1 Thursday, January 06

Topics: Localization and completion, Nakayama’s lemma, Dedekind domains, Hilbert’s basis theorem, Hilbert’s Nullstellensatz, Krull dimension, depth and Cohen-Macaulay rings, regular local rings.

References:

Some examples of module morphisms:

Show that \(\operatorname{Id}(R/I)\rightleftharpoons\left\{{J\in \operatorname{Id}(R) {~\mathrel{\Big\vert}~}J \supseteq I}\right\}\) using \begin{align*} \Pi: R &\to R/I \\ x &\mapsto [x] \\ \pi^{-1}(J) &\mapsfrom J .\end{align*} Show that \(\pi^{-1}(J)\) is in fact an ideal, construct a proposed inverse \(\Pi^{-1}\), and show \(\Pi\circ \Pi^{-1}= \Pi^{-1}\circ \Pi = \operatorname{id}\).

Show that \(R\) is a field iff \(R\) is a simple ring iff any ring morphism \(R\to S\) is injective. For \(3\implies 1\), directly shows that every nonzero element is a unit.

Chapter 1 of A&M:

2 Thursday, January 13

Last time: fields are simple rings.

Let \(k\in \mathsf{Field}\) and show that

Prove that if \(f: R\to S\) is a ring morphism then there is a well-defined map \begin{align*} f^*: \operatorname{Spec}S &\to \operatorname{Spec}R \\ {\mathfrak{p}}&\mapsto f^{-1}({\mathfrak{p}}) ,\end{align*} i.e. if \({\mathfrak{p}}\) is prime in \(S\) then \(f^{-1}({\mathfrak{p}})\) is prime in \(R\). Show that this doesn’t generally hold with \(\operatorname{Spec}\) replaced by \(\operatorname{mSpec}\).

Show that defining \(V(I) \mathrel{\vcenter{:}}=\left\{{{\mathfrak{p}}\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}{\mathfrak{p}}\supseteq I}\right\}\) as closed sets defines a topology on \(\operatorname{Spec}R\). Show that \(\operatorname{Spec}R\) is Hausdorff iff it’s discrete, and \(V\) is functorial.

Describe

Show that every \(I\in \operatorname{Id}(R)\) is contained in some \({\mathfrak{m}}\in \operatorname{mSpec}R\).

Show that the following rings are local:

For \((R, {\mathfrak{m}})\) a local ring, show that \({\mathfrak{m}}= R\setminus(R^{\times})\).

Recall that

Show that \({\sqrt{0_{R}} }\) is the intersection of all prime ideals.

Regarding elements \(f\in R\) as functions on \(\operatorname{Spec}R\), \(f\) nilpotent is like being zero at every point of \(\operatorname{Spec}R\).

Show that \(x\in {J ({R}) } \iff 1-xy\in R^{\times}\) for all \(y\in R\).

3 Tuesday, January 18

A reference for pictures: Mumford’s red book. Note the typo in A&M problem 10: it is false for the zero ring.

Recall some definitions:

Show that if \(M\leq N \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), then the following action makes \(M/N\) into an \(R{\hbox{-}}\)module: \begin{align*} r\cdot [x] \mathrel{\vcenter{:}}=[rx] ,\end{align*} i.e. that if \([x] = [y]\) then \(r\cdot [x] = r\cdot [y]\).

Show the universal properties of kernels and cokernels: given \(f: M\to N\) and \(Q\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), \begin{align*} \mathop{\mathrm{Hom}}_R(Q, \ker f) &= \left\{{ s\in \mathop{\mathrm{Hom}}_R(Q, M) {~\mathrel{\Big\vert}~}f\circ s = 0}\right\} \\ \mathop{\mathrm{Hom}}_R(\operatorname{coker}f, Q) &= \left\{{t\in \mathop{\mathrm{Hom}}_R(N, Q) {~\mathrel{\Big\vert}~}t\circ f = 0 }\right\} .\end{align*}

Show that if \(I\) is a finite indexing set, there is an isomorphism \begin{align*} \bigoplus _{i\in I} M_i { { \, \xrightarrow{\sim}\, }}\prod_{i\in I} M_i , \end{align*} and that \begin{align*} \mathop{\mathrm{Hom}}_R\qty{ T, \prod M_i} &\cong \prod_{i\in I} \mathop{\mathrm{Hom}}_R\qty{T, M_i} \\ \mathop{\mathrm{Hom}}_R\qty{ \bigoplus M_i, T} &\cong \bigoplus \mathop{\mathrm{Hom}}_R\qty{ M_i, T} .\end{align*}

Show that by Yoneda, this satisfies the universal property.

Sketch:

Show that \(\prod\) is a categorical product and \(\bigoplus\) is a categorical coproduct. What are the product and coproduct in \({\mathsf{Top}}\)?

Given \(N \leq M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), define the colon ideal \begin{align*} (N: M) \mathrel{\vcenter{:}}=\left\{{a\in R{~\mathrel{\Big\vert}~}aM \subseteq N}\right\} \end{align*} and the annihilator of \(M\) \begin{align*} \operatorname{Ann}(M) = (0: M) = \left\{{a\in R {~\mathrel{\Big\vert}~}aM = 0}\right\} .\end{align*}

Some annihilators:

An \(R{\hbox{-}}\)module is free iff \(R \cong \bigoplus _{i\in I} R\), where \(I\) can be infinite but (importantly) we need the direct sum instead of the direct product. Note that generally \(\prod_{i\in I} R\) may not be free as an \(R{\hbox{-}}\)module.

A module is finitely generated if there exists a generating set \(X \mathrel{\vcenter{:}}=\left\{{x_i}\right\}_{i\leq n} \subseteq M\) such that any submodule containing \(X\) is all of \(M\), or equivalently \(x\in M \implies x = \sum r_i x_i\) for some \(r_i \in R\).

Show that \(M\) is finitely-generated iff there is a surjective morphism \(R^n \to M\) for some \(n\in {\mathbb{Z}}_{\geq 0}\).

Sketch:

Thus every \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) is the quotient of a free module: find a surjection \(f: F\to M\), so \(\operatorname{im}f = M\), then use that \(\operatorname{im}f \cong /\ker f\). For example, one can take \(F \mathrel{\vcenter{:}}=\bigoplus _{m\in M} R \to M\).

Recall

Check that \(({-}) \oplus M\) is exact.

Show that \(\mathop{\mathrm{Hom}}(N, {-})\) and \(\mathop{\mathrm{Hom}}({-}, N)\) are both left-exact for any \(N\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), where \(0 \to A \xrightarrow{f} B \xrightarrow{g} C\to 0\) is sent to \(0 \to \mathop{\mathrm{Hom}}(N, A) \xrightarrow{f\circ {-}} \cdots\). Give an example of when right-exactness fails.

Hint: try \(0\to {\mathbb{Z}}\xrightarrow{\cdot 2} {\mathbb{Z}}\to C_2 \to 0\) and apply \(\mathop{\mathrm{Hom}}_{\mathbb{Z}}({-}, C_2)\).

Recall that \(\mathop{\mathrm{Hom}}_{{\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}}}({\mathbb{Z}}, {-}) \cong \operatorname{id}\) and \(\mathop{\mathrm{Hom}}_{{\mathsf{{\mathbb{Z}}}{\hbox{-}}\mathsf{Mod}}}(C_n, {-}) \cong ({-})[n]\) picks out the \(n{\hbox{-}}\)torsion.

4 Thursday, January 20

Last time: \(\mathop{\mathrm{Hom}}(N, {-})\) and \(\mathop{\mathrm{Hom}}({-}, N)\) are left exact. We can explicitly describe \begin{align*} \mathop{\mathrm{Hom}}_A(A/I, M) = \left\{{m\in M {~\mathrel{\Big\vert}~}im = 0 \forall i\in I}\right\} ,\end{align*} which is the \(I{\hbox{-}}\)torsion in \(M\). Using that \(0\to I\to A \to A/I \to 0\) is short exact, we’ll get a long exact sequence \begin{align*} 0 \to \mathop{\mathrm{Hom}}(I, M) \to \mathop{\mathrm{Hom}}(A, M) \xrightarrow{f} \left\{{I{\hbox{-}}\text{torsion in } M}\right\} \to \operatorname{Ext} ^1_A(I, M) \to \cdots ,\end{align*} where the \(\operatorname{Ext}\) term measures failure of surjectivity of \(f\).

Show that \(\mathop{\mathrm{Hom}}(N, {-})\) and \(\mathop{\mathrm{Hom}}({-}, N)\) are left exact.

Recall the snake lemma:

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The recipe for the connecting morphism:

An object \(M\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) is of finite presentation iff there is an exact sequence \begin{align*} R^m \to R^n\to M \to 0 ,\end{align*} i.e. there are finitely many generators and finitely many relations.

A nice application of the snake lemma: for finitely presented modules \(M, N\), one can extend a morphism \(M\to N\) to

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Since \(R\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) is free, the extended maps can be represented by matrices, which is a significant simplification. In fact, \(f\) can be recovered uniquely by knowing the map on generators.

Prove the snake lemma, and show exactness at all 6 places.

Recall the universal property of \(M\otimes_R N\) in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) in terms of bilinear morphisms.

Prove uniqueness of any object satisfying a universal property using the Yoneda lemma.

Prove using universal properties:

5 Tuesday, January 25

Do A&M:

See website: https://www.daniellitt.com/commutative-algebra

Last time:

Show that if \(A \xrightarrow{f} B \xrightarrow{g} C\to 0\) is exact and \(A\otimes_R N, B\otimes_R N\) exist, then \(C\otimes_R N\) also exists.

Hint: Use the universal property to produce a map \(\psi: A\otimes N\to B\otimes N\) and set \(C\otimes N \mathrel{\vcenter{:}}=\operatorname{coker}\psi\).

Prove that \(M\otimes_R N\) exists by constructing it.

Some hints: Construction 1: use \(0\to \ker f\to R{ {}^{ \scriptscriptstyle\oplus^{I} } } \xrightarrow{f} M\to 0\), and find \(R{ {}^{ \scriptscriptstyle\oplus^{J} } }\to \ker f\) to assemble an exact sequence \begin{align*} R{ {}^{ \scriptscriptstyle\oplus^{J} } }\to R{ {}^{ \scriptscriptstyle\oplus^{I} } }\to M\to 0 .\end{align*} Now apply \(({-})\otimes_R N\) to exhibit \(M\otimes_R N\) as a cokernel \(N{ {}^{ \scriptscriptstyle\oplus^{J} } }\to N{ {}^{ \scriptscriptstyle\oplus^{I} } }\to M\otimes_R N\to 0\). Separately, use the hands-on construction and prove it satisfies the universal property.

Prove \(C_2 \otimes_{\mathbb{Z}}C_3 \cong 0\) using construction 1 above.

Sketch:

Take \({\mathbb{Z}}\xrightarrow{\times 2} {\mathbb{Z}}\to C_2\to 0\), apply \(({-})\otimes C_3\), and check that \(\operatorname{coker}(C_3 \xrightarrow{\times 2} C_3) = 0\) since multiplication by 2 is invertible. Alternatively, use bilinearity: \begin{align*} B(x,y) = 3B(x, y) - 2B(x, y) = B(x, 3y) - B(2x, y) = B(x, 0) - B(0, y) = 0 .\end{align*}

Show that

\({\mathsf{Alg}}_{/ {A}}\) is the coslice category \(\mathsf{CRing}_{{A/}}\): objects are rings \(B\) equipped with ring morphisms \(A\to B\), and morphisms are cones under \(A\):

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Examples of algebras:

An object \(B\in{\mathsf{Alg}}_{/ {A}}\) is finitely generated if it is a quotient of some \(A[t_1, \cdots, t_n]\). Equivalently, there exist \(x_1,\cdots, x_n\in B\) such that any subring containing the \(x_i\) and the image of \(A\) is all of \(B\).

\(B\) is finite if \(B\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\).

Examples:

6 Thursday, January 27

Nakayama: called a lemma, but arguably the most important statement in commutative algebra! Recall that \({J ({A}) } = \cap_{{\mathfrak{m}}\in \operatorname{mSpec}A} {\mathfrak{m}}\), and being zero mod every \({\mathfrak{m}}\in \operatorname{mSpec}A\) means being zero mod \({J ({A}) }\).

Let \(A \in \mathsf{CRing}\) with \(I \subseteq {J ({I}) }\) and let \(M\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\). Then \begin{align*} M=IM \implies M=0 .\end{align*}

This reduces statements about local rings to statements about fields. Commonly used examples:

If \(A\) is local and \(M\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\), \begin{align*} M/{\mathfrak{m}}_A M = 0 \iff M = 0 .\end{align*}

\(M\) yields a sheaf (or vector bundle) on \(\operatorname{Spec}A\), so think of \(m\in M\) as a function on \(\operatorname{Spec}A\) in the following way: if \(m\in M\), \begin{align*} m: {\mathfrak{p}}\mapsto m \operatorname{mod}{\mathfrak{p}}\in M/{\mathfrak{p}}M .\end{align*}

If \(n\in N\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\) and \(n\equiv 0 \operatorname{mod}{\mathfrak{m}}\) for all \({\mathfrak{m}}\in \operatorname{mSpec}A\), then \(M=0\).

Show that if \(A \in \mathsf{Loc}\mathsf{Ring}\) and \(M\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\) with \(\left\{{x_i}\right\}_{i\leq n} \subseteq M\), then \begin{align*} \left\langle{x_1,\cdots, x_n}\right\rangle = M \iff \left\langle{\mkern 1.5mu\overline{\mkern-1.5mux_1\mkern-1.5mu}\mkern 1.5mu, \cdots, \mkern 1.5mu\overline{\mkern-1.5mux_n\mkern-1.5mu}\mkern 1.5mu}\right\rangle = M/{\mathfrak{m}}M, \qquad \mkern 1.5mu\overline{\mkern-1.5mux_i\mkern-1.5mu}\mkern 1.5mu \mathrel{\vcenter{:}}= x_i \operatorname{mod}{\mathfrak{m}} .\end{align*}

\(\implies\): If \(\mkern 1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern 1.5mu\in M/{\mathfrak{m}}M\), lift to \(y\in M\) to write \(y = \sum c_i x_i\) and thus \(\mkern 1.5mu\overline{\mkern-1.5muy\mkern-1.5mu}\mkern 1.5mu = \sum c_i \mkern 1.5mu\overline{\mkern-1.5mux_i\mkern-1.5mu}\mkern 1.5mu\).

\(\impliedby\): Present \(M\) by \(A^n \xrightarrow{f} M \to C =\operatorname{coker}f \to 0\) where \(f(e_i) = x_i\) – we want to show \(C = 0\). Reduce mod \({\mathfrak{m}}\) by applying \(({-})\otimes_A A/{\mathfrak{m}}\) to get \begin{align*} (A/{\mathfrak{m}})^n \xrightarrow{\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu} M/{\mathfrak{m}}M \to C/{\mathfrak{m}}C\to 0 .\end{align*} This is surjective so \(C/{\mathfrak{m}}C = 0\). By Nakayama, \(C=0\).

Let the local ring \(R = k { \left[ {t} \right] }\) with \({\mathfrak{m}}_R = \left\langle{t}\right\rangle\), and let \(M = R{ {}^{ \scriptscriptstyle\oplus^{3} } }\). Consider \begin{align*} {\left[ {1+t^3 + t^5, t+t^2, t^3} \right]}, {\left[ {t^{22}, 1+t^9, t^{10} + t^{10^{10}}} \right]}, {\left[ {1+ t^{10^{10^{10}}}, 1+t^5 + t^9, 1+t^7 + t^{11} } \right]} ,\end{align*} which reduced \(\operatorname{mod}t\) yields \begin{align*} {\left[ {1,0,0} \right]}, {\left[ {0,1,0} \right]}, {\left[ {1,1,1} \right]} .\end{align*} So the original elements generate \(M\).

Next goal: proving Nakayama. We’ll need a version of Cayley-Hamilton. Recall the definition of multiplicative subsets and localization, along with its universal property.

Examples:

Prove \(S^{-1}A\) exists and is unique for \(A\in \mathsf{CRing}\). Give an example where \(s'a = sb\) is not sufficient.

Remarks on localization:

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Some exercises:

7 Tuesday, February 01

For \(A\in \mathsf{CRing}, M\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\), \({\mathfrak{a}}\in \operatorname{Id}(A)\), and \(\phi: M\to {\mathfrak{a}}M \subseteq M\), there exists \(\left\{{a_i}\right\}_{i\leq n} \subseteq {\mathfrak{a}}\) such that \begin{align*} \phi^r + a_1 \phi^{r-1} + \cdots + a_r \operatorname{id}= 0 .\end{align*}

Reduce to showing this for \(M = A^r\) a free module. Use the diagram:

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Lift by sending \(e_i\) to any element \(a_i \in {\mathfrak{a}}\). Then: STS \(\tilde f\) satisfies some polynomial, since \(\phi\) will satisfy the same polynomial and map to zero by commutativity of the square above. \(\tilde f: A^r\to A^r\) can be written as a matrix \((a_{ij})\). Now reduce to \({\mathbb{C}}\): consider the map \begin{align*} {\mathbb{Z}}[ \left\{{ x_{ij}}\right\}_{i, j \leq r} ] &\to A \\ x_{ij} &\mapsto a_{ij} .\end{align*}

Forming the matrix \(M = (x_{ij})\) yields a commutative diagram:

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Since \({\mathbb{Z}}[\left\{{ x_{ij} }\right\}]\) is an integral domain, we can pass to the fraction field \({\mathbb{Q}}(\left\{{x_{ij}}\right\})\), where by linear algebra \(M\) has a characteristic polynomial in the \(x_{ij}\). So for some homogeneous polynomials \(p_i\) in the \(x_{ij}\), we have \begin{align*} \tilde f^r + p_1(x_{ij}) \tilde f^{r-1} + \cdots + p_r(x_{ij}) = 0 .\end{align*} Now choose an arbitrary embedding \({\mathbb{Q}}(x_{ij}) \hookrightarrow{\mathbb{C}}\), and prove Cayley-Hamilton here using (e.g.) Jordan normal form.

Write a careful proof of Cayley-Hamilton for arbitrary fields.

Useful strategy: reduce to the “universal matrix.”

Recall that there exists an adjugate of any square matrix satisfying \(M \operatorname{adj}(M) = \operatorname{adj}(M) M = \operatorname{det}(M) \operatorname{id}\). Apply this to \(M \mathrel{\vcenter{:}}= tI - \tilde f \in \mathop{\mathrm{End}}_{A[t]}(A(t){ {}^{ \scriptscriptstyle\times^{r} } })\). Write \(\operatorname{adj}(tI-\tilde f) = \sum B_i t^i\) with \(B_i \in \operatorname{Mat}_{n\times n}(A)\). We have \begin{align*} (tI-\tilde f) \operatorname{adj}(tI - \tilde f) = \operatorname{det}(\tilde f)I ,\end{align*} but we can’t plug in \(\tilde f\) here because we don’t know if this lands in a commutative ring, so e.g. \(t m t^r \neq mt^{r+1}\) doesn’t necessarily hold.

Note that \(B_i\) commutes with \(tI - \tilde f \iff B_i\) commutes with \(\tilde f\), e.g. by equating coefficients. Write \(R = Z(\tilde f)\) for the centralizer, those matrices commuting with \(\tilde f\). Then \((tI - \tilde f) \in R[t]\), which reduces us to the world of commutative rings. Then \({ \left.{{ \operatorname{det}(tI - \tilde f) }} \right|_{{\tilde f}} } = (\tilde f - \tilde f)\cdot g = 0\).

Show that if \(M\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\) and \({\mathfrak{a}}\in \operatorname{Id}(A)\), \begin{align*} M = {\mathfrak{a}}M \implies \exists x\cong 1{\mathsf{{\mathfrak{a}}}{\hbox{-}}\mathsf{Mod}}\text{ such that } xM = 0 .\end{align*}

Hint: apply Cayley-Hamilton to \(\operatorname{id}_M\).

For \(M\in {\mathsf{A}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}, I \in \operatorname{Id}(A)\) with \(I \in {J ({A}) }\), \begin{align*} M = IM \implies M = 0 .\end{align*}

Apply the corollary to get \(x\equiv 1 \operatorname{mod}I\) with \(xM = 0\). But then \(x \equiv 1 \operatorname{mod}{J ({A}) }\), so \(x\) is a unique and \(xM = M\).

Alternatively, pick a minimal set of generators \(\left\{{x_i}\right\}\) of \(M\), so \(m = \sum a_i x_i\) with \(a_i\in I\) since \(M=IM\). Since \(1-a_n\in {J ({A}) }\) and is a unit, so \begin{align*} (1-a_n)x_n = \sum_{i\leq n-1} a_i x_i \implies x_n = (1-a_n)^{-1}\sum_{i\leq n-1}a_i x_i .\end{align*} \(\contradiction\)

Notes:

8 Thursday, February 03

Exercises:

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9 Tuesday, February 08

Show that chain-homotopic maps induce the same map in homology.

\({ {\operatorname{Tor}}_{\scriptscriptstyle \bullet}} ^{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N)\) for \(M, N\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) is well-defined, since the identity \(M \xrightarrow{\operatorname{id}_M} M\) induces an isomorphism on \({ {P}_{\scriptscriptstyle \bullet}} \otimes N \to { {P}_{\scriptscriptstyle \bullet}} '\otimes N\) for any two projective resolution \({ {P}_{\scriptscriptstyle \bullet}} , { {P}_{\scriptscriptstyle \bullet}} ' \rightrightarrows M\).

If \(f\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(M, N)\), then there is an induced morphism \(\tilde f\in \mathsf{Ch}{\mathsf{R}{\hbox{-}}\mathsf{Mod}}({ {P}_{\scriptscriptstyle \bullet}} ^M, { {P}_{\scriptscriptstyle \bullet}} ^N)\) between resolutions \({ {P}_{\scriptscriptstyle \bullet}} ^M\rightrightarrows M, { {P}_{\scriptscriptstyle \bullet}} ^N\rightrightarrows N\), where \(\tilde f\) is unique up to homotopy.

For existence, use projectivity to lift through surjections onto kernels. For uniqueness, create \(s_0\) such that \(d_1 s_0 = d_0-g_0\) and check that \(\operatorname{im}f_0 - g_0 \subseteq \ker d_0\) after lifting through \(P_1^N\to \ker d_0\):

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Show that a SES of modules induces a SES of chain complexes between their projective resolutions.

Hint: use the following diagram.

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Show that a SES of chain complexes induces a LES in their homology. There are about eight conditions one needs to check here (as in the snake lemma, of which this is a special case for two-term complexes).

Show that given \(0\to M_1 \to M_2\to M_3\to 0\), taking projective resolutions and applying \(({-})\otimes N\) yields a SES \begin{align*} 0\to { {P}_{\scriptscriptstyle \bullet}} ^1\otimes N \to { {P}_{\scriptscriptstyle \bullet}} ^2\otimes N \to { {P}_{\scriptscriptstyle \bullet}} ^3\otimes N\to 0 ,\end{align*} so there is an induced LES in \({ {\operatorname{Tor}}_{\scriptscriptstyle \bullet}} ^{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}\).

Show

Hint: \begin{align*} \operatorname{Tor}_0^{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N) &= H_0({ {P}_{\scriptscriptstyle \bullet}} ^M \otimes_R N) \\ &= \operatorname{coker}\qty{P_1\otimes N \to P_0\otimes_R N}\\ &= \operatorname{coker}(P_1\to P_0) \otimes_R N \\ &= M\otimes_R N .\end{align*}

For vanishing, use that projective implies flat and exact complexes have zero higher homology. Note that if \(M\) is projective, it is its own projective resolution.

For uniqueness, induct on \(i\): write \(0\to K \to R{ {}^{ \scriptscriptstyle\oplus^{J} } }\to M\to 0\), use that free implies projective, and consider the LES:

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Now induct up using the isomorphisms in the LES.

Show that \begin{align*} { {\operatorname{Tor}}_{\scriptscriptstyle \bullet}} ^{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N) \cong {{ {H}_{\scriptscriptstyle \bullet}} }({ {P}_{\scriptscriptstyle \bullet}} ^M \otimes_R N) \cong {{ {H}_{\scriptscriptstyle \bullet}} }(M\otimes_R { {P}_{\scriptscriptstyle \bullet}} ^N )\cong { { \operatorname{Tor}}_{\scriptscriptstyle \bullet}} ^{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(N, M) .\end{align*}

10 Thursday, February 10

Let \(R= k[x,y]\) and \(M = k \cong k[x,y]/\left\langle{x, y}\right\rangle\), and compute \({ {\operatorname{Tor}}_{\scriptscriptstyle \bullet}} ^{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(k, k)\).

Hint: use the following resolution

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Compute \({ {\operatorname{Tor}}_{\scriptscriptstyle \bullet}} ^{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(k, k)\) for \(R=k[x_1, \cdots, x_{n}]\) and \(M = k = k[x_1, \cdots, x_{n}]/\left\langle{x_1,\cdots, x_n}\right\rangle\).

\(\operatorname{Tor}\) measures failure of injectivity of tensoring against a module \(M\), \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}}\) measures failure of surjectivity when mapping against \(M\).

Show that for \(R\in\mathsf{CRing}\), every \(M\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) admits an injective resolution.

Hint: it suffices to show any \(M\) injects into an injective object. Use the following diagram:

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First show this for \(R={\mathbb{Z}}\) using that \({\mathbb{Q}}{ {}^{ \scriptscriptstyle\oplus^{J} } }/K\) is divisible and thus injective:

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Now reduce to \(R={\mathbb{Z}}\) using that \(M\hookrightarrow D\) for \(D\) a divisible abelian group, and show \(M\hookrightarrow I \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{{\mathbb{Z}{\hbox{-}}\mathsf{Mod}}}(R, D)\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\). Form the map as the composition:

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Then show that \(I\mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{{\mathbb{Z}{\hbox{-}}\mathsf{Mod}}}(R, D)\) is injective using the universal property:

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Show \({\mathbb{Z}}\in {\mathbb{Z}{\hbox{-}}\mathsf{Mod}}\) has an injective resolution.

\begin{align*} 0 \to {\mathbb{Z}}\hookrightarrow{\mathbb{Q}}\twoheadrightarrow{\mathbb{Q}}/{\mathbb{Z}}\to 0 .\end{align*}

Show \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{{\mathbb{Z}{\hbox{-}}\mathsf{Mod}}}(C_3, {\mathbb{Z}}) \cong C_3[1]\).

Hint: apply \(\mathop{\mathrm{Hom}}_{{\mathbb{Z}{\hbox{-}}\mathsf{Mod}}}(C_3, {-})\) to the above resolution and use \(\mathop{\mathrm{Hom}}_{{\mathbb{Z}{\hbox{-}}\mathsf{Mod}}}(C_3, {\mathbb{Q}}/{\mathbb{Z}}) \cong C_3\)

Show that if \(f\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(M, N)\) then there is an induced chain map \(\tilde f\in \mathsf{Ch}{\mathsf{R}{\hbox{-}}\mathsf{Mod}}( { {I}^{\scriptscriptstyle \bullet}} _M, { {I}^{\scriptscriptstyle \bullet}} _N)\) which is unique up to homotopy. Conclude that \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N)\) is independent of injective resolution.

Hint: take \(f=\operatorname{id}_M\).

Show that if \(0\to A\to B\to C\to 0\) is a SES in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) with \({ {I}^{\scriptscriptstyle \bullet}} _A \leftleftarrows A, { {I}^{\scriptscriptstyle \bullet}} _C\leftleftarrows B\), then there is a complex \({ {I}^{\scriptscriptstyle \bullet}} _B\leftleftarrows B\) making \(0\to { {I}^{\scriptscriptstyle \bullet}} _A \to { {I}^{\scriptscriptstyle \bullet}} _B \to { {I}^{\scriptscriptstyle \bullet}} _C \to 0\) a SES in \(\mathsf{Ch}{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).

Show that a SES in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) induces a LES in \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}\). Do this for both homs: start with \(0\to A\to B\to C\to 0\), and produce a LES for \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}} (M, {-})\) and \({ { \operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}({-}, M)\).

Hint: for the first case, apply \(\mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, {-})\) to the SES of chain complexes of injective resolutions, and use that if \(I_1\) is injective then the SES splits. For the second, use that \(\mathop{\mathrm{Hom}}({-}, I)\) is exact iff \(I\) is injective and take an injective resolution of \(M\).

Show that \({ { \operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N)\) is uniquely characterized by

  1. \(\operatorname{Ext} ^0_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N) \cong \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N)\)
  2. \(\tau_{\geq 1} { {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M, N) = 0\) if \(M\) is projective or \(N\) is injective.
  3. The two LESs above exist.

Hints:

11 Tuesday, February 15

Check that \(\operatorname{Ext} _R^i(M, {-})\) is independent of injective resolutions, and \(\operatorname{Ext} _R^i({-}, N)\) is independent of projective resolutions.

Check that \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}}\) is determined by

Show \begin{align*} { {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{k[t]/\left\langle{t^2}\right\rangle } (k, k) = \bigoplus_{i\geq 0} k[i] .\end{align*} Use the projective resolution with entries \(k[t]/\left\langle{t^2}\right\rangle\) with differential \({\partial}= \cdot t\)

Compute \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{k[x_1, \cdots, x_{n}]}(k, k)\) using the Koszul resolution \({ {\bigwedge\nolimits}^{\scriptscriptstyle \bullet}} k[x_1, \cdots, x_{n}]\rightrightarrows k\).

Defining Noetherian rings and modules:

Thank you Emmy Noether!!

Show that TFAE:

For \(1\implies 2\), it STS that \(R^n\) is Noetherian. To reduce, use the diagram