Difference between revisions of "Math 764  Algebraic Geometry II  Homeworks"
(Created page with "== Homeworks == 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. * Homewo...") 

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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.  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 ([Media:Math764S17HW1.pdf PDF]), due February 3rd.  +  * [[Homework 1Homework 1]] ([[Media:Math764S17HW1.pdf PDF]]), due February 3rd. 
=== Homework 1 ===  === 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>?  
+  # <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>. 
Revision as of 13:40, 29 January 2017
Homeworks
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 (Media:Math764S17HW1.pdf PDF), due February 3rd.
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].