# Problem Set 4 (Tuesday, October 06)

::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 3.20"}
Let $X\subset {\mathbb{A}}^n$be an affine variety and $a\in X$. Show that `<span class="math display"> \begin{align*}   {\mathcal{O}}_{X, a} = {\mathcal{O}}_{{\mathbb{A}}^n, a} / I(X) {\mathcal{O}}_{A^n,a} ,\end{align*} <span>`{=html} 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 $f\in I(X)$.
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 3.21"}
Let $a\in {\mathbb{R}}$, and consider sheaves $\mathcal{F}$ on ${\mathbb{R}}$ with the standard topology:

1.  $\mathcal{F} \coloneqq$ the sheaf of continuous functions
2.  $\mathcal{F} \coloneqq$ 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.
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 3.22"}
Let $\phi, \psi \in \mathcal{F}(U)$ be two sections of some sheaf $\mathcal{F}$ on an open $U\subseteq X$ and show that

a.  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.

b.  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.

c.  For a general sheaf $\mathcal{F}$ on $X$, (b) is false.
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::: {.definition .proofenv .proofenv .proofenv .proofenv .proofenv title="Stalk at a subspace"}
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.
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 3.23: Geometry of a Certain Localization"}
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 $A(X)_{I(Y)}$.
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::: {.problem .proofenv .proofenv .proofenv .proofenv .proofenv title="Gathmann 3.24"}
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.
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