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 [math]X[/math]; all sheaves and presheaves are sheaves on [math]X[/math].

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

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

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

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

    Show that if [math]\pi:Y\to X[/math] is a covering space, its sheaf of sections [math]{\mathcal{F}}[/math] 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 [math]X[/math] is pathwise connected, both categories are equivalent to the category of sets with an action of the fundamental group of [math]X[/math].)

  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 [math]{\mathcal{F}}[/math] be a presheaf on [math]X[/math], and let [math]\tilde{\mathcal{F}}[/math] be its sheafification. Then every section [math]s\in\tilde{\mathcal{F}}(U)[/math] can be represented as (the equivalence class of) the following gluing data: an open cover [math]U=\bigcup U_i[/math] and a family of sections [math]s_i\in{\mathcal{F}}(U_i)[/math] such that [math]s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j}[/math].

Homework 2

Due Friday, February 10th

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

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

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

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

    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 [math]R_\alpha[/math] be a finite collection of rings. Put [math]R=\prod_\alpha R_\alpha[/math]. Describe the topological space [math]{\mathop{\mathrm{Spec}}}(R)[/math] in terms of [math]{\mathop{\mathrm{Spec}}}(R_\alpha)[/math]’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 [math]R\to S[/math] such that the image of a map [math]{\mathop{\mathrm{Spec}}}(S)\to{\mathop{\mathrm{Spec}}}(R)[/math] 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 [math]X[/math] be a variety (over an algebraically closed field [math]k[/math]) and let [math]Y\subset X[/math] be a closed subvariety. Our goal is to construct a [math]{k}[/math]-ringed space [math]Z=(Z,{\mathcal{O}}_Z)=X/Y[/math] that is in some sense the result of ‘gluing’ together the points of [math]Y[/math]. While [math]Z[/math] can be described by a universal property, we prefer an explicit construction:

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

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

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

Homework 3

Due Friday, February 17th

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

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

  2. (Gluing morphisms of ringed spaces) Let [math]X[/math] and [math]Y[/math] be ringed spaces. Denote by [math]\underline{Mor}(X,Y)[/math] the following pre-sheaf on [math]X[/math]: its sections over an open subset [math]U\subset X[/math] are morphisms of ringed spaces [math]U\to Y[/math] where [math]U[/math] is considered as a ringed space. (And the notion of restriction is the natural one.) Show that [math]\underline{Mor}(X,Y)[/math] is in fact a sheaf.
  3. (Affinization of a scheme) Let [math]X[/math] be an arbitrary scheme. Prove that there exists an affine scheme [math]X_{aff}[/math] and a morphism [math]X\to X_{aff}[/math] that is universal in the following sense: any map form [math]X[/math] 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 [math]R_i[/math] be a collection of rings ([math]i\gt 0[/math]) together with homomorphisms [math]R_i\to R_{i+1}[/math]. Consider the direct limit [math]R:=\lim\limits_{\longrightarrow} R_i[/math]. Show that in the category of schemes, [math]{\mathop{\mathrm{Spec}}}(R)=\lim\limits_{\longleftarrow}{\mathop{\mathrm{Spec}}}R_i.[/math]

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

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

    Construct a ring [math]R[/math] such that [math]\mathop{\mathrm{Spec}}(R)[/math] is homeomorphic to [math]S[/math].

  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 [math]S[/math] are equivalent:
    1. The intersection of any two quasi-compact open subsets of [math]S[/math] is quasi-compact;
    2. There is a cover of [math]S[/math] by affine open subsets whose (pairwise) intersections are quasi-compact.
  2. In class, we gave the following definition: a scheme [math]S[/math] 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 [math]U\subset S[/math], [math]O_S(U)[/math] is a domain.
  3. Let us call a scheme [math]X[/math] 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 [math]X[/math] are equivalent:
    1. [math]X[/math] is a finite union of open affine sets, each of which is the spectrum of a Noetherian ring;
    2. [math]X[/math] is quasi-compact and locally Noetherian.
  6. Show that any Noetherian scheme [math]X[/math] is a disjoint union of finitely many connected open subsets (the connected components of [math]X[/math].) (A problem from the last homework shows that things might go wrong if we do not assume that [math]X[/math] is Noetherian.)
  7. A locally closed subscheme [math]X\subset Y[/math] is defined as a closed subscheme of an open subscheme of [math]Y[/math]. 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 [math]Y[/math]. 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.