# Tuesday, September 06

## Group varieties 

:::{.remark}
Last time: group varieties.
Most of today will work over $\CC, k\neq \kbar$, or $\ZZ$.
There is a correspondence:

| Affine varieties        	| Rings and k-algebras       	|
|-------------------------	|----------------------------	|
| Group varieties/schemes 	| Hopf coalgebras            	|
| $\mu: G\times G\to G$   	| $\mu^*: R\to R\tensor_k R$ 	|
| $e: \pt\to G$           	| $e^*: R\to k$              	|
| $i: G\to G$             	| $i^*: R\to R$              	|

Recall:

- For $M, N\in\rmod$, there is a tensor product $M\tensor_R N$,
- A morphism $f\in \CRing(R, S)$ yields a functor $f^*: \mods{S}\to \bimods{R}{S}$ given by the base change/scalar extension $(\wait)\tensor_R S$
- If $S_1, S_2 \in \ralg$ then $S_1\tensor_R S_2 \in \rmod$ is in fact a ring, using the product $(u_1\tensor v_1)\cdot(u_2\tensor v_2) \da u_1u_2\tensor v_1 v_2$.
- For any $N\in \rmod$, the functor $(\wait) \tensor_R N$ is **right** exact, so
\[
A\injects B\surjects &C \\ 
\leadsto \\ 
A\tensor_R N\to B\tensor_R N \surjects &C\tensor_R N
.\]

:::

:::{.corollary title="?"}
If $M$ is finitely generated, there is a generator/relation exact sequence $R\sumpower{m} \to R\sumpower{n}\surjects C$.
Tensoring with any $N\in \rmod$ yields
\[
N \sumpower{m} \to N\sumpower{n} \surjects C\tensor_R N
.\]
In particular, this works for base change $\rmod\to\mods{S}$ -- the new module is generated as a module by the same generators but new scalars.
:::

:::{.example title="?"}
Consider $\CC\tensor_\RR \CC$, which has a ring structure.
Write $\CC = \RR[x]/\gens{x^2+1}$, then the base change is
\[
\CC[x]/\gens{x^2+1}
= \CC[x]/\gens{x-i} \oplus \CC[x]/\gens{x+i} \cong \CC \oplus \CC
,\]
which is a ring with zero divisors and idempotents since $(1, 0)^2 = (1^2, 0^2) = (1, 0)$.
:::

:::{.slogan}
For tensor products: same generators, same relations, extend scalars.
:::

:::{.remark}
Recall:

- $\GG_m = \spec k[x^{\pm 1}] \approx \CCstar$.
- $\GG_m^n = \spec k[x_1^{\pm 1}, \cdots, x_n^{\pm 1}] \approx (\CCstar)^n$.
- $\GG_a = \spec k[x] \approx (\CC, +)$.
- $\mu_n \subseteq \GG_m = \spec k[x]/\gens{x^n-1} \approx \ts{\xi \in \CC \st \xi^n=1}$.
- In characteristic $p$, $\mu_p = \spec k[x]/\gens{x^p-1} = \spec k[x]/\gens{x-1}^p$.

:::

:::{.example title="?"}
Of using the tensor product slogan: identifying the map
\[
{ k[z] \over \gens{z^n-1}} \to
{k[x] \over \gens{x^n-1}} \tensor_k
{k[y]\over \gens{y^n-1} } \cong
{k[x,y]\over \gens{x^n-1, y^n-1} }
,\]
realizing this as $z\mapsto x\tensor y\mapsto xy$, checking that $z^n=1 \implies (xy)^n = 1$.
:::

:::{.remark}
If $G$ is an arbitrary finite group it can be made into an affine algebraic group variety over $k$.
Give the underlying set of $G = \disjoint_{g\in G}\ts{\pt}$ the discrete topology to get an algebraic variety.
To get the algebraic group structure, note that any map of finite sets is algebraic.
Define a ring $R \da \bigoplus _{g\in G} k$, and a comultiplication as follows: note that 
\[
R\tensor_k R \cong \bigoplus _{(a, b) \in G\times G } k e_{a,b}
\]
where the $e_{a, b}$ just tracks which summand we're in.
So define
\[
R &\to R\tensor_k R \\
e_g &\mapsto \sum_{ab=g} e_{a, b}
.\]

:::

:::{.remark}
Note that we could have let $\pt = \spec k$.
E.g. $C_p \neq \mu_p$ but are Cartier dual.
However, the ring $k[x]/\gens{x-1}^p$ is much easier to understand than the $R\tensor_k R$ from above, even for very small groups like $C_2$.
:::

:::{.example title="?"}
Recall $\GL_n \subseteq \AA^{n^2}$ is the open subspace which is the complement of $V(\det)$, so a principal open subset.
The ring is $k[x_{ij}, 1/\det]$ for $1\leq i,j\leq k$, which is obtained by localizing at the determinant.
Thus we can embed it as a closed subset in $\AA^{n^2+1}$ using $V(y\det(x_{ij}) = 1)$, i.e. introducing a new free variable for $1/\det$ and ensuring it's nonzero.
:::

:::{.definition title="Affine algebraic groups"}
An **affine algebraic group** is a closed subgroup $G$ of $\GL_n$, and the coordinate ring $R_G$ is a quotient of $R_{\GL_n}$.
:::

:::{.example title="?"}
For $\SL_n$, the ring is $k[x_{ij}]/\gens{\det - 1}$, and define $\PGL_n$ as $\SL_n/\mu_n$ or $\GL_n/\GG_m$.
Although it's not obvious, these are affine -- for $\PGL_n$, the ring is the $\mu_n$ invariants of the coordinate ring of $\SL_n$, so one gets the ring of polynomials in $R_{\SL_n}$ whose powers are multiples of $n$.
:::

## Algebraic group actions

:::{.definition title="Algebraic group action"}
An **action of an algebraic group $G$ on a variety (or scheme) $X$** is a map $G\fiberprod{k}X \mapsvia{a} X$ satisfying the usual axioms encoded in commuting diagrams:

- $(gh).x = g.(h.x)$:

\begin{tikzcd}
	{G\times G\times X} && {G\times X} \\
	\\
	{G\times X} && X
	\arrow["{(\id_G, a)}", from=1-1, to=1-3]
	\arrow["a", from=1-3, to=3-3]
	\arrow["{(\mu, \id_X)}"', from=1-1, to=3-1]
	\arrow["a"', from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJHXFx0aW1lcyBHXFx0aW1lcyBYIl0sWzAsMiwiR1xcdGltZXMgWCJdLFsyLDAsIkdcXHRpbWVzIFgiXSxbMiwyLCJYIl0sWzAsMiwiKFxcaWRfRywgYSkiXSxbMiwzLCJhIl0sWzAsMSwiKFxcbXUsIFxcaWRfWCkiLDJdLFsxLDMsImEiLDJdXQ==)

- $1.x = x$ for all $x$:

\begin{tikzcd}
	{\ts{1}\times X} && {G\times X} \\
	\\
	X && X
	\arrow[from=1-1, to=1-3]
	\arrow["a", from=1-3, to=3-3]
	\arrow[Rightarrow, no head, from=1-1, to=3-1]
	\arrow[Rightarrow, no head, from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXHRzezF9XFx0aW1lcyBYIl0sWzAsMiwiWCJdLFsyLDIsIlgiXSxbMiwwLCJHXFx0aW1lcyBYIl0sWzAsM10sWzMsMiwiYSJdLFswLDEsIiIsMix7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzEsMiwiIiwyLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=)

Note that one can now reverse these diagrams to get the coaction on rings $a^*: A\to R\tensor_k A$.

:::

:::{.example title="?"}
Let $\mu_n \actson \AA^2$ by $\xi.(x,y) \da (\xi x, \xi^k y)$, then the coaction is
\[
k[x,y] &\to {k[x,y,\xi] \over \gens{\xi^n-1} } = {k[\xi] \over \xi^n-1}[x, y] \\
x &\mapsto \xi x \\
y &\mapsto \xi^k y
.\]
:::

:::{.exercise title="?"}
Check that this satisfies the axioms for a coaction.
:::

:::{.definition title="Linear coactions"}
Let $G\in \Grp\Var\slice k, X\in \Vect\slice k$, then a **linear coaction** is a homomorphism $V \mapsvia{a^*} R\tensor_k V$ satisfying the duals of the axioms above.
:::

:::{.example title="?"}
If $V = kx \oplus ky$ then $V\to R\tensor_k V = Rx \oplus Ry$.
:::

:::{.remark}
There is a coaction on $A = \Sym^* V = k \oplus V \oplus \Sym^2 V \oplus \cdots$, where $V = \gens{x,y}, \Sym^2 V = \gens{x^2, xy, y^2}$.
This is the same as an action $G\actson \AA^N$ for some $N$ acting on an affine space.
:::