Difference between revisions of "Math 764 -- Algebraic Geometry II -- Homeworks"

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* [[#Homework 1|Homework 1]] ([[Media:Math764S17HW1.pdf|PDF]]), due February 3rd.
 
* [[#Homework 1|Homework 1]] ([[Media:Math764S17HW1.pdf|PDF]]), due February 3rd.
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* [[#Homework 2|Homework 2]] ([[Media:Math764S17HW2.pdf|PDF]]), due February 10th.
  
 
=== Homework 1 ===
 
=== Homework 1 ===
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# <p> '''Locally constant sheaves:'''</p> <p>'''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>.</p> <p><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.</p><p>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>.)</p> <p>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>.)</p>
 
# <p> '''Locally constant sheaves:'''</p> <p>'''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>.</p> <p><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.</p><p>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>.)</p> <p>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>.)</p>
 
# '''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>.
 
# '''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>.
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 +
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=== Homework 2 ===
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 +
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>
 +
 +
# Is the sheafication necessary in this definition? (Or maybe <math>{\mathcal{G}}</math> is a sheaf automatically?)
 +
# 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>.
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# 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><p> (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>.) </p><p> ''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’)? </p> <p>'''Examples of affine schemes.'''</p>
 +
# Let <math>R_\alpha</math> be a possibly infinite collection of rings. Describe the topological space <math>{\mathop{\mathrm{Spec}}}(R)</math> in terms of <math>{\mathop{\mathrm{Spec}}}(R_\alpha)</math>’s.
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# 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 <p> (a) An infinite intersection of open sets, but not constructible. </p><p> (b) An infinite union of closed sets, but not constructible. </p><p> '''Contraction of a subvariety.'''</p> <p> 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: </p>
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#* 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>.
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#* 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>.)<p> In each of the following examples, determine whether the quotient <math>X/Y</math> is an algebraic variety; if it is, describe it explicitly.</p>
 +
#<math>X={\mathbb{P}}^2</math>, <math>Y={\mathbb{P}}^1</math> (embedded as a line in <math>X</math>).
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#<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>.
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#<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).

Revision as of 22:39, 3 February 2017

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 possibly infinite collection of rings. Describe the topological space [math]{\mathop{\mathrm{Spec}}}(R)[/math] in terms of [math]{\mathop{\mathrm{Spec}}}(R_\alpha)[/math]’s.
  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.

    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).