Math 764 -- Algebraic Geometry II -- Homeworks: Difference between revisions

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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.
* [[#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>.
=== 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>
# 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>.
# 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.
# 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>
#* 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>.)<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>).
#<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>.
#<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 03:39, 4 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]\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 possibly infinite collection of rings. Describe the topological space [math]\displaystyle{ {\mathop{\mathrm{Spec}}}(R) }[/math] in terms of [math]\displaystyle{ {\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]\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.

    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).