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]\displaystyle{ X }[/math]; all sheaves and presheaves are sheaves on [math]\displaystyle{ X }[/math].

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

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

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

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

    Show that if [math]\displaystyle{ \pi:Y\to X }[/math] is a covering space, its sheaf of sections [math]\displaystyle{ {\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]\displaystyle{ X }[/math] is pathwise connected, both categories are equivalent to the category of sets with an action of the fundamental group of [math]\displaystyle{ 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]\displaystyle{ {\mathcal{F}} }[/math] be a presheaf on [math]\displaystyle{ X }[/math], and let [math]\displaystyle{ \tilde{\mathcal{F}} }[/math] be its sheafification. Then every section [math]\displaystyle{ s\in\tilde{\mathcal{F}}(U) }[/math] can be represented as (the equivalence class of) the following gluing data: an open cover [math]\displaystyle{ U=\bigcup U_i }[/math] and a family of sections [math]\displaystyle{ s_i\in{\mathcal{F}}(U_i) }[/math] such that [math]\displaystyle{ 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]\displaystyle{ X }[/math] be a topological space, let [math]\displaystyle{ U\subset X }[/math] be an open subset, and let [math]\displaystyle{ {\mathcal{F}} }[/math] be a sheaf of abelian groups on [math]\displaystyle{ U }[/math].

The extension by zero [math]\displaystyle{ j_{!}{\mathcal{F}} }[/math] of [math]\displaystyle{ {\mathcal{F}} }[/math] (here [math]\displaystyle{ j }[/math] is the embedding [math]\displaystyle{ U\hookrightarrow X }[/math]) is the sheaf on [math]\displaystyle{ X }[/math] that can be defined as the sheafification of the presheaf [math]\displaystyle{ {\mathcal{G}} }[/math] such that [math]\displaystyle{ {\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]\displaystyle{ {\mathcal{G}} }[/math] is a sheaf automatically?)
  2. Describe the stalks of [math]\displaystyle{ j_!{\mathcal{F}} }[/math] over all points of [math]\displaystyle{ X }[/math] and the espace étalé of [math]\displaystyle{ j_!{\mathcal{F}} }[/math].
  3. Verify that [math]\displaystyle{ j_! }[/math] is the left adjoint of the restriction functor from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ U }[/math]: that is, for any sheaf [math]\displaystyle{ {\mathcal{G}} }[/math] on [math]\displaystyle{ X }[/math], there exists a natural isomorphism [math]\displaystyle{ {\mathop{\mathrm{Hom}}}({\mathcal{F}},{\mathcal{G}}|_U)\simeq{\mathop{\mathrm{Hom}}}(j_!{\mathcal{F}},{\mathcal{G}}). }[/math]

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

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

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

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

Homework 3

Due Friday, February 17th

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

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

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

    (b) Let [math]\displaystyle{ R_i }[/math] be a collection of rings ([math]\displaystyle{ i\gt 0 }[/math]) together with homomorphisms [math]\displaystyle{ R_{i+1}\to R_i }[/math]. Consider the inverse limit [math]\displaystyle{ R:=\lim\limits_{\longleftarrow} R_i }[/math]. Show that generally speaking, in the category of schemes, [math]\displaystyle{ {\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]\displaystyle{ k }[/math] be a field, and let [math]\displaystyle{ R_i=k[t]/(t^i) }[/math], so that [math]\displaystyle{ \lim\limits_{\longleftarrow} R_i=k[[t]] }[/math]. Describe the direct limit [math]\displaystyle{ \lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i }[/math] in the category of ringed spaces. Is the direct limit a scheme?
  6. Let [math]\displaystyle{ S }[/math] be a finite partially ordered set. Consider the following topology on [math]\displaystyle{ S }[/math]: a subset [math]\displaystyle{ U\subset S }[/math] is open if and only if whenever [math]\displaystyle{ x\in U }[/math] and [math]\displaystyle{ y\gt x }[/math], it must be that [math]\displaystyle{ y\in U }[/math].

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

Homework 5

Due Friday, March 3rd

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

    Verify that this defines an affine morphism of schemes. Assuming [math]\displaystyle{ X }[/math] is a scheme locally of finite type over a perfect field, verify that [math]\displaystyle{ Fr_X }[/math] is a morphism of finite type (it is in fact finite, if you know what it means).

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

    Consider the extension of scalars [math]\displaystyle{ 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). }[/math] Then [math]\displaystyle{ Fr_X }[/math] naturally extends to a morphism of [math]\displaystyle{ \overline{\mathbb{F}}_p }[/math]-schemes [math]\displaystyle{ X'\to X' }[/math]. Compare the map [math]\displaystyle{ X'\to X' }[/math] with the relative Frobenius of [math]\displaystyle{ X' }[/math] over [math]\displaystyle{ \overline{\mathbb{F}}_p }[/math].

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

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

Homework 6

Due Friday, March 10th

Sheaves of modules on ringed spaces.

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

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

    Coherent sheaves on a noetherian scheme

  4. Let [math]\displaystyle{ {\mathcal{F}} }[/math] be a coherent sheaf on a loclly noetherian scheme [math]\displaystyle{ X }[/math].

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

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

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

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

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

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

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

Homework 7

Due Friday, March 31st

Proper and separated morphisms.

Each scheme [math]\displaystyle{ X }[/math] has a maximal closed reduced subscheme [math]\displaystyle{ X^{red} }[/math]; the ideal sheaf of [math]\displaystyle{ X^{red} }[/math] is the nilradical (the sheaf of all nilpotents in [math]\displaystyle{ {\mathcal{O}}_X }[/math]).

  1. Let [math]\displaystyle{ f:X\to Y }[/math] be a morphism of schemes of finite type. Consider the induced map [math]\displaystyle{ f^{red}:X^{red}\to Y^{red} }[/math]. Prove that [math]\displaystyle{ f }[/math] is separated (resp. proper) if and only if [math]\displaystyle{ f^{red} }[/math] is separated (resp. proper).

    Vector bundles.

    Fix an algebraically closed field [math]\displaystyle{ k }[/math]. Any vector bundle on [math]\displaystyle{ {\mathbb{A}}^1_k={\mathop{Spec}}(k[t]) }[/math] is trivial, you can use this without proof. Let [math]\displaystyle{ X }[/math] be the ‘affine line with a doubled point’ obtained by gluing two copies of [math]\displaystyle{ {\mathbb{A}}^1_k }[/math] away from the origin.

  2. Classify line bundles on [math]\displaystyle{ X }[/math] up to isomorphism.
  3. (Could be hard) Prove that any vector bundle on [math]\displaystyle{ X }[/math] is a direct sum of several line bundles.

    Tangent bundle.

  4. Let [math]\displaystyle{ X }[/math] be an irreducible affine variety, not necessarily smooth. Let [math]\displaystyle{ M }[/math] be the [math]\displaystyle{ k[X] }[/math]-module of [math]\displaystyle{ k }[/math]-linear derivations [math]\displaystyle{ k[X]\to k[X] }[/math]. (These are globally defined vector fields on [math]\displaystyle{ X }[/math], but keep in mind that [math]\displaystyle{ X }[/math] may be singular.) Consider its generic rank [math]\displaystyle{ r:=\dim_{k(X)}M\otimes_{k[X]}k(X) }[/math]. Show that [math]\displaystyle{ r=\dim(X) }[/math].
  5. Suppose now that [math]\displaystyle{ X }[/math] is smooth. Show that the module [math]\displaystyle{ M }[/math] is a locally free coherent module; the corresponding vector bundle is the tangent bundle [math]\displaystyle{ TX }[/math].
  6. Let [math]\displaystyle{ f:X\to Y }[/math] be a morphism of algebraic varieties. Recall that a vector bundle [math]\displaystyle{ E }[/math] over [math]\displaystyle{ Y }[/math] gives a vector bundle [math]\displaystyle{ f^*E }[/math] on [math]\displaystyle{ X }[/math] whose total space is the fiber product [math]\displaystyle{ E\times_YX }[/math].
  7. Suppose now that [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are affine and [math]\displaystyle{ Y }[/math] is smooth. Let [math]\displaystyle{ E=TY }[/math] be the tangent bundle to [math]\displaystyle{ Y }[/math]. Show that the space of [math]\displaystyle{ k }[/math]-linear derivations [math]\displaystyle{ k[Y]\to k[X] }[/math] (where [math]\displaystyle{ f }[/math] is used to equip [math]\displaystyle{ k[X] }[/math] with the structure of a [math]\displaystyle{ k[Y] }[/math]-module) is identified with [math]\displaystyle{ \Gamma(X,f^*(TY)) }[/math].
  8. Let [math]\displaystyle{ X }[/math] be a smooth affine variety. Let [math]\displaystyle{ I_\Delta\subset k[X\times X] }[/math] be the ideal sheaf of the diagonal [math]\displaystyle{ \Delta\subset X\times X }[/math]. Prove that there is a bijection [math]\displaystyle{ I_\Delta/I_\Delta^2=\Gamma(X,\Omega^1_X), }[/math] where [math]\displaystyle{ \Omega^1_X }[/math] is the sheaf of differential 1-forms (that is, the sheaf of sections of the cotangent bundle [math]\displaystyle{ T^\vee X }[/math], which is the dual vector bundle of [math]\displaystyle{ TX }[/math]).

Homework 8

Due Friday, April 7th

  1. (Hartshorne, II.4.4) Fix a Noetherian scheme [math]\displaystyle{ S }[/math], let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be schemes of finite type and separated over [math]\displaystyle{ S }[/math], and let [math]\displaystyle{ f:X\to Y }[/math] be a morphism of [math]\displaystyle{ S }[/math]-schemes. Suppose that [math]\displaystyle{ Z\subset X }[/math] be a closed subscheme that is proper over [math]\displaystyle{ S }[/math]. Show that [math]\displaystyle{ f(Z)\subset Y }[/math] is closed.
  2. In the setting of the previous problem, show that if we consider [math]\displaystyle{ f(Z) }[/math] as a closed subscheme (its ideal of functions consists of all functions whose composition with [math]\displaystyle{ f }[/math] is zero), then [math]\displaystyle{ f }[/math] induces a proper map fro [math]\displaystyle{ Z }[/math] to [math]\displaystyle{ f(Z) }[/math].

    (Galois descent, inspired by Hartshorne II.4.7) Let [math]\displaystyle{ F/k }[/math] be a finite Galois extension of fields. The Galois group [math]\displaystyle{ G:=Gal(F/k) }[/math] acts on the scheme [math]\displaystyle{ {\mathop{Spec}}(F) }[/math]. Given any [math]\displaystyle{ k }[/math]-scheme [math]\displaystyle{ X }[/math], we let [math]\displaystyle{ X_F:={\mathop{Spec}}(F)\times_{{\mathop{Spec}}(k)}X }[/math] be its extension of scalars; the group [math]\displaystyle{ G }[/math] acts on [math]\displaystyle{ X_F }[/math] in a way compatible with its action on [math]\displaystyle{ {\mathop{Spec}}(F) }[/math] (i.e., this action is ‘semilinear’).

  3. Show that [math]\displaystyle{ X }[/math] is affine if and only if [math]\displaystyle{ X_F }[/math] is affine.
  4. Prove that this operation gives a fully faithful functor from the category of [math]\displaystyle{ k }[/math]-schemes into the category of [math]\displaystyle{ F }[/math]-schemes with a semi-linear action of [math]\displaystyle{ G }[/math].
  5. Suppose that [math]\displaystyle{ Y }[/math] is a separated [math]\displaystyle{ F }[/math]-scheme such that any finite subset of [math]\displaystyle{ Y }[/math] is contained in an affine open chart (this holds, for instance, if [math]\displaystyle{ Y }[/math] is quasi-projective). Then for any semi-linear action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ Y }[/math], there exists a [math]\displaystyle{ k }[/math]-scheme [math]\displaystyle{ X }[/math] and an isomorphism [math]\displaystyle{ X_F\simeq Y }[/math] that agrees with an action of [math]\displaystyle{ G }[/math]. (That is, the action of [math]\displaystyle{ G }[/math] gives a [math]\displaystyle{ k }[/math]-structure on the scheme [math]\displaystyle{ Y }[/math].)
  6. Suppose [math]\displaystyle{ X }[/math] is an [math]\displaystyle{ {\mathbb{R}} }[/math]-scheme such that [math]\displaystyle{ X_{\mathbb{C}}\simeq{\mathbb{A}}_{\mathbb{C}}^1 }[/math]. Show that [math]\displaystyle{ X\simeq{\mathbb{A}}^1_{\mathbb{R}} }[/math].
  7. Suppose [math]\displaystyle{ X }[/math] is an [math]\displaystyle{ {\mathbb{R}} }[/math]-scheme such that [math]\displaystyle{ X_{\mathbb{C}}\simeq{\mathbb{P}}_{\mathbb{C}}^1 }[/math]. Show that there are two possibilities for the isomorphism class of [math]\displaystyle{ X }[/math].

Homework 9

Due Friday, April 21st

  1. Let [math]\displaystyle{ X }[/math] be a singular cubic in [math]\displaystyle{ {\mathbb{P}}^2 }[/math], given (in non-homogeneous coordinates) either by [math]\displaystyle{ y^2=x^3+x^2 }[/math] (nodal cubic) or by [math]\displaystyle{ y^2=x^3 }[/math] (cuspidal cubic). Compute the class group of Cartier divisors on [math]\displaystyle{ X }[/math].
  2. Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be schemes over some base scheme [math]\displaystyle{ S }[/math]. For any map [math]\displaystyle{ f:X\to Y }[/math], use the functoriality of the module of Kähler differentials to construct a morphism [math]\displaystyle{ f^*\Omega_{Y/S}\to\Omega_{X/S} }[/math] and verify that [math]\displaystyle{ \Omega_{X/Y}=\mathrm{coker}(f^*\Omega_{Y/S}\to\Omega_{X/S}) }[/math].
  3. Suppose now that [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be schemes over an algebraically closed field [math]\displaystyle{ k }[/math]. A morphism [math]\displaystyle{ f:X\to Y }[/math] is unramified if [math]\displaystyle{ \Omega_{X/Y}=0 }[/math]. Show that this is equivalent to the following condition: given [math]\displaystyle{ D={\mathop{Spec}}k[\epsilon]/{\epsilon^2} }[/math], the map [math]\displaystyle{ f }[/math] induces an injection [math]\displaystyle{ \mathrm{Maps}(D,X)\to\mathrm{Maps}(D,Y) }[/math].
  4. Let us compute the algebraic de Rham cohomology of the affine space. Put [math]\displaystyle{ X={\mathop{Spec}}R }[/math], [math]\displaystyle{ R=k[t_1,\dots,t_n] }[/math]. Since [math]\displaystyle{ X }[/math] is a smooth [math]\displaystyle{ k }[/math]-scheme, [math]\displaystyle{ \Omega^1_R=\Omega_{R/k} }[/math] is a locally free [math]\displaystyle{ R }[/math]-module. Denote by [math]\displaystyle{ \Omega^\bullet_R }[/math] the exterior algebra of [math]\displaystyle{ \Omega^1_R }[/math], so that [math]\displaystyle{ \Omega^i_R=\bigwedge^i\Omega^1_R }[/math]. Define the de Rham differential [math]\displaystyle{ d:\Omega^i_R\to\Omega^{i+1}_R }[/math] by starting with the Kähler differential [math]\displaystyle{ d:R\to\Omega^1_R }[/math] and then extending it by the graded Leibniz rule: [math]\displaystyle{ d(\omega_1\wedge\omega_2)=(d\omega_1)\wedge\omega_2+(-1)^i\omega_1\wedge d(\omega_2),\qquad \omega_1\in\Omega^i_R. }[/math]

    Compute the cohomology of the complex [math]\displaystyle{ \Omega^\bullet_R }[/math] equipped with the differential [math]\displaystyle{ d }[/math]. The answer will depend on the characteristic of [math]\displaystyle{ k }[/math].

  5. Let [math]\displaystyle{ X }[/math] be a Noetherian scheme.Let [math]\displaystyle{ K(X) }[/math] be the [math]\displaystyle{ K }[/math]-group of [math]\displaystyle{ X }[/math]: it is generated by elements [math]\displaystyle{ [F] }[/math] for each coherent sheaf [math]\displaystyle{ F }[/math] with relations [math]\displaystyle{ [F]=[F_1]+[F_2] }[/math] whenever there is a short exact sequence [math]\displaystyle{ 0\to F_1\to F_2\to F_3\to 0. }[/math] Prove that [math]\displaystyle{ K({\mathbb{A}}^n)={\mathbb{Z}} }[/math]. (This is much easier if you know Hilbert’s Syzygy Theorem.)
  6. Let [math]\displaystyle{ X }[/math] be a smooth curve over an algebraically closed field. Show that [math]\displaystyle{ K(X) }[/math] is generated by [math]\displaystyle{ [L] }[/math] for line bundles [math]\displaystyle{ L }[/math].
  7. Let [math]\displaystyle{ X }[/math] be a smooth curve over an algebraically closed field. Show that [math]\displaystyle{ K(X) }[/math] is isomorphic to [math]\displaystyle{ {\mathbb{Z}}\oplus \mathrm{Pic}(X) }[/math]. (If this problem is too hard, look at Hartshorne’s II.6.11 for a step-by-step approach.)