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...