# Localization in Equivariant \(\K\dash\)theory (Friday, September 24)

## Localization Theorems

> Reference: Thomason.

:::{.definition title="Localization theorems"}
Suppose $A\in \Ab\Alg\Grp$ is reductive, and $X \subseteq G/P$ is contained in a flag variety (so $X$ is quasiprojective).
Fix $a\in A$, and consider the fixed point set $X^a$ and the inclusion $\iota: X^a \mapsvia{\subseteq } X$.
We'll say the **localization theorem holds for $X$** if the following induced hom is an isomorphism:
\[
i_*: \K^A(X^a)\localize{\mfm_a} \to \K^A(X)\localize{\mfm_a}
.\]
:::

:::{.remark}
Thomason shows that this is true in this situation.
Recall that we identified $R(A) = \K^A(\pt)$.
Taking the trace of a representation yields a map $R(A) \injects \CC[A]$, the ring of regular functions.
For varieties, we can obtain $\OO_{X, x}$ by localizing rings at their maximal ideals, thinking of these as functions on $X$.
Let 
\[
R_a &\da R(A)\localize{ \qty{ R(A)\sm\mfm_a} } \\
M_a &\da R(A) \tensor_{R(A)} M
.\]
:::

## Proper Pushforward

:::{.remark}
We'll need proper maps for the ever-popular *decomposition theorem*.
However, almost every scheme we use in this class will be reduced, although one does rarely have to worry about this.
:::

:::{.definition title="Proper Maps (and prerequisite notions)"}
**Pullbacks** are universal with respect to the following squares, and have a concrete description for us:

\begin{tikzcd}
	{\ts{(x, z) \in X\times Z \st f(x) = z(g)}} \\
	{X\fiberprod{Y}Z} && Z \\
	\\
	X && Y
	\arrow["g", from=2-3, to=4-3]
	\arrow["f"', from=4-1, to=4-3]
	\arrow["{g'}"', from=2-1, to=4-1]
	\arrow["{f'}", from=2-1, to=2-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-1, to=4-3]
	\arrow[Rightarrow, no head, from=1-1, to=2-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwxLCJYXFxmaWJlcnByb2R7WX1aIl0sWzIsMSwiWiJdLFswLDMsIlgiXSxbMiwzLCJZIl0sWzAsMCwiXFx0c3soeCwgeikgXFxpbiBYXFx0aW1lcyBaIFxcc3QgZih4KSA9IHooZyl9Il0sWzEsMywiZyJdLFsyLDMsImYiLDJdLFswLDIsImcnIiwyXSxbMCwxLCJmJyJdLFswLDMsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFs0LDAsIiIsMCx7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d)

The **diagonal** is the unique morphism $\Delta: X\to X\fiberprod{Y} X$ whose compositions with projections are the identity:

\begin{tikzcd}
	X \\
	\\
	&& {X\fiberprod{Y} X} && X \\
	\\
	&& X && Y
	\arrow[from=3-5, to=5-5]
	\arrow[from=5-3, to=5-5]
	\arrow[from=3-3, to=5-3]
	\arrow[from=3-3, to=3-5]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-3, to=5-5]
	\arrow["{\id_X}", from=1-1, to=3-5]
	\arrow["{\id_X}"', from=1-1, to=5-3]
	\arrow["\Delta"{description}, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwyLCJYXFxmaWJlcnByb2R7WX0gWCJdLFs0LDIsIlgiXSxbMiw0LCJYIl0sWzQsNCwiWSJdLFswLDAsIlgiXSxbMSwzXSxbMiwzXSxbMCwyXSxbMCwxXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbNCwxLCJcXGlkX1giXSxbNCwyLCJcXGlkX1giLDJdLFs0LDAsIlxcRGVsdGEiLDFdXQ==)

A morphism is **separated** if the diagonal is a closed embedding.

A morphism $f:X\to Y$ is **universally closed** if for any $g:Z\to Y$, the base change $f': X\fiberprod{Y} Z\to Z$ is a closed morphism.
This replaces the notion of "$K$ compact $\implies f\inv(K)$ compact" for analytic varieties.

A morphism $f$ is **proper** if $f$ is separated, finite type, and universally closed.
:::

:::{.example title="?"}
\envlist

- Closed embeddings are proper, and open maps are usually not.
- If $f$ is proper, its base change $f'$ is always proper.
- Compositions of proper morphisms are again proper.
- Any morphism between projective varieties is proper.

:::

:::{.theorem title="18.8.1, Rising Sea"}
Let $f:X\to Y$ be proper and $\mcf \in \Coh(X)$.
Note that $\globsec{X, \wait}$ is exact and $\Coh(X)$ is abelian, so we can take its derived functor.
Let $f_*: \Sh\slice X\to \Sh\slice Y$, then e.g. 
\[
\RR^i f_* \mcf(U) = H^i(f\inv(U); \mcf)
.\]

This satisfies several properties:

1. $\RR^if_*: \Coh(X) \to \Coh(Y)$ is a covariant functor.
  Without properness, one can just replace $\Coh$ with $\QCoh$.

2. $\RR^0 f_* = f_*$ 

3. A SES $0\to\mcf_1\to\mcf_2\to\mcf_3\to 0$ induces a LES.

:::

:::{.theorem title="Rising Sea, 18.8.5"}
If $f:X\to Y$ is a proper projective morphism, then $\RR^{i>d} f_* \mcf = 0$ for $d$ defined as the maximum dimension of the fiber, $d\da \max_{y\in Y} \dim f\inv(y)$.
:::

:::{.definition title="Proper Pushforward"}
Let $X, Y$ be arbitrary quasiprojective varieties and $f:X\to Y$ be proper and $G\dash$equivariant.
Then there is a natural direct image morphism $f_*: \K^G(X) \to \K_G(Y)$.

We define it as follows: note that a map such as $f_*([\mcf]) \da [f_* \mcf]$ won't necessarily be well-defined, since SESs are additive in the Grothendieck group.
For $\mcf \in \Coh^G(X)$, then it turns out that $\RR f_* \mcf \in \Coh^G(Y)$ and the higher direct images vanish in large enough degree.
We then define
\[
f_*: \K^G(X) &\to \K_G(Y) \\
[\mcf] &\mapsto \sum (-1)^i [\RR^i f_* \mcf]
.\]

:::

:::{.example title="?"}
Let $G$ be connected reductive with $A \da T$ a maximal torus, which is abelian reductive.
Then take $a\in A$ a *regular* element, so $X^a = X^T$.
In our case, $X^T = W_Y'$, and $X = G/P_Y$.
Then \(\K\dash\)theory is concentrated on the fixed locus:
\[
i_* \K^T(X^T)\localize{\mfm_a} \iso \K^T(X)\localize{\mfm_a}
.\]
:::