# Difference between revisions of "Math 764 -- Algebraic Geometry II -- Homeworks"

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## Homeworks (Spring 2017)

Here are homework problems for Math 764 from Spring 2017 (by Dima Arinkin). I tried to convert the homeworks into the wiki format with pandoc. This does not always work as expected; in case of doubt, check the pdf files.

### Homework 1

Due Friday, February 3rd

In all these problems, we fix a topological space $X$; all sheaves and presheaves are sheaves on $X$.

1. Example: Let $X$ be the unit circle, and let ${\mathcal{F}}$ be the sheaf of $C^\infty$-functions on $X$. Find the (sheaf) image and the kernel of the morphism $\frac{d}{dt}:{\mathcal{F}}\to{\mathcal{F}}.$ Here $t\in{\mathbb{R}}/2\pi{\mathbb{Z}}$ is the polar coordinate on the circle.
2. Sheaf operations: Let ${\mathcal{F}}$ and ${\mathcal{G}}$ be sheaves of sets. Recall that a morphism $\phi:{\mathcal{F}}\to {\mathcal{G}}$ is a (categorical) monomorphism if and only if for any sheaf ${\mathcal{F}}'$ and any two morphisms $\psi_1,\psi_2:{\mathcal{F}}'\to {\mathcal{F}}$, the equality $\phi\circ\psi_1=\phi\circ\psi_2$ implies $\psi_1=\psi_2$. Show that $\phi$ is a monomorphism if and only if it induces injective maps on all stalks.
3. Let ${\mathcal{F}}$ and ${\mathcal{G}}$ be sheaves of sets. Recall that a morphism $\phi:{\mathcal{F}}\to{\mathcal{G}}$ is a (categorical) epimorphism if and only if for any sheaf ${\mathcal{G}}'$ and any two morphisms $\psi_1,\psi_2:{\mathcal{G}}\to{\mathcal{G}}'$, the equality $\psi_1\circ\phi=\psi_2\circ\phi$ implies $\psi_1=\psi_2$. Show that $\phi$ is a epimorphism if and only if it induces surjective maps on all stalks.
4. Show that any morphism of sheaves can be written as a composition of an epimorphism and a monomorphism. (You should know what order of composition I mean here.)
5. Let ${\mathcal{F}}$ be a sheaf, and let ${\mathcal{G}}\subset{\mathcal{F}}$ be a sub-presheaf of ${\mathcal{F}}$ (thus, for every open set $U\subset X$, ${\mathcal{G}}(U)$ is a subset of ${\mathcal{F}}(U)$ and the restriction maps for ${\mathcal{F}}$ and ${\mathcal{G}}$ agree). Show that the sheafification $\tilde{\mathcal{G}}$ of ${\mathcal{G}}$ is naturally identified with a subsheaf of ${\mathcal{F}}$.
6. Let ${\mathcal{F}}_i$ be a family of sheaves of abelian groups on $X$ indexed by a set $I$ (not necessarily finite). Show that the direct sum and direct product of this family exists in the category of sheaves of abelian groups. (E.g., a direct sum would be a sheaf of abelian groups ${\mathcal{F}}$ together with a universal family of homomorphisms ${\mathcal{F}}_i\to {\mathcal{F}}$.) Do these operations agree with (a) taking stalks at a point $x\in X$ (b) taking sections over an open subset $U\subset X$?
7. Locally constant sheaves:

Definition. A sheaf ${\mathcal{F}}$ is constant over an open set $U\subset X$ if there is a subset $S\subset F(U)$ such that the map ${\mathcal{F}}(U)\to{\mathcal{F}}_x:s\mapsto s_x$ (the germ of $s$ at $x$) gives a bijection between $S$ and ${\mathcal{F}}_x$ for all $x\in U$.

${\mathcal{F}}$ is locally constant (on $X$) if every point of $X$ has a neighborhood on which ${\mathcal{F}}$ is constant.

Recall that a covering space $\pi:Y\to X$ is a continuous map of topological spaces such that every $x\in X$ has a neighborhood $U\ni x$ whose preimage $\pi^{-1}(U)\subset U$ is homeomorphic to $U\times Z$ for some discrete topological space $Z$. ($Z$ may depend on $x$; also, the homeomorphism is required to respect the projection to $U$.)

Show that if $\pi:Y\to X$ is a covering space, its sheaf of sections ${\mathcal{F}}$ is locally constant. Moreover, prove that this correspondence is an equivalence between the category of covering spaces and the category of locally constant sheaves. (If $X$ is pathwise connected, both categories are equivalent to the category of sets with an action of the fundamental group of $X$.)

8. Sheafification: (This problem may be hard, but it is still a good idea to try it) Prove or disprove the following statement (contained in the lecture notes). Let ${\mathcal{F}}$ be a presheaf on $X$, and let $\tilde{\mathcal{F}}$ be its sheafification. Then every section $s\in\tilde{\mathcal{F}}(U)$ can be represented as (the equivalence class of) the following gluing data: an open cover $U=\bigcup U_i$ and a family of sections $s_i\in{\mathcal{F}}(U_i)$ such that $s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j}$.

### Homework 2

Due Friday, February 10th

Extension of a sheaf by zero. Let $X$ be a topological space, let $U\subset X$ be an open subset, and let ${\mathcal{F}}$ be a sheaf of abelian groups on $U$.

The extension by zero $j_{!}{\mathcal{F}}$ of ${\mathcal{F}}$ (here $j$ is the embedding $U\hookrightarrow X$) is the sheaf on $X$ that can be defined as the sheafification of the presheaf ${\mathcal{G}}$ such that ${\mathcal{G}}(V)=\begin{cases}{\mathcal{F}}(V),&V\subset U\\0,&V\not\subset U.\end{cases}$

1. Is the sheafication necessary in this definition? (Or maybe ${\mathcal{G}}$ is a sheaf automatically?)
2. Describe the stalks of $j_!{\mathcal{F}}$ over all points of $X$ and the espace étalé of $j_!{\mathcal{F}}$.
3. Verify that $j_!$ is the left adjoint of the restriction functor from $X$ to $U$: that is, for any sheaf ${\mathcal{G}}$ on $X$, there exists a natural isomorphism ${\mathop{\mathrm{Hom}}}({\mathcal{F}},{\mathcal{G}}|_U)\simeq{\mathop{\mathrm{Hom}}}(j_!{\mathcal{F}},{\mathcal{G}}).$

(The restriction ${\mathcal{G}}|_U$ of a sheaf ${\mathcal{G}}$ from $X$ to an open set $U$ is defined by ${\mathcal{G}}|_U(V)={\mathcal{G}}(V)$ for $V\subset U$.)

Side question (not part of the homework): What changes if we consider the version of extension by zero for sheaves of sets (‘the extension by empty set’)?

Examples of affine schemes.

4. Let $R_\alpha$ be a possibly infinite collection of rings. Put $R=\prod_\alpha R_\alpha$. Describe the topological space ${\mathop{\mathrm{Spec}}}(R)$ in terms of ${\mathop{\mathrm{Spec}}}(R_\alpha)$’s.
5. Recall that the image of a regular map of varieties is constructible (Chevalley’s Theorem); that is, it is a union of locally closed sets. Give an example of a map of rings $R\to S$ such that the image of a map ${\mathop{\mathrm{Spec}}}(S)\to{\mathop{\mathrm{Spec}}}(R)$ is

(a) An infinite intersection of open sets, but not constructible.

(b) An infinite union of closed sets, but not constructible.

Contraction of a subvariety.

Let $X$ be a variety (over an algebraically closed field $k$) and let $Y\subset X$ be a closed subvariety. Our goal is to construct a ${k}$-ringed space $Z=(Z,{\mathcal{O}}_Z)=X/Y$ that is in some sense the result of ‘gluing’ together the points of $Y$. While $Z$ can be described by a universal property, we prefer an explicit construction:

• The topological space $Z$ is the ‘quotient-space’ $X/Y$: as a set, $Z=(X-Y)\sqcup \{z\}$; a subset $U\subset Z$ is open if and only if $\pi^{-1}(U)\subset X$ is open. Here the natural projection $\pi:X\to Z$ is identity on $X-Y$ and sends all of $Y$ to the ‘center’ $z\in Z$.
• The structure sheaf ${\mathcal{O}}_Z$ is defined as follows: for any open subset $U\subset Z$, ${\mathcal{O}}_Z(U)$ is the algebra of functions $g:U\to{k}$ such that the composition $g\circ\pi$ is a regular function $\pi^{-1}(U)\to{k}$ that is constant along $Y$. (The last condition is imposed only if $z\in U$, in which case $Y\subset\pi^{-1}(U)$.)

In each of the following examples, determine whether the quotient $X/Y$ is an algebraic variety; if it is, describe it explicitly.

6. $X={\mathbb{P}}^2$, $Y={\mathbb{P}}^1$ (embedded as a line in $X$).
7. $X=\{(s_0,s_1;t_0:t_1)\in{\mathbb{A}}^2\times{\mathbb{P}}^1:s_0t_1=s_1t_0\}$, $Y=\{(s_0,s_1;t_0:t_1)\in X:s_0=s_1=0\}$.
8. $X={\mathbb{A}}^2$, $Y$ is a two-point set (if you want a more challenging version, let $Y\subset{\mathbb{A}}^2$ be any finite set).