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 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>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>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>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.