# 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 $(K, \tensor, 1)$ where

- $K$ is a triangulated category
- $(K, \tensor)$ is a symmetric monoidal category
- $1$ is a unit, so $X\tensor 1 \iso X \iso 1\tensor X$ for all $X$ in $K$.

:::

:::{.remark}
We'll have notions of ideals, thick ideals, and prime ideals in $K$.
Define $\spc K$ to be the set of prime ideals with the following topology: for a collection $C \subseteq \spec K$, define $Z(C) = \ts{p\in \spc K \st C \intersect p = \emptyset}$.
Note that there is a universal categorical construction of $\spc K$ 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 $\spc K^c$?
- 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 $X\cong_\Top \Spc(K^c)$, e.g. a Zariski space: Noetherian, and every irreducible contains a generic point.
We'll also need an assignment $V: K^c\leadsto X_{\cl}$ (the closed sets of $X$) satisfying certain properties, which is called a *support datum*.
For $I$ a thick tensor ideal, define 
\[
\Gamma(I) \da \Union_{M\in I} V(M) \in X_{\mathrm{sp}}
,\]
a union of closed sets which is called *specialization closed*.
Conversely, for $W$ a specialized closed set, define a thick tensor ideal
\[
\Theta(W) \da \ts{M\in K^c \st V(M) \subseteq W}
.\]


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

:::{.remark}
Define 
\[
f: X\to \Spc K^c \\
x &\mapsto P_x \da \ts{M \in K^c \st x\not\int V(M)}
.\]
This is a prime ideal: if $M\tensor N\in P_x$, then $x\not \in V(M\tensor N) = V(M) \intersect V(N)$, so $M\in P_x$ or $N\in P_x$.
:::

## Zariski spaces

:::{.definition title="Zariski spaces"}
A space $X\in \Top$ 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 $X = \Union_{1\leq i \leq t} W_i$.
:::

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

- For $R$ a unital Noetherian commutative ring, $X = \spec R$ is Zariski.
- For $R$ a graded unital Noetherian ring, taking homogeneous prime ideals $\Proj R$.
- For $G\in \Aff\Alg\Grp$ with $G\actson R$ a graded ring by automorphisms (permuting the graded pieces), the stack $X \da \Proj_G(R)$ (which is not Proj of the fixed points) is the set of $G\dash$invariant homogeneous prime ideals.
  There's a map $\rho: \Proj R\to \Proj_G R$ where $P\mapsto \intersect _{g\in G} gP$ which gives $\Proj_G R$ the quotient topology: $W\in \Proj_G R$ is closed iff $\rho\in R$ is close din $\Proj R$. 
  This topologizes orbit closures.
:::

:::{.remark}
Notation:

- $\mcx = 2^X$ for the powerset of $X$, 
- $\mcx_{\cl}$ the closed sets,
- $\mcx_{\irr}$ the irreducible closed sets,
- $\mcx_{\mathrm{sp} }$ the specialization-closed sets.

:::

## Support data

:::{.remark}
Recall 

- $M = \mods{kG}$
- $R = H^{\even}(G; k)$
- $V_G(M) = \ts{p\in \Proj R \st \Ext_{kG}(M, M)_p\neq 0 }$.

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 $V: K \to \mcx$ such that 

1. $V(0) = \emptyset$ and $V(1) = X$.
2. $V\qty{\bigoplus _{i\in I} M_i = \Union_{i\in I} V(M_i) }$
3. $V(\Sigma M) = V(M)$ (similar to $\Omega$)
4. For any distinguished triangle $M\to N\to Q\to \Sigma M, V(N) \subseteq V(M) \union V(Q)$.
5. $V(M\tensor N) = V(M) \intersect V(N)$.

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

6. Faithfulness: $V(M) = \emptyset \iff M \cong 0$.
7. Realization: for any $W\in \mcx_{\cl}$ 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 $V:K\to \mcx$ 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 $\Loc(C)$.
:::

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

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

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

## Extension of support data

:::{.remark}
Let $X$ be a Zariski space and let $K\contains K^c$ be a compactly generated TTC.
Let $V: K^c\to \mcx_{\cl}$ be a support data on compact objects, we then seek an *extension*: a support datum $\mcv$ on $K$ forming a commutative diagram:

\begin{tikzcd}
	K && \mcx \\
	\\
	{K^c} && {\mcx_{\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["\mcv", 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 $V: K^c\to \mcx_{\cl}$ be a support datum.
Then $\mcv: K\to \mcx$ **extends** $V$ iff 

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

:::

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