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

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(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.
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* [[Homework 1|Homework 1]] ([[Media:Math764S17HW1.pdf PDF]]), due February 3rd.
  
 
=== Homework 1 ===
 
=== Homework 1 ===
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Due Friday, February 3rd
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In all these problems, we fix a topological space <math>X</math>; all sheaves and presheaves are sheaves on <math>X</math>.
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# '''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.
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# '''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.
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# 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.
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# 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.)
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# 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>.
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# 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>?
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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>
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# '''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

Due Friday, February 3rd

In all these problems, we fix a topological space X; all sheaves and presheaves are sheaves on X.

  1. Example: Let X be the unit circle, and let {\mathcal{F}} be the sheaf of C^\infty-functions on X. Find the (sheaf) image and the kernel of the morphism \frac{d}{dt}:{\mathcal{F}}\to{\mathcal{F}}. Here t\in{\mathbb{R}}/2\pi{\mathbb{Z}} is the polar coordinate on the circle.
  2. Sheaf operations: Let {\mathcal{F}} and {\mathcal{G}} be sheaves of sets. Recall that a morphism \phi:{\mathcal{F}}\to {\mathcal{G}} is a (categorical) monomorphism if and only if for any sheaf {\mathcal{F}}' and any two morphisms \psi_1,\psi_2:{\mathcal{F}}'\to {\mathcal{F}}, the equality \phi\circ\psi_1=\phi\circ\psi_2 implies \psi_1=\psi_2. Show that \phi is a monomorphism if and only if it induces injective maps on all stalks.
  3. Let {\mathcal{F}} and {\mathcal{G}} be sheaves of sets. Recall that a morphism \phi:{\mathcal{F}}\to{\mathcal{G}} is a (categorical) epimorphism if and only if for any sheaf {\mathcal{G}}' and any two morphisms \psi_1,\psi_2:{\mathcal{G}}\to{\mathcal{G}}', the equality \psi_1\circ\phi=\psi_2\circ\phi implies \psi_1=\psi_2. Show that \phi 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 {\mathcal{F}} be a sheaf, and let {\mathcal{G}}\subset{\mathcal{F}} be a sub-presheaf of {\mathcal{F}} (thus, for every open set U\subset X, {\mathcal{G}}(U) is a subset of {\mathcal{F}}(U) and the restriction maps for {\mathcal{F}} and {\mathcal{G}} agree). Show that the sheafification \tilde{\mathcal{G}} of {\mathcal{G}} is naturally identified with a subsheaf of {\mathcal{F}}.
  6. Let {\mathcal{F}}_i be a family of sheaves of abelian groups on X indexed by a set I (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 {\mathcal{F}} together with a universal family of homomorphisms {\mathcal{F}}_i\to {\mathcal{F}}.) Do these operations agree with (a) taking stalks at a point x\in X (b) taking sections over an open subset U\subset X?
  7. Locally constant sheaves:

    Definition. A sheaf {\mathcal{F}} is constant over an open set U\subset X if there is a subset S\subset F(U) such that the map {\mathcal{F}}(U)\to{\mathcal{F}}_x:s\mapsto s_x (the germ of s at x) gives a bijection between S and {\mathcal{F}}_x for all x\in U.

    {\mathcal{F}} is locally constant (on X) if every point of X has a neighborhood on which {\mathcal{F}} is constant.

    Recall that a covering space \pi:Y\to X is a continuous map of topological spaces such that every x\in X has a neighborhood U\ni x whose preimage \pi^{-1}(U)\subset U is homeomorphic to U\times Z for some discrete topological space Z. (Z may depend on x; also, the homeomorphism is required to respect the projection to U.)

    Show that if \pi:Y\to X is a covering space, its sheaf of sections {\mathcal{F}} 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 X is pathwise connected, both categories are equivalent to the category of sets with an action of the fundamental group of X.)

  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 {\mathcal{F}} be a presheaf on X, and let \tilde{\mathcal{F}} be its sheafification. Then every section s\in\tilde{\mathcal{F}}(U) can be represented as (the equivalence class of) the following gluing data: an open cover U=\bigcup U_i and a family of sections s_i\in{\mathcal{F}}(U_i) such that s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j}.