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 14:40, 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 ; all sheaves and presheaves are sheaves on .
 Example: Let be the unit circle, and let be the sheaf of functions on . Find the (sheaf) image and the kernel of the morphism Here is the polar coordinate on the circle.
 Sheaf operations: Let and be sheaves of sets. Recall that a morphism is a (categorical) monomorphism if and only if for any sheaf and any two morphisms , the equality implies . Show that is a monomorphism if and only if it induces injective maps on all stalks.
 Let and be sheaves of sets. Recall that a morphism is a (categorical) epimorphism if and only if for any sheaf and any two morphisms , the equality implies . Show that 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 be a sheaf, and let be a subpresheaf of (thus, for every open set , is a subset of and the restriction maps for and agree). Show that the sheafification of is naturally identified with a subsheaf of .
 Let be a family of sheaves of abelian groups on indexed by a set (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 together with a universal family of homomorphisms .) Do these operations agree with (a) taking stalks at a point (b) taking sections over an open subset ?

Locally constant sheaves:
Definition. A sheaf is constant over an open set if there is a subset such that the map (the germ of at ) gives a bijection between and for all .
is locally constant (on ) if every point of has a neighborhood on which is constant.
Recall that a covering space is a continuous map of topological spaces such that every has a neighborhood whose preimage is homeomorphic to for some discrete topological space . ( may depend on ; also, the homeomorphism is required to respect the projection to .)
Show that if is a covering space, its sheaf of sections 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 is pathwise connected, both categories are equivalent to the category of sets with an action of the fundamental group of .)
 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 be a presheaf on , and let be its sheafification. Then every section can be represented as (the equivalence class of) the following gluing data: an open cover and a family of sections such that .