# Thursday, September 02

## Decomposition using characters

:::{.remark}
Last time: for $G\actson R \contains R^G$ for $G\in\Fin\Grp$, the Todd-Shepherd(-Chevalley) theorem states that if $G\actson \AA^n$ and $G$ is generated by pseudoreflections then $\kxn^G$ is again a polynomial ring.
Consider now $G\actson R$ for $G$ finite abelian and $\characteristic k = 0$.
This yields a grading $R = \bigoplus _{\chi \in \hat G} R_\chi$ where $\hat G = \Hom(G, \CC\units) = \Hom(G, \QQ/\ZZ)$ and $R_\chi R_{\chi'} \subseteq R_{\chi + \chi'}$
Note that if $G \cong \bigoplus \ZZ/n_i \ZZ$ then $\hat{G} \cong \bigoplus \mu_{n_i}$, which is non-canonically isomorphic to \( \bigoplus \ZZ/n_i\ZZ \).
Recall that reflections have eigenvalues $\ts{1, 1,\cdots, 1, \alpha\neq 1}$.
:::

:::{.example title="?"}
Let $C_2 \actson \AA^2\slice\CC$ by $(x,y)\mapsto (-x, -y)$.
What are the invariants $k[x,y]^{C_2}$?
Check that $p(x,y) = \sum a_{ij} x^i y^j \mapsto \sum (-1)^{i+j} a_{ij} x^i y^j$, which equals $p(x,y)$ when all of the $i,j$ are even.
Write $k[x,y] = \bigoplus _{i+j\equiv_2 0} x^iy^j \oplus \bigoplus _{i+j \equiv_2 1} x^i y^j \da R_0 \oplus R_1$ and note $\hat G \cong \mu_2 \cong C_2$ and $?$.
Also note that for $r \in R_\chi$ we have $g.r = \chi(g) r$
We can write
\[
k[x,y]^{C_2} = R_0 = k[x^2, xy, y^2] = k[u,v,w]/\gens{uw=v^2}
,\]
which is a singular cone $V(uw-v^2) \subseteq \AA^3$:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
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\end{tikzpicture}

Shepherd's theorem does not apply here since the action is given by $\matt {-1} 0 0 {-1}$, which is not a reflection.
:::

:::{.example title="?"}
Take $C^2\actson \AA^2$ by $(x,y)\mapsto (x,-y)$, then $k[x,y]^{C_2} = k[x, y^2]$.
:::

:::{.remark}
Note that in general, $\AA^n/G = \mspec \kxn^G$ has quotient singularities.
Three types of varieties we work with in AG:

- Affine $\mapstofrom$ rings,
- Projective $\mapstofrom$ graded rings,
- General: covered by affines, not necessarily projective.

Upshot: we can think of projective varieties not as covered by affines, but rather as a "spectrum" of a single *graded* ring.
Given a subset $Z = V(f_1,\cdots, f_m) \subseteq \PP^n\slice k$ cut out by homogeneous polynomials of degree $d_i$ in the homogeneous degree 1 coordinates $x_0,\cdots, x_n$, one can take the affine cone $C(Z) \subseteq \AA^{n+1}$.
A linear action of $G\actson \PP^n\slice k$ descends to $G\actson Z$, where linear means that $g.\tv{x_0:\cdots:x_n} = M g\tv{x_0: \cdots: x_n}$.
Not every action is of this form: take $G=\CCstar\actson \PP^1$ by \( \lambda \tv{x_0: x_1} = \tv{x_0: \lambda x_1} \).
This is linear; to make a nonlinear action glued the fixed points $\ts{0}$ and $\ts{\infty}$ to get a rational nodal curve:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/Moduli/sections/figures}{2022-09-01_13-23.pdf_tex} };
\end{tikzpicture}

Note that $\Pic(C) = \ZZ \bigoplus \CCstar$.
:::

:::{.remark}
For a linear action by a finite group $G$, writing $Z = \mproj R$ with $R = \kxn/\sqrt{\gens{f_i}}$ then $Z/G = \mproj R^G$.
Such actions can be lifted from $Z$ to $\ro{\OO_{\PP^n}(1) }{Z} = \OO_Z(1)$.
:::

## Group Varieties

:::{.definition title="Group variety"}
A variety $G\in\Var\slice k$ is a **group variety** if it admits morphisms

- Multiplication $\mu \in \Alg\Var(G\fiberprod{k} G, G)$,
- Units: $e\in \Alg\Var(\spec k, G)$,
- Inverses: $i\in \Alg\Var(G\to G)$.

These are required to satisfy some axioms.

Encoding associativity:

\begin{tikzcd}
	{G\cartpower{3}} && {G\cartpower{2}} \\
	\\
	{G\cartpower{2}} && G
	\arrow["{(\mu_{12}, \id_3)}", from=1-1, to=1-3]
	\arrow["\mu", from=1-3, to=3-3]
	\arrow["{(\id_1, \mu_{23})}"', from=1-1, to=3-1]
	\arrow["\mu"', from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJHXFxjYXJ0cG93ZXJ7M30iXSxbMiwwLCJHXFxjYXJ0cG93ZXJ7Mn0iXSxbMCwyLCJHXFxjYXJ0cG93ZXJ7Mn0iXSxbMiwyLCJHIl0sWzAsMSwiKFxcbXVfezEyfSwgXFxpZF8zKSJdLFsxLDMsIlxcbXUiXSxbMCwyLCIoXFxpZF8xLCBcXG11X3syM30pIiwyXSxbMiwzLCJcXG11IiwyXV0=)

Encoding $1a=a$:

\begin{tikzcd}
	{\spec k\times G} && {G\times G} \\
	\\
	&& G
	\arrow[Rightarrow, no head, from=1-1, to=3-3]
	\arrow["{(e, \id_2)}", from=1-1, to=1-3]
	\arrow["\mu", from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHNwZWMga1xcdGltZXMgRyJdLFsyLDAsIkdcXHRpbWVzIEciXSxbMiwyLCJHIl0sWzAsMiwiIiwwLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCwxLCIoZSwgXFxpZF8yKSJdLFsxLDIsIlxcbXUiXV0=)

Encoding $aa\inv = 1$:

\begin{tikzcd}
	G && {G\cartpower{2}} \\
	\\
	{\spec k} && G
	\arrow["{\text{structure map}}"', from=1-1, to=3-1]
	\arrow["\mu", from=1-3, to=3-3]
	\arrow["e"', from=3-1, to=3-3]
	\arrow["{a\mapsto (a, a\inv)}", from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJHIl0sWzIsMCwiR1xcY2FydHBvd2VyezJ9Il0sWzAsMiwiXFxzcGVjIGsiXSxbMiwyLCJHIl0sWzAsMiwiXFx0ZXh0e3N0cnVjdHVyZSBtYXB9IiwyXSxbMSwzLCJcXG11Il0sWzIsMywiZSIsMl0sWzAsMSwiYVxcbWFwc3RvIChhLCBhXFxpbnYpIl1d)

:::

:::{.remark}
Suppose that $G = \spec R$ is affine, then there are dual notions:

- Comultiplication: $\mu^*: R\to R\tensor_k R$.
- Counits: $e^*: R\to k$.
- Coinverses: $i^*: R\to R$.

:::

:::{.example title="?"}
The additive group $\GG_a = \spec k[x]$, whose underlying variety is $\AA^1$.
In coordinates, the group law is written additively as 

- $\mu(x, y) = x+y$
- $e = 0$
- $i(x) = -x$

Write $z=x+y$, then on the ring side we have

- Comultiplication:
\[
\mu^*: k[z] &\to k[x]\tensor_k k[y] \cong k[x+y] \\
z &\mapsto x\tensor 1 + 1\tensor y &\mapsto x+y
.\]

- Counit: $e^*: k[x] \to k$ where $x\mapsto 0$
- Coinverse: $i^*: k[x] \to k[x]$ where $x\mapsto -x$.

:::

:::{.example title="?"}
The multiplicative group $\GG_m = \spec k[x, x\inv]$ whose underlying variety is $\AA^2\smz$

The group law is:

- $\mu(x, y) = xy$
- $e = 1$
- $i(x) = x\inv$

For rings:

- Comultiplication:
\[
\mu^*: k[z] &\mapsto k[x,x\inv]\tensor_k k[y,y\inv] \cong k[x^{\pm 1}, y^{\pm 1}] \\
z &\mapsto x\tensor y \mapsto xy
.\]

- Counit: $e^*: k[x^{\pm 1} ] \to k$ where $x\mapsto 1$.
- Coinverse: $i^*: k[x^{\pm 1}] \to k[x^{\pm 1}]$ where $x\mapsto x\inv$.

:::

:::{.example title="?"}
Roots of units $\mu_n = \spec k[x]/\gens{x^n-1}$.
Note that there is a closed embedding $\mu_n \injects \GG_m$ since there is a surjection $\spec k[x, x\inv] \surjects k[x]/\gens{x^n-1}$.
Note that in $\characteristic k = p$, this yields a scheme that is not a variety since it is not reduced: one has $\mu_p = \spec k[x]/\gens{x^p-1} = \spec k[x]/\gens{(x-1)^p}$ which contains nilpotents.
This is the first example of a *group scheme* which is not a group variety.

The group operations agree with that on $\GG_m$, e.g. comultiplication is 
\[
\mu^*: k[z]/\gens{z^n-1} \to k[x]/\gens{x^n-1}\tensor_k k[y]/\gens{y^n-1} \cong k[x,y]/\gens{x^n-1, y^n-1}
.\]
One can similarly define $\alpha_p = \ker\Frob \embeds \GG_m = \spec k[x]/\gens{x^p}$.
:::

:::{.remark}
Upcoming: more group varieties and schemes, especially $\GL_n, \SL_n$, and their actions/coactions.
:::