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.

Homework 5

Due Friday, March 3rd

1. Fix a field $k$, and put $X={\mathop{Spec}}k[x]$ and $Y={\mathop{Spec}}k[y]$. Consider the morphism $f:X\to Y$ given by $y=x^2$. Describe the fiber product $X\times_YX$ as explicitly as possible. (The answer may depend on $k$.)
2. (The Frobenius morphism.) Let $X$ be a scheme of characteristic $p$: by definition, this means that $p=0$ in the structure sheaf of $X$. Define the (absolute) Frobenius morphism $Fr_X:X\to X$ as follows: it is the identity map on the underlying set, and the pullback $Fr_X^*(f)$ equals $f^p$ for any (local) function $f\in{\mathcal{O}}_X$.

Verify that this defines an affine morphism of schemes. Assuming $X$ is a scheme locally of finite type over a perfect field, verify that $Fr_X$ is a morphism of finite type (it is in fact finite, if you know what it means).

3. (The relative Frobenius morphism.) Let $X\to Y$ be a morphism of schemes of characteristic $p$. Put $\overline X:=X\times_{Y,Fr_Y}Y,$ where the notation means that $Y$ is considered as a $Y$-scheme via the Frobenius map.
1. Show that the Frobenius morphism $Fr_X$ naturally factors as the composition $X\to\overline{X}\to X$, where the first map $X\to\overline{X}$ is naturally a morphism of schemes over $Y$ (while the second map, generally speaking, is not). The map $X\to\overline{X}$ is called the relative Frobenius morphism.
2. Suppose $Y={\mathop{Spec}}(\overline{\mathbb{F}}_p)$, and $X$ is an affine variety (that is, an affine reduced scheme of finite type) over $\overline{\mathbb{F}}_p$. Describe $\overline X$ and the relative Frobenius $X\to\overline{X}$ explicitly in coordinates.
4. Let $X$ be a scheme over $\mathbb{F}_p$. In this case, the absolute Frobenius $Fr_X:X\to X$ is a morphism of schemes over $\mathbb{F}_p$ (and it coincides with the relative Frobenius of $X$ over $\mathbb{F}_p$.

Consider the extension of scalars $X'=X_{\overline{\mathbb{F}}_p}=X\otimes_{\mathbb{F}_p}\overline{\mathbb{F}}_p=X\times_{{\mathop{Spec}}(\mathbb{F}_p)}{\mathop{Spec}}(\overline{\mathbb{F}}_p).$ Then $Fr_X$ naturally extends to a morphism of $\overline{\mathbb{F}}_p$-schemes $X'\to X'$. Compare the map $X'\to X'$ with the relative Frobenius of $X'$ over $\overline{\mathbb{F}}_p$.

5. A morphism of schemes is surjective if it is surjective as a morphism of sets. Show that surjectivity is preserved by base changes. That is, if $f:X\to Z$ is surjective and $g:Y\to Z$ is arbitrary, then $X\times_ZY\to Y$ is surjective.
6. (Normalization) A scheme is normal if all of its local rings are integrally closed domains. Let $X$ be an integral scheme. Show that there exists a normal integral scheme $\tilde{X}$ together with a morphism $\tilde{X}\to X$ that is universal in the following sense: any dominant morphism $Y\to X$ from a normal integral scheme to $X$ factors through $\tilde{X}$. (Just like in the case of varieties, a morphism is dominant if its image is dense.)
7. Let $X$ be a scheme of finite type over a field $k$. For every field extension $K\supset k$, put $X_K:=X\otimes_kK={\mathop{Spec}}(K)\times_{{\mathop{Spec}}(k)}X.$ </p>

Show that $X$ is geometrically irreducible (that is, the morphism $X\to{\mathop{Spec}}(k)$ has geometrically irreducible fibers) if and only if $X_K$ is irreducible for all finite extensions $K\supset k$.

Homework 6

Due Friday, March 10th

Sheaves of modules on ringed spaces.

Let $(X,{\mathcal{O}}_X)$ be a ringed space, and let ${\mathcal{F}}$ and ${\mathcal{G}}$ be sheaves of ${\mathcal{O}}_X$-modules. The tensor product of ${\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}}$ is the sheafification of the presheaf $U\mapsto{\mathcal{F}}(U)\otimes_{{\mathcal{O}}_X(U)}{\mathcal{G}}(U).$

1. Prove that the stalks of ${\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}}$ are given by the tensor product: $({\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}})_x={\mathcal{F}}_x\otimes_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x,$ where $x\in X$. Conclude that the tensor product is a right exact functor (in each of the two arguments).
2. Suppose that ${\mathcal{F}}$ is locally free of finite rank. (That is to say, every point $x\in X$ has a neighborhood $U$ such that ${\mathcal{F}}|_U\simeq({\mathcal{O}}_U)^n$. Prove that there exists a natural isomorphism ${{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{G}})={\mathcal{G}}\otimes{\mathcal{F}}^\vee.$ Here ${\mathcal{F}}^\vee={\mathop{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{O}}_X)$ is the dual of the locally free sheaf ${\mathcal{F}}$, and ${\mathop{\mathcal{H}\mathit{om}}}$ is the sheaf of homomorphisms. (Note that ${\mathcal{G}}$ is not assumed to be quasi-coherent.)
3. (Projection formula) Let $f:(X,{\mathcal{O}}_X)\to(Y,{\mathcal{O}}_Y)$ be a morphism of ringed spaces. Suppose ${\mathcal{F}}$ is an ${\mathcal{O}}_X$-module and ${\mathcal{G}}$ is a locally free ${\mathcal{O}}_Y$-module of finite rank. Construct a natural isomorphism $f_*({\mathcal{F}}\otimes_{{\mathcal{O}}_X} f^*{\mathcal{G}})\simeq f_*({\mathcal{F}})\otimes_{{\mathcal{O}}_Y}{\mathcal{G}}.$

Coherent sheaves on a noetherian scheme

4. Let ${\mathcal{F}}$ be a coherent sheaf on a loclly noetherian scheme $X$.

Show that ${\mathcal{F}}$ is locally free if and only if its stalks ${\mathcal{F}}_x$ are free ${\mathcal{O}}_{X,x}$-modules for all $x\in X$.

(b) Show that ${\mathcal{F}}$ is locally free of rank one if and only if it is invertible: there exists a coherent sheaf ${\mathcal{G}}$ such that ${\mathcal{F}}\otimes{\mathcal{G}}\simeq{\mathcal{O}}_X$.

5. As in the previous problem, supposed ${\mathcal{F}}$ be a coherent sheaf on a locally noetherian scheme $X$. The fiber of ${\mathcal{F}}$ at a point $x\in X$ is the $k(x)$-vector space $i^*{\mathcal{F}}$ for the natural map $i:{\mathop{Spec}}(k(x))\to X$ (where $k(x)$ is the residue field of $x\in X$). Denote by $\phi(x)$ the dimension $\dim_{k(x)} i^*{\mathcal{F}}$.

(a) Show that the function $\phi(x)$ is upper semi-continuous: for every $n$, the set $\{x\in X:\phi(x)\ge n\}$ is closed.

(b) Suppose $X$ is reduced. Show that ${\mathcal{F}}$ is locally free if and only if $\phi(x)$ is constant on each connected component of $X$. (Do you see why we impose the assumption that $X$ is reduced here?)

6. Let $X$ be a locally noetherian scheme and let $U\subset X$ be an open subset. Show that any coherent sheaf ${\mathcal{F}}$ on $U$ can be extended to a coherent sheaf on $\overline{{\mathcal{F}}}$ on $X$. (We say that $\overline{{\mathcal{F}}}$ is an extension of ${\mathcal{F}}$ if $\overline{{\mathcal{F}}}|_U\simeq{\mathcal{F}}$.)

(If you need a hint for this problem, look at Problem II.5.15 in Hartshorne.)