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

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* [[#Homework 1|Homework 1]] ([[Media:Math764S17HW1.pdf|PDF]]), due February 3rd.
 
* [[#Homework 1|Homework 1]] ([[Media:Math764S17HW1.pdf|PDF]]), due February 3rd.
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* [[#Homework 2|Homework 2]] ([[Media:Math764S17HW2.pdf|PDF]]), due February 10th.
  
 
=== Homework 1 ===
 
=== Homework 1 ===
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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>
 
# <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>.
 
# '''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>.
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=== Homework 2 ===
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Due Friday, February 10th
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'''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>.
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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>
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# Is the sheafication necessary in this definition? (Or maybe <math>{\mathcal{G}}</math> is a sheaf automatically?)
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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>.
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# 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>
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# Let <math>R_\alpha</math> be a possibly infinite collection of rings. Describe the topological space <math>{\mathop{\mathrm{Spec}}}(R)</math> in terms of <math>{\mathop{\mathrm{Spec}}}(R_\alpha)</math>’s.
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# 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>
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#* The topological space <math>Z</math> is the ‘quotient-space’ <math>X/Y</math>: as a set, <math>Z=(X-Y)\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>X-Y</math> and sends all of <math>Y</math> to the ‘center’ <math>z\in Z</math>.
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#* 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>
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#<math>X={\mathbb{P}}^2</math>, <math>Y={\mathbb{P}}^1</math> (embedded as a line in <math>X</math>).
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#<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>.
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#<math>X={\mathbb{A}}^2</math>, <math>Y</math> is a two-point set (if you want a more challenging version, let <math>Y\subset{\mathbb{A}}^2</math> be any finite set).

Revision as of 22:39, 3 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

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}.


Homework 2

Due Friday, February 10th

Extension of a sheaf by zero. Let X be a topological space, let U\subset X be an open subset, and let {\mathcal{F}} be a sheaf of abelian groups on U.

The extension by zero j_{!}{\mathcal{F}} of {\mathcal{F}} (here j is the embedding U\hookrightarrow X) is the sheaf on X that can be defined as the sheafification of the presheaf {\mathcal{G}} such that {\mathcal{G}}(V)=\begin{cases}{\mathcal{F}}(V),&V\subset U\\0,&V\not\subset U.\end{cases}

  1. Is the sheafication necessary in this definition? (Or maybe {\mathcal{G}} is a sheaf automatically?)
  2. Describe the stalks of j_!{\mathcal{F}} over all points of X and the espace étalé of j_!{\mathcal{F}}.
  3. Verify that j_! is the left adjoint of the restriction functor from X to U: that is, for any sheaf {\mathcal{G}} on X, there exists a natural isomorphism {\mathop{\mathrm{Hom}}}({\mathcal{F}},{\mathcal{G}}|_U)\simeq{\mathop{\mathrm{Hom}}}(j_!{\mathcal{F}},{\mathcal{G}}).

    (The restriction {\mathcal{G}}|_U of a sheaf {\mathcal{G}} from X to an open set U is defined by {\mathcal{G}}|_U(V)={\mathcal{G}}(V) for V\subset U.)

    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.

  4. Let R_\alpha be a possibly infinite collection of rings. Describe the topological space {\mathop{\mathrm{Spec}}}(R) in terms of {\mathop{\mathrm{Spec}}}(R_\alpha)’s.
  5. 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 R\to S such that the image of a map {\mathop{\mathrm{Spec}}}(S)\to{\mathop{\mathrm{Spec}}}(R) is

    (a) An infinite intersection of open sets, but not constructible.

    (b) An infinite union of closed sets, but not constructible.

    Contraction of a subvariety.

    Let X be a variety (over an algebraically closed field k) and let Y\subset X be a closed subvariety. Our goal is to construct a {k}-ringed space Z=(Z,{\mathcal{O}}_Z)=X/Y that is in some sense the result of ‘gluing’ together the points of Y. While Z can be described by a universal property, we prefer an explicit construction:

    • The topological space Z is the ‘quotient-space’ X/Y: as a set, Z=(X-Y)\sqcup \{z\}; a subset U\subset Z is open if and only if \pi^{-1}(U)\subset X is open. Here the natural projection \pi:X\to Z is identity on X-Y and sends all of Y to the ‘center’ z\in Z.
    • The structure sheaf {\mathcal{O}}_Z is defined as follows: for any open subset U\subset Z, {\mathcal{O}}_Z(U) is the algebra of functions g:U\to{k} such that the composition g\circ\pi is a regular function \pi^{-1}(U)\to{k} that is constant along Y. (The last condition is imposed only if z\in U, in which case Y\subset\pi^{-1}(U).)

      In each of the following examples, determine whether the quotient X/Y is an algebraic variety; if it is, describe it explicitly.

  6. X={\mathbb{P}}^2, Y={\mathbb{P}}^1 (embedded as a line in X).
  7. X=\{(s_0,s_1;t_0:t_1)\in{\mathbb{A}}^2\times{\mathbb{P}}^1:s_0t_1=s_1t_0\}, Y=\{(s_0,s_1;t_0:t_1)\in X:s_0=s_1=0\}.
  8. X={\mathbb{A}}^2, Y is a two-point set (if you want a more challenging version, let Y\subset{\mathbb{A}}^2 be any finite set).