Math 764 -- Algebraic Geometry II -- Homeworks

From Math
Jump to: navigation, search

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.

Homework 5

Due Friday, March 3rd

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

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

    Consider the extension of scalars [math]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]Fr_X[/math] naturally extends to a morphism of [math]\overline{\mathbb{F}}_p[/math]-schemes [math]X'\to X'[/math]. Compare the map [math]X'\to X'[/math] with the relative Frobenius of [math]X'[/math] over [math]\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]f:X\to Z[/math] is surjective and [math]g:Y\to Z[/math] is arbitrary, then [math]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]X[/math] be an integral scheme. Show that there exists a normal integral scheme [math]\tilde{X}[/math] together with a morphism [math]\tilde{X}\to X[/math] that is universal in the following sense: any dominant morphism [math]Y\to X[/math] from a normal integral scheme to [math]X[/math] factors through [math]\tilde{X}[/math]. (Just like in the case of varieties, a morphism is dominant if its image is dense.)
  7. Let [math]X[/math] be a scheme of finite type over a field [math]k[/math]. For every field extension [math]K\supset k[/math], put [math]X_K:=X\otimes_kK={\mathop{Spec}}(K)\times_{{\mathop{Spec}}(k)}X.[/math] </p>

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

Homework 6

Due Friday, March 10th

Sheaves of modules on ringed spaces.

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

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

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

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

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

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

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

  6. Let [math]X[/math] be a locally noetherian scheme and let [math]U\subset X[/math] be an open subset. Show that any coherent sheaf [math]{\mathcal{F}}[/math] on [math]U[/math] can be extended to a coherent sheaf on [math]\overline{{\mathcal{F}}}[/math] on [math]X[/math]. (We say that [math]\overline{{\mathcal{F}}}[/math] is an extension of [math]{\mathcal{F}}[/math] if [math]\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]X[/math] has a maximal closed reduced subscheme [math]X^{red}[/math]; the ideal sheaf of [math]X^{red}[/math] is the nilradical (the sheaf of all nilpotents in [math]{\mathcal{O}}_X[/math]).

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

    Vector bundles.

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

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

    Tangent bundle.

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

Homework 8

Due Friday, April 7th

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

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

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

Homework 9

Due Friday, April 21st

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

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