Math 764 -- Algebraic Geometry II -- Homeworks

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 finite 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. What changes if the collection is infinite?
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. (This part may be very hard.)

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).

Homework 3

Due Friday, February 17th

1. (Gluing morphisms of sheaves) Let $F$ and $G$ be two sheaves on the same space $X$. For any open set $U\subset X$, consider the restriction sheaves $F|_U$ and $G|_U$, and let $Hom(F|_U,G|_U)$ be the set of sheaf morphisms between them.

Prove that the presheaf on $X$ given by the correspondence $U\mapsto Hom(F|_U,G|_U)$ is in fact a sheaf.

2. (Gluing morphisms of ringed spaces) Let $X$ and $Y$ be ringed spaces. Denote by $\underline{Mor}(X,Y)$ the following pre-sheaf on $X$: its sections over an open subset $U\subset X$ are morphisms of ringed spaces $U\to Y$ where $U$ is considered as a ringed space. (And the notion of restriction is the natural one.) Show that $\underline{Mor}(X,Y)$ is in fact a sheaf.
3. (Affinization of a scheme) Let $X$ be an arbitrary scheme. Prove that there exists an affine scheme $X_{aff}$ and a morphism $X\to X_{aff}$ that is universal in the following sense: any map form $X$ to an affine scheme factors through it.
4. Let us consider direct and inverse limits of affine schemes. For simplicity, we will work with limits indexed by positive integers.

(a) Let $R_i$ be a collection of rings ($i\gt 0$) together with homomorphisms $R_i\to R_{i+1}$. Consider the direct limit $R:=\lim\limits_{\longrightarrow} R_i$. Show that in the category of schemes, ${\mathop{\mathrm{Spec}}}(R)=\lim\limits_{\longleftarrow}{\mathop{\mathrm{Spec}}}R_i.$

(b) Let $R_i$ be a collection of rings ($i\gt 0$) together with homomorphisms $R_{i+1}\to R_i$. Consider the inverse limit $R:=\lim\limits_{\longleftarrow} R_i$. Show that generally speaking, in the category of schemes, ${\mathop{\mathrm{Spec}}}(R)\neq\lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i.$

5. Here is an example of the situation from 4(b). Let $k$ be a field, and let $R_i=k[t]/(t^i)$, so that $\lim\limits_{\longleftarrow} R_i=k[[t]]$. Describe the direct limit $\lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i$ in the category of ringed spaces. Is the direct limit a scheme?
6. Let $S$ be a finite partially ordered set. Consider the following topology on $S$: a subset $U\subset S$ is open if and only if whenever $x\in U$ and $y\gt x$, it must be that $y\in U$.

Construct a ring $R$ such that $\mathop{\mathrm{Spec}}(R)$ is homeomorphic to $S$.

7. Show that any quasi-compact scheme has closed points. (It is not true that any scheme has closed points!)
8. Give an example of a scheme that has no open connected subsets. In particular, such a scheme is not locally connected. Of course, my convention here is that the empty set is not connected...

Homework 4

Due Friday, February 24th

1. Show that the following two definitions of quasi-separated-ness of a scheme $S$ are equivalent:
1. The intersection of any two quasi-compact open subsets of $S$ is quasi-compact;
2. There is a cover of $S$ by affine open subsets whose (pairwise) intersections are quasi-compact.
2. In class, we gave the following definition: a scheme $S$ is integral if it is irreducible and reduced. Show that this is equivalent to the definition from Vakil’s notes: a scheme is integral if for any non-empty open $U\subset S$, $O_S(U)$ is a domain.
3. Let us call a scheme $X$ locally irreducible if every point has an irreducible neighborhood. (Since a non-empty open subset of an irreducible space is irreducible, this implies that all smaller neighborhoods of this point are irreducible as well.) Prove or disprove the following claim: a scheme is irreducible if and only if it is connected and locally irreducible.
4. Show that a locally Noetherian scheme is quasi-separated.
5. Show that the following two definitions of a Noetherian scheme $X$ are equivalent:
1. $X$ is a finite union of open affine sets, each of which is the spectrum of a Noetherian ring;
2. $X$ is quasi-compact and locally Noetherian.
6. Show that any Noetherian scheme $X$ is a disjoint union of finitely many connected open subsets (the connected components of $X$.) (A problem from the last homework shows that things might go wrong if we do not assume that $X$ is Noetherian.)
7. A locally closed subscheme $X\subset Y$ is defined as a closed subscheme of an open subscheme of $Y$. Accordingly, a locally closed embedding is a composition of a closed embedding followed by an open embedding (in this order). In principle, one can try to reverse the order, and consider open subschemes of closed subschemes of $Y$. Does this yield an equivalent definition?

Remark. The difficulty of such questions (and, sometimes, the answer to them) depends on the class of schemes one works with: often, very mild assumptions (such as, say, quasicompactness) would make the question easy. A complete answer to this problem would include both the mild assumptions that would make the two versions equivalent, and a description of what happens for general schemes.