# Problem Set 4 (Tuesday, October 06)

Let $$X\subset {\mathbb{A}}^n$$be an affine variety and $$a\in X$$. Show that \begin{align*} {\mathcal{O}}_{X, a} = {\mathcal{O}}_{{\mathbb{A}}^n, a} / I(X) {\mathcal{O}}_{A^n,a} ,\end{align*} where $$I(X) {\mathcal{O}}_{{\mathbb{A}}^n, a}$$ denotes the ideal in $${\mathcal{O}}_{{\mathbb{A}}^n, a}$$ generated by all quotients $$f/1$$ for $$f\in I(X)$$.

Let $$a\in {\mathbb{R}}$$, and consider sheaves $$\mathcal{F}$$ on $${\mathbb{R}}$$ with the standard topology:

• $$\mathcal{F} \coloneqq$$ the sheaf of continuous functions
• $$\mathcal{F} \coloneqq$$ the sheaf of locally polynomial functions.

For which is the stalk $$\mathcal{F}_a$$ a local ring?

Recall that a local ring has precisely one maximal ideal.

Let $$\phi, \psi \in \mathcal{F}(U)$$ be two sections of some sheaf $$\mathcal{F}$$ on an open $$U\subseteq X$$ and show that

• If $$\phi, \psi$$ agree on all stalks, so $$\mkern 1.5mu\overline{\mkern-1.5mu(U, \phi)\mkern-1.5mu}\mkern 1.5mu = \mkern 1.5mu\overline{\mkern-1.5mu(U, \psi)\mkern-1.5mu}\mkern 1.5mu \in \mathcal{F}_a$$ for all $$a\in U$$, then $$\phi$$ and $$\psi$$ are equal.

• If $$\mathcal{F} \coloneqq{\mathcal{O}}_X$$ is the sheaf of regular functions on some irreducible affine variety $$X$$, then if $$\psi = \phi$$ on one stalk $$\mathcal{F}_a$$, then $$\phi = \psi$$ everywhere.

• For a general sheaf $$\mathcal{F}$$ on $$X$$, (b) is false.

Let $$Y\subset X$$ be a nonempty and irreducible subspace of $$X$$ a topological space with a sheaf $$\mathcal{F}$$ on $$X$$. Then the stalk of $$\mathcal{F}$$ at $$Y$$ is defined by the pairs $$(U, \phi)$$ such that $$U\subset X$$ is open, $$U\cap Y$$ is nonempty, and $$\phi \in \mathcal{F}(U)$$, where we identify $$(U, \phi) \sim (U',\phi')$$ iff there is a small enough open set such that the restrictions agree.

Let $$Y\subset X$$ be a nonempty and irreducible subvariety of an affine variety $$X$$, and show that the stalk $${\mathcal{O}}_{X, Y}$$ of $${\mathcal{O}}_X$$ at $$Y$$ is a $$k{\hbox{-}}$$algebra which is isomorphic to the localization $$A(X)_{I(Y)}$$.

Let $$\mathcal{F}$$ be a sheaf on $$X$$ a topological space and $$a\in X$$. Show that the stalk $$\mathcal{F}_a$$ is a local object, i.e. if $$U\subset X$$ is an open neighborhood of $$a$$, then $$\mathcal{F}_a$$ is isomorphic to the stalk of $${ \left.{{ \mathcal{F} }} \right|_{{U}} }$$ at $$a$$ on $$U$$ viewed as a topological space.