Difference between revisions of "Math 764  Algebraic Geometry II  Homeworks"
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* [[#Homework 1Homework 1]] ([[Media:Math764S17HW1.pdfPDF]]), due February 3rd.  * [[#Homework 1Homework 1]] ([[Media:Math764S17HW1.pdfPDF]]), due February 3rd.  
−  * [[#Homework 2Homework 2]] ([[Media:Math764S17HW2.pdfPDF]]), due February 10th.  +  * [[#Homework 2Homework 2]] ([[Media:Math764S17HW2.pdfPDF]]), due February 10th. (Two problems corrected.) 
=== Homework 1 ===  === Homework 1 ===  
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# 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>.  # 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>  # 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  +  # 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? 
−  # 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>  +  # 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. (This part may be very hard.) </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 ‘quotientspace’ <math>X/Y</math>: as a set, <math>Z=(XY)\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>XY</math> and sends all of <math>Y</math> to the ‘center’ <math>z\in Z</math>.  #* The topological space <math>Z</math> is the ‘quotientspace’ <math>X/Y</math>: as a set, <math>Z=(XY)\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>XY</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>  #* 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> 
Revision as of 12:10, 9 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 (PDF), due February 3rd.
 Homework 2 (PDF), due February 10th. (Two problems corrected.)
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].
 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.
 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.
 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.
 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.)
 Let [math]{\mathcal{F}}[/math] be a sheaf, and let [math]{\mathcal{G}}\subset{\mathcal{F}}[/math] be a subpresheaf 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].
 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]?

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].)
 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]
(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.
 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?
 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 ‘quotientspace’ [math]X/Y[/math]: as a set, [math]Z=(XY)\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]XY[/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.
 [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 twopoint set (if you want a more challenging version, let [math]Y\subset{\mathbb{A}}^2[/math] be any finite set).