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.