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

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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 1|Homework 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 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>.
# 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 19:40, 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 [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].