-3]
    \arrow[from=1-3, to=3-3]
    \arrow["{(a, b)\mapsto a\smile b}"', from=1-1, to=3-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXGJpZ29wbHVzIF97cyt0PW19IFxcRXh0X3trR31ecyhrLCBOKSBcXHRlbnNvcl9rIFxcRXh0XnRfe2tHfShrLCBNKSJdLFsyLDAsIlxcRXh0XnttfV97a0dcXHRlbnNvcnBvd2VyIGsgMn0oa1xcdGVuc29yX2sgTiwga1xcdGVuc29yX2sgTSkgIl0sWzIsMiwiXFxFeHRfe2tHfV5tKE4sIE0pIl0sWzAsMSwiXFxjb25nIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn19fV0sWzEsMl0sWzAsMiwiKGEsIGIpXFxtYXBzdG8gYVxcY3VwcHJvZCBiIiwyXV0=)

It is a theorem that this coincides with the Yoneda product.
:::

::: {.theorem title="?"}
```{=tex}
\envlist
```
-   \( H^0(G, k) \) is a graded commutative ring, so \( xy = (-1)^{{\left\lvert {x} \right\rvert} {\left\lvert {y} \right\rvert}} yx \)
-   The even part \(  { {H}^{\scriptscriptstyle \bullet}} ^{\text{even}}(G; k) \) is a (usual) commutative ring.
:::

::: {.theorem title="Evans-Venkov, 61"}
```{=tex}
\envlist
```
-   \( H^0(G; k) \) is a finitely generated in \( {\mathsf{Alg}_{/k} } \)
-   If \( M\in {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \) then \( \operatorname{Ext} ^0_{kG}(k, M) \in {\mathsf{  { {H}^{\scriptscriptstyle \bullet}} (G; k) }{\hbox{-}}\mathsf{Mod}} \).
:::

::: {.remark}
Quillen described \( \operatorname{mSpec} { {H}^{\scriptscriptstyle \bullet}} (G, k)^{ \text{red} } \) in the 70s. Idea: look at \( E \hookrightarrow G \) the elementary abelian subgroups, so \( E \cong C_p{ {}^{ \scriptscriptstyle\times^{m} }  } \) where \( p = \operatorname{ch}k \), and consider \( V_G(k) = \displaystyle\bigcup_{E\leq G} V_E(k)/\sim \) the union of all elementary abelian subgroups, where \( V_G(k) \coloneqq\operatorname{mSpec} { {H}^{\scriptscriptstyle \bullet}} ^{}(G; k)^{ \text{red} } \). Note that in characteristic zero, this is semisimple and only \( H^0=k \) survives.
:::

::: {.example title="?"}
```{=tex}
\envlist
```
-   For \( A = C_p \) with \( \operatorname{ch}k = p > 0 \), then
    \[
    R \coloneqq H^0(C_p; k) \cong
    \begin{cases}
    k[x,y]/\left\langle{y^2}\right\rangle, {\left\lvert {x} \right\rvert} = 2, {\left\lvert {y} \right\rvert} = 1 & p\geq 3 
    \\
     k[x], {\left\lvert {x} \right\rvert} = 1 & p = 2.
    \end{cases}, \qquad \operatorname{mSpec}R \cong {\mathbb{A}}^1_{/ {k}} 
    .\]

-   Dan's favorite: \( A = u({\mathfrak{g}}) \) for \( {\mathfrak{g}}= {\mathfrak{sl}}_2 \) with \( \operatorname{ch}k = p \geq 3 \) for \( u \) the *small enveloping algebra*. Friedlander-Parshall show \( \operatorname{mSpec}R = k[{\mathcal{N}}] \) for \( {\mathcal{N}}\coloneqq\left\{{M {
    \begin{bmatrix}
      {a} & {b} 
    \\
      {c} & {-a}
    \end{bmatrix}
    } {~\mathrel{\Big\vert}~}M\text{ is nilpotent}}\right\} \). This can be presented as
    \[
    k[{\mathcal{N}}] \cong k[x,y,z] / \left\langle{z^2 + xy}\right\rangle, {\left\lvert {x} \right\rvert}, {\left\lvert {y} \right\rvert}, {\left\lvert {z} \right\rvert} = 2
    ,\]
    and we'll see how finite generation is used in this setting.
:::

# Tuesday, March 29

::: {.remark}
Setup: for \( G \in {\mathsf{Fin}}{\mathsf{Grp}}, k\in \mathsf{Field} \) with \( \operatorname{ch}k = p \divides {\sharp}G \). For \( M\in {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \), we associate \( V_G(M) \subseteq \operatorname{mSpec}(R) \) for \( R\coloneqq H^0(G; k) \). There is a ring morphism \( \Phi_M: R\to \operatorname{Ext} ^0_{kG}(M, M) \), we set \( I_G(M) = \left\{{x\in R {~\mathrel{\Big\vert}~}\Phi_M(x) = 0}\right\} \) and define the support variety as \( V_G(M) = \operatorname{mSpec}(R/I_G(M)) \).
:::

::: {.example title="?"}
Let \( G = C_p{ {}^{ \scriptscriptstyle\times^{n} }  } \), then

-   \( H^2(G; k) = k[x_1, \cdots, x_{n}] \) for \( \operatorname{ch}k = p \geq 3 \).
-   \( \operatorname{mSpec}R = {\mathbb{A}}^n \supseteq V_E(M) \)
:::

## Rank Varieties

::: {.definition title="Rank varieties"}
For \( kG = k[z_1,\cdots, z_n]/\left\langle{z_1^p,\cdots, z_n^p}\right\rangle \), let \( x_{\mathbf{a}} \coloneqq\sum a_i z_i \) for \( a_i\in k \). Define the **rank variety**
\[
V_E^r(M) = \left\{{\mathbf{a} {~\mathrel{\Big\vert}~}\mathop{\mathrm{Res}}^{kG}_{ \left\langle{x_{\mathbf{a}}}\right\rangle } \text{ is not free} }\right\} \cup\left\{{0}\right\}
.\]
:::

::: {.theorem title="Carlson"}
\[
V_E(M) \cong V_E^r(M)
.\]
:::

::: {.remark}
Note that \( \operatorname{Ext} ^0(M, M)\curvearrowright\operatorname{Ext} ^0(M', M) \) by splicing, so we can define \( I_G(M', M) \coloneqq\operatorname{Ann}_R \operatorname{Ext} _{kG}^1(M', M) \) and the **relative support** variety \( V_G(M', M) = \operatorname{mSpec}(R/ I_G(M', M)) \). This recovers the previous notion by \( V_G(M, M) = V_G(M) \).
:::

::: {.remark}
Since \( I_G(M', M) \supseteq I_G(M) + I_G(M') \),
\[
V_G(M', M) \subseteq V_G(M) \cap V_G(M')
,\]
which relates relative support varieties to the usual support varieties.
:::

::: {.remark}
If \( 0\to A\to B\to C\to 0 \) is a SES, there is a LES in \( \operatorname{Ext} _{kG} \) and by considering annihilators we have
\[
I_G(A, M)\cdot I_G(B, M) \subseteq I_G(C, M) \implies V_G(C, M) \subseteq V_G(A, M)\cup V_G(C, M)
.\]
:::

::: {.proposition title="?"}
Let \( M\in \mathsf{kG}{\hbox{-}}\mathsf{Mod} \), then
\[
V_G(M) \subseteq \displaystyle\bigcup_{S\leq M \text{ simple}} V_G(S, M)
.\]
:::

::: {.proof title="?"}
Take the SES \( 0\to S_1 \to M \to M/S_1\to 0 \), then \( V_G(M) = V_G(M, M) \subseteq V_G(S_1, M) \cup V_G(M/S_1, M) \). Continuing this way yields a union of \( V(T, M) \) over all composition factors \( T \). Conversely, by the intersection formula above, this union is contained in \( V_G(M) \), so these are all equal.
:::

::: {.theorem title="?"}
Let \( M \in {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \), then

1.  \( c_G(M) = \dim V_G(M) \)
2.  \( V_G(M) = \left\{{0}\right\} \) (as a conical varieties) iff \( M \) is projective.
:::

::: {.proof title="?"}
Note (2) follows from (1), since complexity zero modules are precisely projectives. Consider \( \Phi_M: R\to \operatorname{Ext} ^0_{kG}(M, M) \), which induces \( R/I_G(M) \hookrightarrow\operatorname{Ext} _{kG}^0(M, M) \) which is finitely generated over \( R/I_{G}(M) \). A computation shows
\[
c_G(M) 
&= \gamma(\operatorname{Ext} _{kG}^0(M, M)) \\
&= \gamma( R/I_G(M) ) \\
&= \operatorname{krulldim}(R/I_G(M)) \\
&= \dim V_G(M)
.\]
:::

::: {.remark}
Consider a LES \( 0\to M\to M_1\to \cdots \to M_n \to M\to 0 \in \operatorname{Ext} _{kG}^n(M, M) \). Apply \( \Omega^n({-}) \), which arises from projective covers \(  { {P}^{\scriptscriptstyle \bullet}}  \rightrightarrows M \) and truncating to get \( 0\to \Omega^n \to P^{n-1}\to \cdots \to P_0 \to M\to 0 \). Similarly define \( \Omega^{-n} \) in terms of injective resolutions. There is an isomorphism \( \operatorname{Ext} _{kG}^n(M, M) \cong \operatorname{Ext} _{kG}^n(\Omega^s M, \Omega^s M) \) which is compatible with the \( R \) action. Thus \( V_G(M) \cong V_G (\Omega^s M) \) for any \( s \). Since \( kG \) is a Hopf algebra, dualizing yields \( \operatorname{Ext} _{kG}^n(M, M) \cong \operatorname{Ext} _{kG}^n(M {}^{ \vee }, M {}^{ \vee }) \) and thus \( V_G(M) \cong V_G(M {}^{ \vee }) \).
:::

## Properties of support varieties

::: {.proposition title="?"}
\[
V_G(M_1 \bigoplus M_2) \cong V_G(M_1)\cup V_G(M_2)
.\]
:::

::: {.proof title="?"}
Distribute:
\[
\operatorname{Ext} _{kG}^0(M_1 \oplus M_2, M_1 \oplus M_2) 
& \cong 
\operatorname{Ext} _{kG}^0(M_1, M_1) \oplus 
\operatorname{Ext} _{kG}^0(M_1, M_2) \oplus 
\operatorname{Ext} _{kG}^0(M_2, M_1) \oplus 
\operatorname{Ext} _{kG}^0(M_3, M_2)
.\]

Now \( I_G(M_1 \bigoplus M_2) \subseteq I_G(M_1) \oplus I_G(M_2) \), so \( V_G(M_1) \cup V_G(M_2) \subseteq V_G(M_1 \oplus M_2) \). Applying the 2 out of 3 property, \( V_G(M_1 \oplus M_2) \subseteq V_G(M_1) \cup V_G(M_2) \) since there is a SES \( 0\to M_1 \to M_1 \oplus M_2 \to M_2\to 0 \).
:::

::: {.theorem title="Tensor product property"}
Let \( M, N\in {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \), then
\[
V_G(M\otimes_k N) = V_G(M) \cap V_G(N)
.\]
:::

::: {.remark}
Conjectured by Carlson, proved by Arvrunin-Scott (82). Prove for elementary abelians, piece together using the Quillen stratification.
:::

::: {.theorem title="Carlson"}
Let \( X = \operatorname{mSpec}R \), which is a conical variety, and let \( W \subseteq X \) be a closed conical subvariety (e.g. a line through the origin). Then there exists an \( M\in {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \) such that \( V_G(M) = W \).
:::

::: {.remark}
Take \( \zeta: \Omega^n k \to k \), so \( \zeta\in R/I_G(M) \), and define certain \( L_\zeta \) modules and set \( Z(\zeta) \coloneqq V_G(L_\zeta) \).
:::

::: {.theorem title="Carlson"}
Let \( M \in {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \) be indecomposable. Then the projectivization \( \mathop{\mathrm{Proj}}V_G(M) \) is connected.
:::

## Supports using primes

::: {.remark}
As before, set \( R = H^{\text{even}}(G; k), X= \operatorname{Spec}R \), and now define
\[
V_G(M) = \left\{{p\in X{~\mathrel{\Big\vert}~}\operatorname{Ext} _{kG}^0(M, M)_p \neq 0}\right\}
.\]
All of the theorems mentioned today go through with this new definition.
:::

::: {.exercise title="?"}
Let \( I_G(M) = \operatorname{Ann}_R \operatorname{Ext} _{kG}^0(M, M) {~\trianglelefteq~}R \), and show
\[
V_G(M) = \left\{{p\in X{~\mathrel{\Big\vert}~}p\supseteq I_G(M) }\right\} = V(I_G(M))
\]
is a closed set.
:::

::: {.remark}
Let \( {\mathfrak{g}}\in \mathsf{Lie}{\mathsf{Alg}}_{/ {k}}  \) with \( \operatorname{ch}k = p > 0 \), e.g. \( {\mathfrak{g}}= {\mathfrak{gl}}_n(k) \). Then there is a \( p \)th power operation \( x^{{\left\lceil p \right\rceil}} = x\cdot x\cdots x \). The pair \( ({\mathfrak{g}}, {\left\lceil p \right\rceil}) \) forms a restricted Lie algebra. Consider the enveloping algebra \( U({\mathfrak{g}}) \), and define
\[
u({\mathfrak{g}}) \coloneqq U({\mathfrak{g}})/ \left\langle{x^p - x{ {}^{ \scriptstyle\otimes_{k}^{p} }  } {~\mathrel{\Big\vert}~}x\in {\mathfrak{g}}}\right\rangle
,\]
which is a finite-dimensional Hopf algebra:

-   The counit is \( {\varepsilon}(g) = 0 \) for \( g\in {\mathfrak{g}} \)
-   The antipode is \( \theta(g) = -g \)
-   The comultiplication is \( \Delta(g) = g\otimes 1 + 1\otimes g \).

The dimension is given by \( \dim u({\mathfrak{g}}) = p^{\dim {\mathfrak{g}}} \).
:::

# Tuesday, April 05

## Lie Theory

::: {.remark}
Setup: \( k =  { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }  \), \( \operatorname{ch}k = p > 0 \), \( {\mathfrak{g}} \) a restricted Lie algebra (e.g. \( {\mathfrak{g}}= \mathsf{Lie}(G) \) for \( G\in{\mathsf{Aff}}{\mathsf{Alg}}{\mathsf{Grp}}_{/ {k}}  \)). Write \( A^{{\left\lceil p \right\rceil} } = AA\cdots A \) and set \( A = u({\mathfrak{g}}) = U({\mathfrak{g}})/ J \) where \( J = \left\langle{x{ {}^{ \scriptstyle\otimes_{k}^{p} }  } - x^{{\left\lceil p \right\rceil}}}\right\rangle \) which is an ideal generated by central elements. Note that \( A \) is a finite-dimensional Hopf algebra.

Proved last time: \( H^0(A; k) \in {\mathsf{Alg}_{/k} }^{\mathrm{fg}} \), using a spectral sequence argument. From the spectral sequence, there is a finite morphism
\[
\Phi: S({\mathfrak{g}}^+)^{(1)} \to H^0(A; k)
,\]
making \( H^0(A; k) \) an integral extension of \( \operatorname{im}\Phi \). This induces a map
\[
\Phi: \operatorname{mSpec}H^0(A; k) \hookrightarrow{\mathfrak{g}}
.\]
:::

::: {.theorem title="Jantzen"}
\[
\operatorname{mSpec}H^0(A; k) \cong {\mathcal{N}}_p \coloneqq\left\{{x\in {\mathfrak{g}}{~\mathrel{\Big\vert}~}x^{{\left\lceil p \right\rceil}}}\right\}
.\]
:::

::: {.example title="?"}
For \( {\mathfrak{g}}= {\mathfrak{gl}}_n \), \( {\mathcal{N}}_p \leq {\mathcal{N}} \) is a subvariety of the nilpotent cone. Moreover \( {\mathcal{N}}_p \) is stable under \( G = \operatorname{GL}_n \), and there are only finitely many orbits. There is a decomposition into finitely many irreducible orbit closures
\[
{\mathcal{N}}_p = \displaystyle\bigcup_i \mkern 1.5mu\overline{\mkern-1.5muGx_i\mkern-1.5mu}\mkern 1.5mu
.\]
This corresponds to Jordan decompositions with blocks of size at most \( p \).
:::

::: {.remark}
Using spectral sequences one can show that if \( M, N \in {\mathsf{A}{\hbox{-}}\mathsf{Mod}} \) then \( \operatorname{Ext} ^0_A(M, N) \) is finitely-generated as a module over \( R\coloneqq H^0(A; k) \). So one can define support varieties \( V_{{\mathfrak{g}}}(M) = \operatorname{mSpec}R/J_M \) where \( I_M = \operatorname{Ann}_R \operatorname{Ext} ^0_A(M, M) \). Some facts:

-   \( V_{{\mathfrak{g}}}(M) \subseteq {\mathcal{N}}_p \subseteq {\mathfrak{g}} \)
-   If \( M \) is a \( G{\hbox{-}} \)module in addition to being a \( {\mathfrak{g}}{\hbox{-}} \)module, then \( V_G(M) \) is a \( G{\hbox{-}} \)stable closed subvariety of \( {\mathcal{N}}_p \).
:::

::: {.theorem title="Friedlander-Parshall (Inventiones 86)"}
Given \( M\in {\mathsf{u({\mathfrak{g}})}{\hbox{-}}\mathsf{Mod}} \),
\[
V_{{\mathfrak{g}}}(M) \cong \left\{{x\in {\mathfrak{g}}{~\mathrel{\Big\vert}~}x^{[p]} = 0, M \downarrow_{U(\left\langle{x}\right\rangle)} \text{ is not free over } u(\left\langle{x}\right\rangle) \leq u({\mathfrak{g}}) }\right\} \cup\left\{{0}\right\}
,\]
which is similar to the rank variety for finite groups, concretely realize the support variety.
:::

::: {.remark}
Here \( \left\langle{x}\right\rangle = kx \) is a 1-dimensional Lie algebra, and if \( x^{[p]} = 0 \) then \( u(\left\langle{x}\right\rangle) = k[x] / \left\langle{x^p}\right\rangle \) is a PID. We know how to classify modules over a PID: there are only finitely many indecomposable such modules.
:::

## Reductive algebraic groups

::: {.example title="?"}
For type \( A_n \sim \operatorname{GL}_{n+1} \), \( \alpha_0 = \tilde \alpha_n = \sum_{1\leq i \leq n} \alpha_i \) and \( h=n+1 \). For \( {\mathsf{G}}_2 \), \( \tilde \alpha_n = 3\alpha_1 + 2\alpha_2 \) and \( h=6 \).
:::

::: {.fact}
If \( p\geq h \) then \( {\mathcal{N}}_p({\mathfrak{g}}) = {\mathcal{N}} \).
:::

::: {.definition title="Good and bad primes"}
A prime is *bad* if it divides any coefficient of the highest weight. By type:

  Type        Bad primes
  ----------- ------------
  \( A_n \)   None
  \( B_n \)   2
  \( C_n \)   2
  \( D_n \)   2
  \( E_6 \)   2,3
  \( E_7 \)   2,3
  \( E_8 \)   2,3,5
  \( F_4 \)   2,3
  \( G_2 \)   2,3
:::

::: {.theorem title="Carlson-Lin-Nakano-Parshall (good primes), UGA VIGRE (bad primes)"}
\( {\mathcal{N}}_p = \mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{O}}\mkern-1.5mu}\mkern 1.5mu \) is an orbit closure, where \( {\mathcal{O}} \) is a \( G{\hbox{-}} \)orbit in \( {\mathcal{N}} \). Hence \( {\mathcal{N}}_p({\mathfrak{g}}) \) is an irreducible variety.
:::

::: {.remark}
Let \( X = X(T) \) be the weight lattice and let \( \lambda \in X \), then
\[
\Phi_\lambda \coloneqq\left\{{ \alpha\in \Phi {~\mathrel{\Big\vert}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \in p{\mathbb{Z}}}\right\}
.\]
Under the action of the affine Weyl group, this is empty when \( \lambda \) is on a wall (non-regular) and otherwise contains some roots for regular weights. When \( p \) is a good prime, there exists a \( w\in W \) with \( w(\Phi_\lambda) = \Phi_J \) for \( J \subseteq \Delta \) a subsystem of simple roots. In this case, there is a **Levi decomposition**
\[
{\mathfrak{g}}= u_J \oplus \ell_J \oplus u_J^+
.\]
:::

::: {.remark}
On Levis: consider type \( A_5 \sim \operatorname{GL}_6 \) with simple roots \( \alpha_i \).

```{=html}
```
![](figures/2022-04-05_10-34-18.png)
:::

::: {.remark}
Consider induced/costandard modules \( H^0( \lambda) = \operatorname{Ind}_B^G \lambda = \nabla(\lambda) \), which are nonzero only when \( \lambda \in X_+ \) is a dominant weight. Their characters are given by Weyl's character formula, and their duals are essentially *Weyl modules* which admit Weyl filtrations. What are their support varieties?
:::

::: {.theorem title="Nakano-Parshall-Vella, 2008"}
Let \( \lambda\in X_+ \) and let \( p \) be a good prime, and let \( w\in W \) such that \( w(\Phi_\lambda ) = \Phi_J \) for \( J \subseteq \Delta \). Then
\[
V_{{\mathfrak{g}}} H^0( \lambda) = G\cdot u_J = \mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{O}}\mkern-1.5mu}\mkern 1.5mu
\]
is the closure of a "Richardson orbit".
:::

::: {.remark}
```{=tex}
\envlist
```
-   This theorem was conjectured by Jantzen in 87, proved for type \( A \).
-   For bad primes, \( H^0(\lambda) \) is computed in one of seven VIGRE papers (2007). These still yield orbit closures that are irreducible, but need not be Richardson orbits.

Natural progression: what about tilting modules (good filtrations with costandard sections and good + Weyl filtrations)? We're aiming for the Humphreys conjecture.
:::

::: {.remark}
Let \( T( \lambda) \) be a tilting module for \( \lambda \in X_+ \). A conjecture of Humphreys: \( V_{{\mathfrak{g}}} T( \lambda) \) arises from considering 2-sided cells of the affine Weyl group, which biject with nilpotent orbits.
:::

::: {.example title="?"}
In type \( A_2 \):

```{=html}
```
![](figures/2022-04-05_10-39-31.png)

There are three nilpotent orbits corresponding to Jordan blocks of type \( X\alpha_1: (1,0) \) and \( X_\mathrm{reg}: (1,1) \) in \( {\mathfrak{gl}}_3 \). Three cases:

-   \( V_{{\mathfrak{g}}} T( \lambda) = {\mathcal{N}}= \mkern 1.5mu\overline{\mkern-1.5muG X_\mathrm{reg}\mkern-1.5mu}\mkern 1.5mu \)
-   \( V_{{\mathfrak{g}}} T( \lambda) = \mkern 1.5mu\overline{\mkern-1.5muG X_{ \alpha_1}\mkern-1.5mu}\mkern 1.5mu \)
-   \( V_{{\mathfrak{g}}} T( \lambda) = \left\{{0}\right\} \)

![](figures/2022-04-05_10-44-03.png)
:::

::: {.remark}
The computation of \( V_G T( \lambda) \) is still open. Some recent work:

-   \( p=2, A_n \): done by B. Cooper,
-   \( p > n+1, A_n \) by W. Hardesty,
-   \( p \gg 1 \), Achar, Hardesty, Riche.
:::

::: {.remark}
What about simple \( G{\hbox{-}} \)modules? Recall \( L(\lambda) = \mathop{\mathrm{Soc}}_G \nabla( \lambda) \subseteq \nabla( \lambda) \) -- computing \( V_G L( \lambda) \) is open.
:::

::: {.theorem title="Drupieski-N-Parshall"}
Let \( p > h \) and \( w( \Phi_ \lambda) = \Phi_J \), then
\[
V_{u_q({\mathfrak{g}})} L( \lambda) = G u_J
,\]
i.e. the support varieties in the quantum case are known. This uses that the Lusztig character formula is know for \( u_q( {\mathfrak{g}}) \).
:::

# Tuesday, April 12

## Tensor triangular geometry

::: {.remark}
Last time: tensor categories and triangulated categories. Idea due to Balmer: treat categories like rings.
:::

::: {.definition title="Tensor triangulated categories"}
A **tensor triangulated category** (TTC) is a triple \( (K, \otimes, 1) \) where

-   \( K \) is a triangulated category
-   \( (K, \otimes) \) is a symmetric monoidal category
-   \( 1 \) is a unit, so \( X\otimes 1  { \, \xrightarrow{\sim}\, }X  { \, \xrightarrow{\sim}\, }1\otimes X \) for all \( X \) in \( K \).
:::

::: {.remark}
We'll have notions of ideals, thick ideals, and prime ideals in \( K \). Define \( \operatorname{Spc}K \) to be the set of prime ideals with the following topology: for a collection \( C \subseteq \operatorname{Spec}K \), define \( Z(C) = \left\{{p\in \operatorname{Spc}K {~\mathrel{\Big\vert}~}C \cap p = \emptyset}\right\} \). Note that there is a universal categorical construction of \( \operatorname{Spc}K \) which we won't discuss here.
:::

::: {.remark}
TTC philosophy: let \( K \) be a compactly generated TTC with a generating set \( K^c \). Note that \( K \) can include "infinitely generated" objects, while \( K^c \) should thought of as "finite-dimensional" objects. Problems:

-   What is the homeomorphism type of \( \operatorname{Spc}K^c \)?
-   What are the thick ideals in \( K^c \)?

Although not all objects can be classified, there is a classification of thick tensor ideals. Idea: use the algebraic topology philosophy of passing to infinitely generated objects to simplify classification.
:::

::: {.remark}
We'll need a candidate space \( X\cong_{\mathsf{Top}}\operatorname{Spc}(K^c) \), e.g. a Zariski space: Noetherian, and every irreducible contains a generic point. We'll also need an assignment \( V: K^c\leadsto X_{{ \operatorname{cl}} } \) (the closed sets of \( X \)) satisfying certain properties, which is called a *support datum*. For \( I \) a thick tensor ideal, define
\[
\Gamma(I) \coloneqq\displaystyle\bigcup_{M\in I} V(M) \in X_{\mathrm{sp}}
,\]
a union of closed sets which is called *specialization closed*. Conversely, for \( W \) a specialized closed set, define a thick tensor ideal
\[
\Theta(W) \coloneqq\left\{{M\in K^c {~\mathrel{\Big\vert}~}V(M) \subseteq W}\right\}
.\]

One can check that a tensor product property holds: if \( M\in K^c \) and \( N\in \Theta(W) \), check \( V(M\otimes N) = V(M) \cap V(N) \subseteq W \). Under suitable conditions, a deep result is that \( \Gamma \circ \Theta = \operatorname{id} \) and \( \Theta \circ \Gamma = \operatorname{id} \). This yields a bijection
\[
\left\{{\substack{
  \text{Thick tensor ideals of } K^c
}}\right\}
&\rightleftharpoons
\left\{{\substack{
  \text{Specialization closed sets of } X
}}\right\} \\
I &\mapsto \Gamma(I) \\
\Theta(W) &\mapsfrom W
\]
:::

::: {.remark}
Define
\[
f: X\to \operatorname{Spc}K^c \\
x &\mapsto P_x \coloneqq\left\{{M \in K^c {~\mathrel{\Big\vert}~}x\not\int V(M)}\right\}
.\]
This is a prime ideal: if \( M\otimes N\in P_x \), then \( x\not \in V(M\otimes N) = V(M) \cap V(N) \), so \( M\in P_x \) or \( N\in P_x \).
:::

## Zariski spaces

::: {.definition title="Zariski spaces"}
A space \( X\in {\mathsf{Top}} \) is a **Zariski space** iff

1.  \( X \) is a Noetherian space, and
2.  Every irreducible closed set has a unique generic point.

Note that since \( X \) is Noetherian, it admits a decomposition into irreducible components \( X = \displaystyle\bigcup_{1\leq i \leq t} W_i \).
:::

::: {.example title="?"}
The basic examples:

-   For \( R \) a unital Noetherian commutative ring, \( X = \operatorname{Spec}R \) is Zariski.
-   For \( R \) a graded unital Noetherian ring, taking homogeneous prime ideals \( \mathop{\mathrm{Proj}}R \).
-   For \( G\in {\mathsf{Aff}}{\mathsf{Alg}}{\mathsf{Grp}} \) with \( G\curvearrowright R \) a graded ring by automorphisms (permuting the graded pieces), the stack \( X \coloneqq\mathop{\mathrm{Proj}}_G(R) \) (which is not Proj of the fixed points) is the set of \( G{\hbox{-}} \)invariant homogeneous prime ideals. There's a map \( \rho: \mathop{\mathrm{Proj}}R\to \mathop{\mathrm{Proj}}_G R \) where \( P\mapsto \cap_{g\in G} gP \) which gives \( \mathop{\mathrm{Proj}}_G R \) the quotient topology: \( W\in \mathop{\mathrm{Proj}}_G R \) is closed iff \( \rho\in R \) is close din \( \mathop{\mathrm{Proj}}R \). This topologizes orbit closures.
:::

::: {.remark}
Notation:

-   \( {\mathcal{X}}= 2^X \) for the powerset of \( X \),
-   \( {\mathcal{X}}_{{ \operatorname{cl}} } \) the closed sets,
-   \( {\mathcal{X}}_{{\mathrm{irr}}} \) the irreducible closed sets,
-   \( {\mathcal{X}}_{\mathrm{sp} } \) the specialization-closed sets.
:::

## Support data

::: {.remark}
Recall

-   \( M = {\mathsf{kG}{\hbox{-}}\mathsf{Mod}} \)
-   \( R = H^{\text{even}}(G; k) \)
-   \( V_G(M) = \left\{{p\in \mathop{\mathrm{Proj}}R {~\mathrel{\Big\vert}~}\operatorname{Ext} _{kG}(M, M)_p\neq 0 }\right\} \).

Note that \( V_G(P) = \emptyset \) for any projective and \( V_G(k) = \emptyset \). In general, we'll similarly want \( V_G(0) = \emptyset \) and \( V_G(1) = X \).
:::

::: {.definition title="Support data"}
A **support datum** is an assignment \( V: K \to {\mathcal{X}} \) such that

1.  \( V(0) = \emptyset \) and \( V(1) = X \).
2.  \( V\qty{\bigoplus _{i\in I} M_i = \displaystyle\bigcup_{i\in I} V(M_i) } \)
3.  \( V(\Sigma M) = V(M) \) (similar to \( \Omega \))
4.  For any distinguished triangle \( M\to N\to Q\to \Sigma M, V(N) \subseteq V(M) \cup V(Q) \).
5.  \( V(M\otimes N) = V(M) \cap V(N) \).

We'll need two more properties for the Balmer classification:

6.  Faithfulness: \( V(M) = \emptyset \iff M \cong 0 \).
7.  Realization: for any \( W\in {\mathcal{X}}_{{ \operatorname{cl}} } \) there exists a compact \( M\in K^c \) with \( V(M) = W \).
:::

::: {.remark}
Note that (6) holds for group cohomology, and (7) is Carlson's realization theorem.
:::

::: {.lemma title="?"}
Let \( K \) be a TTC which is closed under set-indexed coproducts and let \( V:K\to {\mathcal{X}} \) be a support datum. Let \( C \) be a collection of objects in \( K \) and suppose \( W \subseteq X \) with \( V(M) \subseteq W \) for all \( M\in C \). Then \( V(M) \subseteq W \) for all \( M \) in \( \mathsf{Loc}(C) \).
:::

::: {.proof title="?"}
Note that \( \mathsf{Loc}(C) \) is closed under

-   Applying \( \Sigma \) or \( \Sigma^{-1} \),
-   2-out-of-3: if two objects in a distinguished triangle are in \( \mathsf{Loc}(C) \), the third is in \( \mathsf{Loc}(C) \),
-   Taking direct summands,
-   Taking set-indexed coproducts.

These follow directly from the properties of support data and properties of \( \mathsf{Loc}(C) \).
:::

## Extension of support data

::: {.remark}
Let \( X \) be a Zariski space and let \( K\supseteq K^c \) be a compactly generated TTC. Let \( V: K^c\to {\mathcal{X}}_{{ \operatorname{cl}} } \) be a support data on compact objects, we then seek an *extension*: a support datum \( {\mathcal{V}} \) on \( K \) forming a commutative diagram:

```{=tex}
\begin{tikzcd}
    K && {\mathcal{X}}\\
    \\
    {K^c} && {{\mathcal{X}}_{{ \operatorname{cl}} }}
    \arrow[hook, from=3-1, to=1-1]
    \arrow[hook, from=3-3, to=1-3]
    \arrow["V", from=3-1, to=3-3]
    \arrow["{\mathcal{V}}", from=1-1, to=1-3]
\end{tikzcd}
```
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJLXmMiXSxbMCwwLCJLIl0sWzIsMCwiXFxtY3giXSxbMiwyLCJcXG1jeF97XFxjbH0iXSxbMCwxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFszLDIsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzAsMywiViJdLFsxLDIsIlxcbWN2Il1d)
:::

::: {.definition title="?"}
Let \( K \) be a compactly generated TTC and \( V: K^c\to {\mathcal{X}}_{{ \operatorname{cl}} } \) be a support datum. Then \( {\mathcal{V}}: K\to {\mathcal{X}} \) **extends** \( V \) iff

-   \( {\mathcal{V}} \) satisfies properties (1) -- (5) above,
-   \( V(M) = {\mathcal{V}}(M) \)for any \( M\in K^c \).
-   If \( V \) is faithful then \( {\mathcal{V}} \) is faithful.
:::

::: {.remark}
We'll need Hopkins' theorem to analyze such extensions.
:::

# Tuesday, April 19

## Hopkins' Theorem

::: {.remark}
Let \( \mathsf{K} \) be a compactly generated tensor triangulated category with \( \mathsf{K}^c \) a subcategory of compact objects. Goal: classify \( \operatorname{Spc}\mathsf{K}^c \). A candidate for its homeomorphism type: we'll build a Zariski space \( X \) and a homeomorphism \( \operatorname{Spc}\mathsf{K}^c \to X \). We'll use support data \( \mathbf{V}: \mathsf{K}^c\to {\mathcal{X}}_{{ \operatorname{cl}} } \) which satisfies the faithfulness and realization properties. We'll extend this to \( \mathcal{V}: \mathsf{K} \to {\mathcal{X}} \). So we need

-   A Zariski space \( X \),
-   Support data \( \mathbf V \),
-   An extension \( {\mathcal{V}} \).
:::

## Localization functors

::: {.remark}
Let \( \mathsf{C} \leq \mathsf{K} \) be a thick subcategory for \( \mathsf{K}\in {\mathsf{triang}}\mathsf{Cat} \). A mysterious sequence:
\[
\Gamma_c(M) \to M \to L_c(M)
.\]

Suppose \( W\in {\mathcal{X}}_{{\mathrm{irr}}} \) is nonempty and let \( Z = \left\{{x\in X{~\mathrel{\Big\vert}~}w\not\subseteq { \operatorname{cl}} _X\left\{{x}\right\}}\right\} \). Define a functor \( \nabla_W = \Gamma_{I_W} L_{I_Z} \) and \( {\mathcal{V}}(M) \coloneqq\left\{{x\in X {~\mathrel{\Big\vert}~}\nabla_{\left\{{x}\right\}} (M) = 0}\right\} \).
:::

::: {.theorem title="Hopkins-Neeman"}
Let \( \mathsf{K} \) be a compactly generated tensor triangulated category, \( X \) a Zariski space, and \( {\mathcal{X}}_{{ \operatorname{cl}} } \) the closed sets. Given a compact object \( M\in \mathsf{K}^c \), let \( \left\langle{M}\right\rangle_{\mathsf{K} ^c} \) be the thick tensor ideal in \( \mathsf{K}^c \) generated by \( M \). Let \( \mathbf V: \mathsf{K}^c\to {\mathcal{X}}_{{ \operatorname{cl}} } \) be support data satisfying the faithfulness condition and suppose \( {\mathcal{V}}: \mathsf{K}\to {\mathcal{X}} \) is an extension. Set \( W = \mathbf V(M) \) and \( I_W = \left\{{N\in \mathsf{K}^c {~\mathrel{\Big\vert}~}V(N) \subseteq W}\right\} \). Then
\[
I_W = \left\langle{M}\right\rangle_{\mathsf{K}^c}
,\]
i.e. this is generated by a single object.
:::

::: {.proof title="?"}
Let \( I \coloneqq I_W \) and \( I' \coloneqq\left\langle{M}\right\rangle_{\mathsf{K}^c} \).

\( I' \subseteq I \): If \( N\in I' \), then \( N \) is obtained by taking direct sums, direct summands, distinguished triangles, shifts, etc. These all preserve support containment, so \( \mathbf V(N) \subseteq W \) and \( N\in I = I_W \).

\( I \subseteq I' \): Let \( N\in {\mathsf{K^c}} \). Apply the functorial triangle \( \Gamma_{I'} \to \operatorname{id}\to L_{I'} \) to \( \Gamma_I(N) \) to obtain
\[
\Gamma_{I'} \Gamma_I N\to \Gamma_I(N) \to L_{I'} \Gamma_I N
.\]
From above, \( I' \subseteq I \) so the first term is in \( \mathsf{Loc}(I) \). Since the second term is as well, the 2-out-of-3 property guarantees that the third term satisfies \( L_{I'} \Gamma_I N \in\mathsf{Loc}(I) \). By the lemma, \( V(L_{I'} \Gamma_I N) \subseteq W \). There are no nonzero maps \( I' \to VL_{I'}\Gamma_I N \), therefore for \( S\in {\mathsf{K^c}} \), noting that \( S\otimes M \in I' \),
\[
0 
= \mathop{\mathrm{Hom}}_{\mathsf{K}}(S\otimes M, L_{I'} \Gamma_I M) 
= \mathop{\mathrm{Hom}}_{\mathsf{K}}(S, M {}^{ \vee }\otimes L_{I'} \Gamma_I N)
,\]
and since \( S \) is an arbitrary compact object, this forces \( M {}^{ \vee }\otimes L_{I'} \Gamma_I N = 0 \). By faithfulness, and the tensor product property,
\[
\emptyset 
&= {\mathcal{V}}(M {}^{ \vee }\otimes L_{I'} \Gamma_I N)\\
&= {\mathcal{V}}(M {}^{ \vee }) \cap{\mathcal{V}}(L_{I'}\Gamma_I N)\\
&= \mathbf{V}(M) \cap{\mathcal{V}}(L_{I'} \Gamma_I N)\\
&= W \cap{\mathcal{V}}(L_{I'} \Gamma_I N)\\
&= {\mathcal{V}}(L_{I'} \Gamma_I N)
,\]
so by faithfulness (again) \( L_{I'} \Gamma_I N = 0 \). Thus by the localization triangle, \( \Gamma_{I'} \Gamma_I N \cong \Gamma_I N \).

Now specialize to \( N\in I \); the localization triangle yields
\[
\Gamma_I N \to N \xrightarrow{0} L_I(N) \implies \Gamma_I N \cong N
.\]
Now replacing \( I \) with \( I' \) yields \( \Gamma_{I'} N \cong N \) since \( L_{I'} N \cong L_{I'} \Gamma_I N \cong 0 \) by the previous part. Thus \( N\in \mathsf{Loc}(I') \) by applying a result of Neeman, implying \( N\in I' \) and \( I \subseteq I' \).
:::

::: {.remark}
Many different takes on classification of thick tensor ideals:

-   Benson, Carlson, Rickard at UGA in the late 90s, for finite group representations (now extended).
-   Benson, Iyengar, Krause: axiomatic approach and description of supports.
-   Dell'Ambrogio
-   Boe, Kujawa, Nakano
:::

::: {.theorem title="?"}
Let

-   \( \mathsf{K} \) be a compactly generated tensor triangulated category,
-   \( X \) be a Zariski space,
-   \( \mathbf{V}: {\mathsf{K^c}}\to {\mathcal{X}}_{{ \operatorname{cl}} } \) be a support datum satisfying both the faithfulness *and* realization properties,
-   \( {\mathcal{V}}: \mathsf{K}\to C \) be an extension of \( \mathbf{V} \).

Let \( \operatorname{Id}({\mathsf{K^c}}) \) be the set of thick tensor ideals in \( {\mathsf{K^c}} \), then there is a bijection

\[
\operatorname{Id}({\mathsf{K^c}})
&\rightleftharpoons
{\mathcal{X}}_{\mathrm{sp} } \\
I &\mapsto \Gamma(I) \coloneqq\displaystyle\bigcup_{M\in I} \mathbf{V}(I) \\
\Theta(W) = I_W \coloneqq\left\{{N\in {\mathsf{K^c}}{~\mathrel{\Big\vert}~}\mathbf V(N) \subseteq W}\right\} &\mapsfrom W
.\]
:::

::: {.exercise title="?"}
Show that \( I_W\in \operatorname{Id}({\mathsf{K^c}}) \) is in fact a thick tensor ideal.
:::

::: {.proof title="?"}
\( \Gamma \circ \Theta = \operatorname{id} \): Check that
\[
\Gamma\Theta W = \Gamma(I_W) = \displaystyle\bigcup_{M\in I_W} \mathbf{V}(M) \subseteq W
.\]
For the reverse inclusion, let \( W = \displaystyle\bigcup_{j\in W} W_j \) where \( W_j\in {\mathcal{X}}_{{ \operatorname{cl}} } \). By the realization property, there exist \( N_j \in {\mathsf{K^c}} \) such that \( \mathbf{V}(N_j) = W_j \), so \( N_j\in I_W \). Now \( W \subseteq \displaystyle\bigcup_{M\in I_W} \mathbf{V}(M) \), so \( W = \displaystyle\bigcup_{M\in I_W} \mathbf{V}(M) \).

------------------------------------------------------------------------

\( \Theta \circ \Gamma = \operatorname{id} \): For \( I\in \operatorname{Id}({\mathsf{K^c}}) \), set \( W \coloneqq\Gamma(I) = \displaystyle\bigcup_{M\in I} \mathbf{V}(M) \), then
\[
\Theta\Gamma I = \Theta(W) = I_W \supseteq I
.\]
For the reverse inclusion \( I_W \subseteq I \): let \( N\in I_W \). Since \( X \) is a Zariski space, \( X \) is Noetherian and there is an irreducible component decomposition \( V(N) = \displaystyle\bigcup_i W_i \) with each \( W_i \) irreducible with a unique generic point, so \( W_i = { \operatorname{cl}} _{W_i} \left\{{x_i}\right\} \). Since each \( W_i \subseteq W \), each \( x_i\in W = \displaystyle\bigcup\mathbf{V}(M) \), so there exist \( M_i \) with \( x_i \in \mathbf{V}(M_i) \). Since supports are closed, \( W_i = { \operatorname{cl}} _{W_i}\left\{{x_i}\right\} \subseteq \mathbf{V}(M_i) \). Setting \( M\coloneqq\bigoplus _i M_i\in I \) yields \( V(N) \subseteq \displaystyle\bigcup V(M_i) = V(M) \subseteq W \).

::: {.claim}
\[
N \in \left\langle{M}\right\rangle_{{\mathsf{K^c}}}
.\]
:::

Proving the claim will complete the proof, since \( I \) is a thick ideal containing \( M \), so \( \left\langle{M}\right\rangle_{{\mathsf{K^c}}} \subseteq I \) and \( N\in I \).

::: {.proof title="of claim"}
By Hopkins' theorem, \( \left\langle{M}\right\rangle_{{\mathsf{K^c}}} = I_Z \) where \( Z = \mathbf{V}(M) \). Since \( V(N) \subseteq V(M) = Z \), we have \( N\in I_Z = \left\langle{M}\right\rangle_{{\mathsf{K^c}}} \).
:::
:::

::: {.remark}
Next time:

-   Showing \( \operatorname{Spc}{\mathsf{K^c}}\underset{{\mathsf{Top}}}{\cong} X \)
-   Examples: \( {\mathsf{kG}{\hbox{-}}\mathsf{stMod}} \), \( {\mathsf{u({\mathfrak{g}})}{\hbox{-}}\mathsf{stMod}} \), and \( {\mathbb{D}}{\mathsf{R}{\hbox{-}}\mathsf{Mod}} \).
:::

# Thursday, April 21

## Classification theorem

::: {.theorem title="?"}
Let \( \mathsf{K} \) be a compactly generated tensor-triangulated category and let \( X \) be a Zariski space. Suppose that

1.  \( \mathbf{V}: \mathsf{K}^c\to {\mathcal{X}}_{{ \operatorname{cl}} } \) is a support datum,
2.  \( \mathbf{V} \) satisfies the faithfulness property,
3.  \( {\mathcal{V}}: \mathsf{K}\to {\mathcal{X}} \) extends \( \mathbf{V} \).

Then there exists a bijective correspondence
\[
\adjunction{\Gamma}{\Theta}{\operatorname{Id}(\mathsf{K}^c) }{{\mathcal{X}}_{\mathrm{sp}} }
\]
where \( \Gamma(I) \coloneqq\displaystyle\bigcup_{M\in I} \mathbf{V}(M) \) and \( \Theta(W) \coloneqq\left\{{N\in\mathsf{K}^c{~\mathrel{\Big\vert}~}\mathbf{V}(N) \subseteq W}\right\} \).
:::

::: {.remark}
This relies on Hopkins' theorem.
:::

## Balmer spectrum

::: {.theorem title="?"}
Let \( \mathsf{K} \) and \( X \) be as in the previous theorem, satisfying the same assumptions. Then there exists a homeomorphism \( f: X\to \operatorname{Spc}\mathsf{K}^c \).
:::

::: {.proof title="?"}
Since \( \mathbf{V}: \mathsf{K}^c\to {\mathcal{X}}_{{ \operatorname{cl}} } \) is a support datum, Balmer shows there exists a continuous map
\[
f: X &\to \operatorname{Spc}\mathsf{K}^c \\
x &\mapsto P_x \coloneqq\left\{{M{~\mathrel{\Big\vert}~}x\not\in\mathbf{V}(M) }\right\} 
.\]

Note that \( P_x \) is a prime ideal:

\[
M\otimes N\in P_x 
&\implies x\not\in\mathbf{V}(M\otimes N) \\
&\implies x\not\in\mathbf{V}(M) \cap\mathbf{V}(N) \\
&\implies x\not\in \mathbf{V}(M) \text{ or } x\not\in \mathbf{V}(N) \\
&\implies M\in P_x \text{ or } N\in P_x
.\]

Applying the classification theorem, this yields a bijection.
:::

::: {.remark}
Examples of classification:

For \( G\in{\mathsf{Fin}}{\mathsf{Grp}}, \operatorname{ch}k = p\divides {\sharp}G \), take \( \mathsf{K} = {\mathsf{kG}{\hbox{-}}\mathsf{stMod}} \), \( R = H^{\mathrm{even}}(G; k) \), and \( X = \mathop{\mathrm{Proj}}R = \mathop{\mathrm{Proj}}(\operatorname{Spec}R) \). Checking that this satisfies the 4 properties in the theorem:

1.  For \( M\in \mathsf{K}^c \), we take \( \mathbf{V}(M) = \left\{{p\in X{~\mathrel{\Big\vert}~} { {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{kG}(M, M) \left[ { \scriptstyle { {p}^{-1}}  } \right] \neq 0 }\right\} \). This yields a support datum.

2.  The tensor product property holds because \( \mathbf{V}_E(M) = \mathbf{V}_E^r(M) \) (the rank variety), and we showed that \( \mathbf{V} \) satisfies faithfulness and (Carlson) realization properties.

3.  We can use localization functors to define \( {\mathcal{V}}: \mathsf{K}\to {\mathcal{X}} \) which satisfies the same support data properties. For this to be an extension, one should check that

-   \( \mathbf{V}(M) = {\mathcal{V}}(M) \) for every compact \( M\in \mathsf{K}^c \).
-   \( \mathbf{V}(M\otimes N) = {\mathcal{V}}(M) \cap{\mathcal{V}}(N) \) for all \( M,N\in \mathsf{K} \)
-   If \( {\mathcal{V}}(M) \) is empty then \( M = 0 \).
:::

::: {.remark}
To prove these properties, Benson-Carlson-Rickard start with \( E \) elementary abelian, so \( E = \left\langle{x_1,\cdots, x_n}\right\rangle \cong C_p{ {}^{ \scriptscriptstyle\times^{n} }  } \) with \( o(x_i) = p \) for all \( i \). Set \( y_i = x_i-1 \in kE \), so \( y_i^p=0 \), and define cyclic subgroups \( \mathbf{\alpha }= {\left[ {\alpha_1,\cdots, \alpha_n} \right]} \in L^n \) where \( L/k \) is a field of large transcendence degree. Define \( y_{\mathbf{\alpha}} \coloneqq\sum_{1\leq i\leq n} \alpha_i y_i \) and define a rank variety
\[
{\mathcal{V}}_E^r(M) = \left\{{ \mathbf{\alpha }\in L^n {~\mathrel{\Big\vert}~}L\otimes_k M \downarrow_{\left\langle{y_{\mathbf{\alpha}}}\right\rangle} \text{ is not free } }\right\}\cup\left\{{0}\right\} 
.\]
:::

::: {.theorem title="?"}
Let \( E \) be as above and suppose \( \operatorname{trdeg}(L/k) \geq n \). Then if \( M\in \mathsf{K} \), \( {\mathcal{V}}_E(M) \cong {\mathcal{V}}_E^r(M) \), and the three properties for (3) above hold for \( E \).
:::

::: {.theorem title="?"}
Let \( A = kG \) for \( G \) a finite group scheme, and let \( R = H^{\mathrm{even}}(G; k) \) and \( X = \mathop{\mathrm{Proj}}(R) \). Then

-   There is a bijective correspondence
    \[
    \adjunction{\Gamma}{\Theta}{{\mathsf{kG}{\hbox{-}}\mathsf{stMod}}}{{\mathcal{X}}_{\mathrm{sp}}}
    .\]

-   \( \operatorname{Spc}({\mathsf{kG}{\hbox{-}}\mathsf{stMod}}) \underset{{\mathsf{Top}}}{\cong} X \).
:::

::: {.remark}
Some remarks:

-   This theorem is an indication of why cohomology is central in understanding the tensor structure of representation categories. If \( G\in {\mathsf{Fin}}{\mathsf{Grp}}{\mathsf{Sch}}_{/ {k}}  \) then the coordinate ring \( k[G] \) is a commutative Hopf algebra, so \( A = kG = k[G] {}^{ \vee } \) is a finite dimensional cocommutative Hopf algebra. So there is an equivalence of categories between \( {\mathsf{Rep}}G \) and \( {\mathsf{Rep}}A \) for \( A \) such a Hopf algebra. By a result of Friedlander-Suslin, \( R \) is finitely generated.

-   The realization of \( \mathbf{V} \) and \( {\mathcal{V}} \) for a general group scheme involve so-called *\( \pi{\hbox{-}} \)points* developed be Friedlander-Pevtsovz and the construction of explicit rank varieties.
:::

::: {.remark}
A special case: let \( {\mathfrak{g}}= \mathsf{Lie}G \) for \( G\in{\mathsf{Alg}}{\mathsf{Grp}}_{/ {k}}  \) reductive and \( k \) positive characteristic. Let \( A = u({\mathfrak{g}}) \), which is a finite-dimensional cocommutative Hopf algebra. If \( p > h \) for \( h \) the Coxeter number,
\[
{\mathcal{N}}_p = \left\{{x\in {\mathfrak{g}}{~\mathrel{\Big\vert}~}x^{[p]} = 0 }\right\} = {\mathcal{N}}, \text{ the nilpotent cone}
,\]
\( R = H^{\mathrm{even}}(u({\mathfrak{g}}); k) = k[{\mathcal{N}}] \), and \( X = \mathop{\mathrm{Proj}}(k[{\mathcal{N}}]) \), then applying the theorem,

-   There is a correspondence
    \[
    \adjunction{}{}{{\mathsf{u({\mathfrak{g}})}{\hbox{-}}\mathsf{stMod}}}{{\mathcal{X}}_{\mathrm{sp}}}
    .\]

-   There is a homeomorphism
    \[
    \operatorname{Spc}\qty{ {\mathsf{u({\mathfrak{g}})}{\hbox{-}}\mathsf{stMod}} } \underset{{\mathsf{Top}}}{\cong} \mathop{\mathrm{Proj}}(k[{\mathcal{N}}])
    .\]
:::

::: {.theorem title="Arkhipov-Bezrukavikov-Ginzburg"}
Let \( \tilde {\mathcal{N}}\to {\mathcal{N}} \) be the Springer resolution. There is an equivalence of derived categories
\[
{\mathbb{D}}^b {\mathsf{ u_\zeta({\mathfrak{g}})_0}{\hbox{-}}\mathsf{Mod}}  { \, \xrightarrow{\sim}\, }{\mathbb{D}}^b {\mathsf{Coh}}^{G\times {\mathbb{C}}^{\times}} k[\tilde{\mathcal{N}}]  { \, \xrightarrow{\sim}\, }{\mathbb{D}}^b \mathsf{Perv}({\Omega}{\operatorname{Gr}})
.\]
where \( \mathsf{Perv}({-}) \) is the category of perverse sheaves and \( {\Omega}{\operatorname{Gr}} \) is the loop Grassmannian.
:::

::: {.remark}
For \( M \) a \( u_\zeta({\mathfrak{g}}){\hbox{-}} \)module and \( R = H^{\mathrm{even}}(u_\zeta({\mathfrak{g}}); M) = {\mathbb{C}}[{\mathcal{N}}] \cong {\mathbb{C}}[\tilde {\mathcal{N}}] \). There is an action of \( R \) on \(  { {H}^{\scriptscriptstyle \bullet}} (u_\zeta({\mathfrak{g}}); M) \). Next time: examples for Lie superalgebras and Thomason's reconstruction theorem for rings.
:::

# Tuesday, April 26

> See Boe-Kujawa-Nakano, Adv. Math 2017.

::: {.remark}
Setup: \( \mathsf{K}^c \leq \mathsf{K}\in {\mathsf{TTC}} \), \( X \) a Zariski space, \( V:\mathsf{K}^c\to {\mathcal{X}}_{{ \operatorname{cl}} } \) with an extension \( {\mathcal{V}}:\mathsf{K}\to {\mathcal{X}} \). Let \( {\mathfrak{g}}= {\mathfrak{g}}_{0} \oplus {\mathfrak{g}}_1 \) be a Lie superalgebra with a \( C_2 \) grading over \( k= {\mathbb{C}} \) where \( {\mathfrak{g}}_0\curvearrowright{\mathfrak{g}}_1 \), e.g. \( {\mathfrak{gl}}_{m, n} = {\mathfrak{gl}}_m \times {\mathfrak{gl}}_n \) with matrices \( {
\begin{bmatrix}
  {{\mathfrak{g}}_0} & {{\mathfrak{g}}_1} 
\\
  {{\mathfrak{g}}_1} & {{\mathfrak{g}}_0}
\end{bmatrix}
} \) with the bracket action. Write \( \mathsf{Lie}G_0 = {\mathfrak{g}}_0 \), and note that \( G_0 \) is reductive. Let \( {\mathcal{F}}({\mathfrak{g}}, {\mathfrak{g}}_0) \) be the category of finite-dimensional \( {\mathfrak{g}}{\hbox{-}} \)supermodules which are completely reducible over \( {\mathfrak{g}}_0 \). Take \( \mathsf{K}^c = {\mathsf{{\mathcal{F}}({\mathfrak{g}},{\mathfrak{g}}_0)}{\hbox{-}}\mathsf{stMod}} \leq \mathsf{K} = {\mathsf{C({\mathfrak{g}}, {\mathfrak{g}}_0)}{\hbox{-}}\mathsf{stMod}} \), where for \( C \) we drop the finite-dimensional condition.

Set \( R = H^0({\mathfrak{g}}_1, {\mathfrak{g}}_0; {\mathbb{C}}) = \operatorname{Ext} ({\mathbb{C}},{\mathbb{C}}) \cong S({\mathfrak{g}}_0 {}^{ \vee })^{G_0} \). By a theorem of Hilbert, \( \operatorname{Ext} (M, M) \) is finitely generated over \( R \). Write \( V_{{\mathfrak{g}},{\mathfrak{g}}_0}(M) = \mathrm{sp}ec R/J_M \) -- for Kac modules \( K(\lambda) = U({\mathfrak{g}}) \otimes_{U(P^0)} L_0(\lambda) \), \( V = 0 \) but not every \( K(\lambda) \) is projective.
:::

::: {.remark}
Idea: use detecting subalgebras. For \( {\mathfrak{g}}= {\mathfrak{gl}}_{n,n} \), let \( f_1 \) be the "torus":

```{=html}
```
![](figures/2022-04-26_09-58-21.png)

Then define \( f_0 = [f_1, f_1] \).
:::

::: {.remark}
Let \( X = N\mathop{\mathrm{Proj}}(S^*(f_1 {}^{ \vee })) \) where \( S^*(f_1 {}^{ \vee }) \cong \operatorname{Ext} _{f_1, f_0}({\mathbb{C}}, {\mathbb{C}}) = R' \) and \( N = { N }_{G_0}(f_1) \), which is a reductive algebraic group. Define a support datum by \( \mathbf{V}(M) = \left\{{p\in X{~\mathrel{\Big\vert}~}\operatorname{Ext} _{f, f_0}(M,M)_p = 0}\right\} \). The goal is to construct \( {\mathcal{V}}: K\to {\mathcal{X}} \) using localization functors -- one needs to show the tensor product formula, and the faithfulness and realization properties, which follows from Dede's lemma. It turns out that \( f_1\cong {\mathfrak{sl}}(1,1){ {}^{ \scriptscriptstyle\times^{m} }  } \) and it suffices to define the rank variety on \( f_1 \). Define
\[
V_{f_1}^{\operatorname{rank}}(M) = \left\{{\mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu = \tilde K\otimes_{\mathbb{Q}}f_1 {~\mathrel{\Big\vert}~}K\otimes_{\mathbb{C}}M\downarrow{\left\langle{\mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu}\right\rangle} \text{ is not projective} }\right\}
\]
where \( \tilde K\supseteq{\mathbb{C}} \) is an extension with \( \operatorname{trdeg}_{\mathbb{C}}\tilde K \geq \dim f_1 \). A theorem shows \( {\mathcal{V}}(M) = V_{f_1}^{\operatorname{rank}}(M) \) for \( M\in K \). This yields a classification for \( {\mathfrak{gl}}_{m, n} \) of thick tensor ideals in \( K^c \) in terms of \( {\mathcal{X}}_{\mathrm{sp}} \).
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::: {.remark}
What is the classification of other Lie superalgebras? This is an open problem.
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## Noncommutative theory

::: {.remark}
How does one extend this theory to noncommutative TTCs? See Nakano-Vashaw-Yakomov, to appear in Amer J. Math.
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::: {.remark}
Let \( K \) be a compactly generated monoidal triangulated category, not necessarily symmetric. One approaches this via noncommutative ring theory, where e.g. even the definition of prime ideals differs. We'll only consider 2-sided ideals.
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::: {.definition title="(Noncommutative) prime ideals"}
A thick triangulated subcategory \( P \) is a **completely prime** ideal iff \( M\otimes N\in P\implies M\in P \) or \( N\in P \). The ideal \( P \) is **prime** iff \( I\otimes J\subseteq P \implies I \subseteq P \) or \( J \subseteq P \), where \( I,J \) are themselves ideals. Define \( \mathrm{sp}c K \) to be prime ideals and \( \mathrm{CP}\operatorname{Spc}K \) to be completely prime ideals.
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::: {.example title="?"}
Let \( A\in \mathsf{Hopf}{\mathsf{Alg}_{/k} }^{{\mathrm{fd}}} \) where the coproduct \( \Delta: A\to A{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \) is not necessarily commutative, e.g. in the setting of quantum groups. Some remarks:

-   Note that \( M\otimes N \not\cong N\otimes M \) in general.
-   Here \( \mathrm{sp}c K^c \) is not known, but there is a conjectural answer.
-   In general \( \mathrm{sp}c K^c\not\cong \mathop{\mathrm{Proj}}R \) for \( R = H(A; k) \).
-   \( R \) is not known to be finitely-generated. Etingof-Ostrik conjecture this in the setting of finite tensor categories.
-   The definition of prime ideals is due to Buan-Krause-Solberg in 2007.
-   A weird example: there are nilpotents where \( M\neq 0 \) (is not projective) but \( M{ {}^{ \scriptstyle\otimes_{k}^{2} }  } = 0 \) (is projective).
-   Being a prime ideal \( P \) is equivalent to \( A\otimes C\otimes B \in P \) for all \( C \) \( \implies A\in P \) or \( B\in P \).
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::: {.definition title="(Noncommutative) support data"}
Let \( K \) be a monoidal triangulated category, \( X \) a Zariski space, and \( {\mathcal{X}}= 2^X \) the subsets of \( X \). A map \( \sigma: K\to{\mathcal{X}} \) is a **weak support datum** iff

-   \( \sigma(0) = \emptyset \) and \( \sigma(\one) = X \)
-   \( \sigma(A\otimes B) = \sigma(A) \cup\sigma(B) \)
-   \( \sigma(\Sigma A) = \sigma(A) \)
-   If \( A\to B\to C \) is exact then \( \sigma(A) \subseteq \sigma(B) \cup\sigma(C) \).

Set \( \Phi_\sigma(I) \coloneqq\displaystyle\bigcup_{M\in I} \sigma(I) \); Then \( \sigma \) is a **support datum** if additionally

-   \( \displaystyle\bigcup_{C\in K} \sigma(A\otimes C\otimes B)= \sigma(A) \cap\sigma(B) \) and
-   \( \Phi_\sigma(I\otimes J) = \Phi_\sigma(I) \cap\Phi_\sigma(J) \).
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::: {.remark}
Next time:

-   Classification theorems
-   The NVY conjecture for finite-dimensional Hopf algebras.
-   Tensor product theorems.
-   Examples of applications.
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