# Problem Set 4 (Tuesday, October 06)


:::{.problem title="Gathmann 3.20"}
Let $X\subset \AA^n$be an affine variety and $a\in X$.
Show that
\[  
\OO_{X, a} = \OO_{\AA^n, a} / I(X) \OO_{A^n,a}
,\]
where $I(X) \OO_{\AA^n, a}$ denotes the ideal in $\OO_{\AA^n, a}$ generated by all quotients $f/1$ for $f\in I(X)$.

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:::{.problem title="Gathmann 3.21"}
Let $a\in \RR$, and consider sheaves $\mathcal{F}$ on $\RR$ with the standard topology:

1. $\mathcal{F} \da$ the sheaf of continuous functions
2. $\mathcal{F} \da$ 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.

:::


:::{.problem 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 $\bar{(U, \phi)} = \bar{(U, \psi)} \in \mathcal{F}_a$ for all $a\in U$, then $\phi$ and $\psi$ are equal.

b. If $\mathcal{F} \da \OO_X$ 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.
:::


:::{.definition 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 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 $\OO_{X, Y}$ of $\OO_X$ at $Y$ is a $k\dash$algebra which is isomorphic to the localization $A(X)_{I(Y)}$.

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:::{.problem 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 $\ro{ \mathcal{F} }{U}$ at $a$ on $U$ viewed as a topological space.
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