Difference between revisions of "Math 764  Algebraic Geometry II  Homeworks"
m (Arinkin moved page Math 764  Algebraic Geometry II  Homeworks'17 to Math 764  Algebraic Geometry II  Homeworks) 

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* [[#Homework 1Homework 1]] ([[Media:Math764S17HW1.pdfPDF]]), due February 3rd.  * [[#Homework 1Homework 1]] ([[Media:Math764S17HW1.pdfPDF]]), due February 3rd.  
+  * [[#Homework 2Homework 2]] ([[Media:Math764S17HW2.pdfPDF]]), 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>.  
+  
+  
+  === Homework 2 ===  
+  
+  Due Friday, February 10th  
+  
+  '''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>.  
+  
+  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>  
+  
+  # Is the sheafication necessary in this definition? (Or maybe <math>{\mathcal{G}}</math> is a sheaf automatically?)  
+  # 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>.  
+  # 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>  
+  # 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.  
+  # 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>  
+  #* The topological space <math>Z</math> is the ‘quotientspace’ <math>X/Y</math>: as a set, <math>Z=(XY)\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>XY</math> and sends all of <math>Y</math> to the ‘center’ <math>z\in Z</math>.  
+  #* 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>  
+  #<math>X={\mathbb{P}}^2</math>, <math>Y={\mathbb{P}}^1</math> (embedded as a line in <math>X</math>).  
+  #<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>.  
+  #<math>X={\mathbb{A}}^2</math>, <math>Y</math> is a twopoint set (if you want a more challenging version, let <math>Y\subset{\mathbb{A}}^2</math> be any finite set). 
Revision as of 21:39, 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 (PDF), due February 3rd.
 Homework 2 (PDF), due February 10th.
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 .
Homework 2
Due Friday, February 10th
Extension of a sheaf by zero. Let be a topological space, let be an open subset, and let be a sheaf of abelian groups on .
The extension by zero of (here is the embedding ) is the sheaf on that can be defined as the sheafification of the presheaf such that
 Is the sheafication necessary in this definition? (Or maybe is a sheaf automatically?)
 Describe the stalks of over all points of and the espace étalé of .
 Verify that is the left adjoint of the restriction functor from to : that is, for any sheaf on , there exists a natural isomorphism
(The restriction of a sheaf from to an open set is defined by for .)
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.
 Let be a possibly infinite collection of rings. Describe the topological space in terms of ’s.
 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 such that the image of a map 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 be a variety (over an algebraically closed field ) and let be a closed subvariety. Our goal is to construct a ringed space that is in some sense the result of ‘gluing’ together the points of . While can be described by a universal property, we prefer an explicit construction:
 The topological space is the ‘quotientspace’ : as a set, ; a subset is open if and only if is open. Here the natural projection is identity on and sends all of to the ‘center’ .
 The structure sheaf is defined as follows: for any open subset , is the algebra of functions such that the composition is a regular function that is constant along . (The last condition is imposed only if , in which case .)
In each of the following examples, determine whether the quotient is an algebraic variety; if it is, describe it explicitly.
 , (embedded as a line in ).
 , .
 , is a twopoint set (if you want a more challenging version, let be any finite set).