# Wednesday, September 27

## Coherent sheaves on toric varieties

:::{.remark}
Recall $\mcf\in \Coh(X) \iff \mcf\in \oxmod$ and is locally of the form $\mcf_M$ for a module $M$ where $\mcf_M(V(f)^c) = M_f$.
We can write $X_ \Sigma = (\CC^r\sm Z)/G$ where $r = \size \Sigma(1)$ and $G = \Hom(\Cl(X_ \Sigma), \cstar)$ and $\CC^r \cong \spec S \da \spec \CC[x_1,\cdots, x_r]$.
The degree of a divisor in $\Cl(X _{\Sigma})$ induces a grading $S = \bigoplus _{\alpha\in \Cl(X_ \Sigma)} S_ \alpha$ where $S_ \alpha$ are all degree \( \alpha \) polynomials.
Trying to define sheaves on $X_ \Sigma$: attempt to define $G\dash$equivariant sheaves on $\CC^r\sm Z$.
Here a $G\dash$equivariant sheaf is any $\mcf$ with an action $G\actson \mcf(U)$ for all $G\dash$invariant open sets of $X_ \Sigma$ which is compatible with the action $G\actson \CC^r$:

\[\begin{tikzcd}
	{\OO_X(U)} && {\mcf(U)} && {\mcf(U)} & {\ni g.f} \\
	\\
	{\OO_X(U)} && {\mcf(U)} && {\mcf(U)} & {\ni f}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=3-5, to=1-5]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-3, to=1-3]
	\arrow[from=3-1, to=1-1]
	\arrow[from=3-6, to=1-6]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsOCxbMCwwLCJcXE9PX1goVSkiXSxbMiwwLCJcXG1jZihVKSJdLFs0LDAsIlxcbWNmKFUpIl0sWzAsMiwiXFxPT19YKFUpIl0sWzIsMiwiXFxtY2YoVSkiXSxbNCwyLCJcXG1jZihVKSJdLFs1LDIsIlxcbmkgZiJdLFs1LDAsIlxcbmkgZy5mIl0sWzAsMV0sWzEsMl0sWzUsMl0sWzMsNF0sWzQsNV0sWzQsMV0sWzMsMF0sWzYsN11d)


Fact: $G\dash$equivariant coherent sheaves on $\CC^r = \spec S$ correspond to graded finitely-generated $S\dash$modules $M$.
:::

:::{.remark}
The $G\dash$equivariant structure sheaf $\OO_{\CC^r}$ on $\CC^r$ induces the coherent sheaf $\OO_{X_ \Sigma}$ on $X_ \Sigma$.
:::