Difference between revisions of "Math 764  Algebraic Geometry II  Homeworks"
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−  == Homeworks ==  +  == Homeworks (Spring 2017) == 
+  
+  Here are homework problems for [[Math 764  Algebraic Geometry IIMath 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 1Homework 1]] ([[Media:Math764S17HW1.pdfPDF]]), due February 3rd.  
+  * [[#Homework 2Homework 2]] ([[Media:Math764S17HW2.pdfPDF]]), due February 10th.  
+  * [[#Homework 3Homework 3]] ([[Media:Math764S17HW3.pdfPDF]]), due February 17th.  
+  * [[#Homework 4Homework 4]] ([[Media:Math764S17HW4.pdfPDF]]), due February 24th.  
+  * [[#Homework 5Homework 5]] ([[Media:Math764S17HW5.pdfPDF]]), due March 3rd.  
+  * [[#Homework 6Homework 6]] ([[Media:Math764S17HW6.pdfPDF]]), due March 10th.  
+  * [[#Homework 7Homework 7]] ([[Media:Math764S17HW7.pdfPDF]]), due March 31th.  
+  * [[#Homework 8Homework 8]] ([[Media:Math764S17HW8.pdfPDF]]), due April 7th.  
+  * [[#Homework 9Homework 9]] ([[Media:Math764S17HW9.pdfPDF]]), due April 21st.  
−  
−  
=== 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>.  
+  
+  
+  === 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 finite collection of rings. Put <math>R=\prod_\alpha R_\alpha</math>. Describe the topological space <math>{\mathop{\mathrm{Spec}}}(R)</math> in terms of <math>{\mathop{\mathrm{Spec}}}(R_\alpha)</math>’s. What changes if the collection is infinite?  
+  # 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. (This part may be very hard.) </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).  
+  
+  === Homework 3 ===  
+  
+  Due Friday, February 17th  
+  
+  #(Gluing morphisms of sheaves) Let <math>F</math> and <math>G</math> be two sheaves on the same space <math>X</math>. For any open set <math>U\subset X</math>, consider the restriction sheaves <math>F_U</math> and <math>G_U</math>, and let <math>Hom(F_U,G_U)</math> be the set of sheaf morphisms between them.<p>Prove that the presheaf on <math>X</math> given by the correspondence <math>U\mapsto Hom(F_U,G_U)</math> is in fact a sheaf.</p>  
+  #(Gluing morphisms of ringed spaces) Let <math>X</math> and <math>Y</math> be ringed spaces. Denote by <math>\underline{Mor}(X,Y)</math> the following presheaf on <math>X</math>: its sections over an open subset <math>U\subset X</math> are morphisms of ringed spaces <math>U\to Y</math> where <math>U</math> is considered as a ringed space. (And the notion of restriction is the natural one.) Show that <math>\underline{Mor}(X,Y)</math> is in fact a sheaf.  
+  #(Affinization of a scheme) Let <math>X</math> be an arbitrary scheme. Prove that there exists an affine scheme <math>X_{aff}</math> and a morphism <math>X\to X_{aff}</math> that is universal in the following sense: any map form <math>X</math> to an affine scheme factors through it.  
+  #Let us consider direct and inverse limits of affine schemes. For simplicity, we will work with limits indexed by positive integers.<p>(a) Let <math>R_i</math> be a collection of rings (<math>i>0</math>) together with homomorphisms <math>R_i\to R_{i+1}</math>. Consider the direct limit <math>R:=\lim\limits_{\longrightarrow} R_i</math>. Show that in the category of schemes, <math>{\mathop{\mathrm{Spec}}}(R)=\lim\limits_{\longleftarrow}{\mathop{\mathrm{Spec}}}R_i.</math></p><p>(b) Let <math>R_i</math> be a collection of rings (<math>i>0</math>) together with homomorphisms <math>R_{i+1}\to R_i</math>. Consider the inverse limit <math>R:=\lim\limits_{\longleftarrow} R_i</math>. Show that generally speaking, in the category of schemes, <math>{\mathop{\mathrm{Spec}}}(R)\neq\lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i.</math>  
+  #Here is an example of the situation from 4(b). Let <math>k</math> be a field, and let <math>R_i=k[t]/(t^i)</math>, so that <math>\lim\limits_{\longleftarrow} R_i=k[[t]]</math>. Describe the direct limit <math>\lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i</math> in the category of ringed spaces. Is the direct limit a scheme?  
+  #Let <math>S</math> be a finite partially ordered set. Consider the following topology on <math>S</math>: a subset <math>U\subset S</math> is open if and only if whenever <math>x\in U</math> and <math>y>x</math>, it must be that <math>y\in U</math>. </p><p>Construct a ring <math>R</math> such that <math>\mathop{\mathrm{Spec}}(R)</math> is homeomorphic to <math>S</math>. </p>  
+  #Show that any quasicompact scheme has closed points. (It is not true that any scheme has closed points!)  
+  #Give an example of a scheme that has no open connected subsets. In particular, such a scheme is not locally connected. Of course, my convention here is that the empty set is not connected...  
+  
+  === Homework 4 ===  
+  
+  Due Friday, February 24th  
+  
+  #Show that the following two definitions of quasiseparatedness of a scheme <math>S</math> are equivalent:  
+  ##The intersection of any two quasicompact open subsets of <math>S</math> is quasicompact;  
+  ##There is a cover of <math>S</math> by affine open subsets whose (pairwise) intersections are quasicompact.  
+  #In class, we gave the following definition: a scheme <math>S</math> is ''integral'' if it is irreducible and reduced. Show that this is equivalent to the definition from Vakil’s notes: a scheme is integral if for any nonempty open <math>U\subset S</math>, <math>O_S(U)</math> is a domain.  
+  #Let us call a scheme <math>X</math> ''locally irreducible'' if every point has an irreducible neighborhood. (Since a nonempty open subset of an irreducible space is irreducible, this implies that all smaller neighborhoods of this point are irreducible as well.) Prove or disprove the following claim: a scheme is irreducible if and only if it is connected and locally irreducible.  
+  #Show that a locally Noetherian scheme is quasiseparated.  
+  #Show that the following two definitions of a Noetherian scheme <math>X</math> are equivalent:  
+  ## <math>X</math> is a finite union of open affine sets, each of which is the spectrum of a Noetherian ring;  
+  ## <math>X</math> is quasicompact and locally Noetherian.  
+  #Show that any Noetherian scheme <math>X</math> is a disjoint union of finitely many connected open subsets (the ''connected components'' of <math>X</math>.) (A problem from the last homework shows that things might go wrong if we do not assume that <math>X</math> is Noetherian.)  
+  #A locally closed subscheme <math>X\subset Y</math> is defined as a closed subscheme of an open subscheme of <math>Y</math>. Accordingly, a locally closed embedding is a composition of a closed embedding followed by an open embedding (in this order). In principle, one can try to reverse the order, and consider open subschemes of closed subschemes of <math>Y</math>. Does this yield an equivalent definition?  
+  
+  <span>''Remark.''</span> The difficulty of such questions (and, sometimes, the answer to them) depends on the class of schemes one works with: often, very mild assumptions (such as, say, quasicompactness) would make the question easy. A complete answer to this problem would include both the mild assumptions that would make the two versions equivalent, and a description of what happens for general schemes.  
+  
+  === Homework 5 ===  
+  
+  Due Friday, March 3rd  
+  
+  # Fix a field <math>k</math>, and put <math>X={\mathop{Spec}}k[x]</math> and <math>Y={\mathop{Spec}}k[y]</math>. Consider the morphism <math>f:X\to Y</math> given by <math>y=x^2</math>. Describe the fiber product <math>X\times_YX</math> as explicitly as possible. (The answer may depend on <math>k</math>.)  
+  # (The Frobenius morphism.) Let <math>X</math> be a scheme of characteristic <math>p</math>: by definition, this means that <math>p=0</math> in the structure sheaf of <math>X</math>. Define the (absolute) Frobenius morphism <math>Fr_X:X\to X</math> as follows: it is the identity map on the underlying set, and the pullback <math>Fr_X^*(f)</math> equals <math>f^p</math> for any (local) function <math>f\in{\mathcal{O}}_X</math>.<p>Verify that this defines an affine morphism of schemes. Assuming <math>X</math> is a scheme locally of finite type over a perfect field, verify that <math>Fr_X</math> is a morphism of finite type (it is in fact finite, if you know what it means).</p>  
+  # (The relative Frobenius morphism.) Let <math>X\to Y</math> be a morphism of schemes of characteristic <math>p</math>. Put <math>\overline X:=X\times_{Y,Fr_Y}Y,</math> where the notation means that <math>Y</math> is considered as a <math>Y</math>scheme via the Frobenius map.  
+  ## Show that the Frobenius morphism <math>Fr_X</math> naturally factors as the composition <math>X\to\overline{X}\to X</math>, where the first map <math>X\to\overline{X}</math> is naturally a morphism of schemes over <math>Y</math> (while the second map, generally speaking, is not). The map <math>X\to\overline{X}</math> is called the ''relative'' Frobenius morphism.  
+  ## Suppose <math>Y={\mathop{Spec}}(\overline{\mathbb{F}}_p)</math>, and <math>X</math> is an affine variety (that is, an affine reduced scheme of finite type) over <math>\overline{\mathbb{F}}_p</math>. Describe <math>\overline X</math> and the relative Frobenius <math>X\to\overline{X}</math> explicitly in coordinates.  
+  # Let <math>X</math> be a scheme over <math>\mathbb{F}_p</math>. In this case, the absolute Frobenius <math>Fr_X:X\to X</math> is a morphism of schemes over <math>\mathbb{F}_p</math> (and it coincides with the relative Frobenius of <math>X</math> over <math>\mathbb{F}_p</math>. <p> Consider the extension of scalars <math>X'=X_{\overline{\mathbb{F}}_p}=X\otimes_{\mathbb{F}_p}\overline{\mathbb{F}}_p=X\times_{{\mathop{Spec}}(\mathbb{F}_p)}{\mathop{Spec}}(\overline{\mathbb{F}}_p).</math> Then <math>Fr_X</math> naturally extends to a morphism of <math>\overline{\mathbb{F}}_p</math>schemes <math>X'\to X'</math>. Compare the map <math>X'\to X'</math> with the relative Frobenius of <math>X'</math> over <math>\overline{\mathbb{F}}_p</math>.</p>  
+  # A morphism of schemes is ''surjective'' if it is surjective as a morphism of sets. Show that surjectivity is preserved by base changes. That is, if <math>f:X\to Z</math> is surjective and <math>g:Y\to Z</math> is arbitrary, then <math>X\times_ZY\to Y</math> is surjective.  
+  # (Normalization) A scheme is ''normal'' if all of its local rings are integrally closed domains. Let <math>X</math> be an integral scheme. Show that there exists a normal integral scheme <math>\tilde{X}</math> together with a morphism <math>\tilde{X}\to X</math> that is universal in the following sense: any dominant morphism <math>Y\to X</math> from a normal integral scheme to <math>X</math> factors through <math>\tilde{X}</math>. (Just like in the case of varieties, a morphism is ''dominant'' if its image is dense.)  
+  # Let <math>X</math> be a scheme of finite type over a field <math>k</math>. For every field extension <math>K\supset k</math>, put <math>X_K:=X\otimes_kK={\mathop{Spec}}(K)\times_{{\mathop{Spec}}(k)}X.</math> </p><p> Show that <math>X</math> is geometrically irreducible (that is, the morphism <math>X\to{\mathop{Spec}}(k)</math> has geometrically irreducible fibers) if and only if <math>X_K</math> is irreducible for all ''finite'' extensions <math>K\supset k</math>.</p>  
+  
+  === Homework 6 ===  
+  
+  Due Friday, March 10th  
+  
+  <span>''Sheaves of modules on ringed spaces.''</span>  
+  
+  Let <math>(X,{\mathcal{O}}_X)</math> be a ringed space, and let <math>{\mathcal{F}}</math> and <math>{\mathcal{G}}</math> be sheaves of <math>{\mathcal{O}}_X</math>modules. The ''tensor product'' of <math>{\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}}</math> is the sheafification of the presheaf <math>U\mapsto{\mathcal{F}}(U)\otimes_{{\mathcal{O}}_X(U)}{\mathcal{G}}(U).</math>  
+  
+  # Prove that the stalks of <math>{\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}}</math> are given by the tensor product: <math>({\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}})_x={\mathcal{F}}_x\otimes_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x,</math> where <math>x\in X</math>. Conclude that the tensor product is a right exact functor (in each of the two arguments).  
+  # Suppose that <math>{\mathcal{F}}</math> is locally free of finite rank. (That is to say, every point <math>x\in X</math> has a neighborhood <math>U</math> such that <math>{\mathcal{F}}_U\simeq({\mathcal{O}}_U)^n</math>. Prove that there exists a natural isomorphism <math>{{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{G}})={\mathcal{G}}\otimes{\mathcal{F}}^\vee.</math> Here <math>{\mathcal{F}}^\vee={\mathop{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{O}}_X)</math> is the dual of the locally free sheaf <math>{\mathcal{F}}</math>, and <math>{\mathop{\mathcal{H}\mathit{om}}}</math> is the sheaf of homomorphisms. (Note that <math>{\mathcal{G}}</math> is not assumed to be quasicoherent.)  
+  # (Projection formula) Let <math>f:(X,{\mathcal{O}}_X)\to(Y,{\mathcal{O}}_Y)</math> be a morphism of ringed spaces. Suppose <math>{\mathcal{F}}</math> is an <math>{\mathcal{O}}_X</math>module and <math>{\mathcal{G}}</math> is a locally free <math>{\mathcal{O}}_Y</math>module of finite rank. Construct a natural isomorphism <math>f_*({\mathcal{F}}\otimes_{{\mathcal{O}}_X} f^*{\mathcal{G}})\simeq f_*({\mathcal{F}})\otimes_{{\mathcal{O}}_Y}{\mathcal{G}}.</math><p><span>''Coherent sheaves on a noetherian scheme''</span></p>  
+  # Let <math>{\mathcal{F}}</math> be a coherent sheaf on a loclly noetherian scheme <math>X</math>.<p>Show that <math>{\mathcal{F}}</math> is locally free if and only if its stalks <math>{\mathcal{F}}_x</math> are free <math>{\mathcal{O}}_{X,x}</math>modules for all <math>x\in X</math>.</p> <p>(b) Show that <math>{\mathcal{F}}</math> is locally free of rank one if and only if it is ''invertible'': there exists a coherent sheaf <math>{\mathcal{G}}</math> such that <math>{\mathcal{F}}\otimes{\mathcal{G}}\simeq{\mathcal{O}}_X</math>.</p>  
+  # As in the previous problem, supposed <math>{\mathcal{F}}</math> be a coherent sheaf on a locally noetherian scheme <math>X</math>. The ''fiber'' of <math>{\mathcal{F}}</math> at a point <math>x\in X</math> is the <math>k(x)</math>vector space <math>i^*{\mathcal{F}}</math> for the natural map <math>i:{\mathop{Spec}}(k(x))\to X</math> (where <math>k(x)</math> is the residue field of <math>x\in X</math>). Denote by <math>\phi(x)</math> the dimension <math>\dim_{k(x)} i^*{\mathcal{F}}</math>.<p>(a) Show that the function <math>\phi(x)</math> is upper semicontinuous: for every <math>n</math>, the set <math>\{x\in X:\phi(x)\ge n\}</math> is closed.</p><p>(b) Suppose <math>X</math> is reduced. Show that <math>{\mathcal{F}}</math> is locally free if and only if <math>\phi(x)</math> is constant on each connected component of <math>X</math>. (Do you see why we impose the assumption that <math>X</math> is reduced here?)</p>  
+  # Let <math>X</math> be a locally noetherian scheme and let <math>U\subset X</math> be an open subset. Show that any coherent sheaf <math>{\mathcal{F}}</math> on <math>U</math> can be extended to a coherent sheaf on <math>\overline{{\mathcal{F}}}</math> on <math>X</math>. (We say that <math>\overline{{\mathcal{F}}}</math> is an ''extension'' of <math>{\mathcal{F}}</math> if <math>\overline{{\mathcal{F}}}_U\simeq{\mathcal{F}}</math>.) <p>(If you need a hint for this problem, look at Problem II.5.15 in Hartshorne.)</p>  
+  
+  ===Homework 7===  
+  
+  Due Friday, March 31st  
+  
+  <span>''Proper and separated morphisms.''</span>  
+  
+  Each scheme <math>X</math> has a maximal closed reduced subscheme <math>X^{red}</math>; the ideal sheaf of <math>X^{red}</math> is the nilradical (the sheaf of all nilpotents in <math>{\mathcal{O}}_X</math>).  
+  
+  # Let <math>f:X\to Y</math> be a morphism of schemes of finite type. Consider the induced map <math>f^{red}:X^{red}\to Y^{red}</math>. Prove that <math>f</math> is separated (resp. proper) if and only if <math>f^{red}</math> is separated (resp. proper). <p><span>''Vector bundles.''</span></p><p>Fix an algebraically closed field <math>k</math>. Any vector bundle on <math>{\mathbb{A}}^1_k={\mathop{Spec}}(k[t])</math> is trivial, you can use this without proof. Let <math>X</math> be the ‘affine line with a doubled point’ obtained by gluing two copies of <math>{\mathbb{A}}^1_k</math> away from the origin.</p>  
+  # Classify line bundles on <math>X</math> up to isomorphism.  
+  # (Could be hard) Prove that any vector bundle on <math>X</math> is a direct sum of several line bundles.<p><span>''Tangent bundle.''</span></p>  
+  # Let <math>X</math> be an irreducible affine variety, not necessarily smooth. Let <math>M</math> be the <math>k[X]</math>module of <math>k</math>linear derivations <math>k[X]\to k[X]</math>. (These are globally defined vector fields on <math>X</math>, but keep in mind that <math>X</math> may be singular.) Consider its generic rank <math>r:=\dim_{k(X)}M\otimes_{k[X]}k(X)</math>. Show that <math>r=\dim(X)</math>.  
+  # Suppose now that <math>X</math> is smooth. Show that the module <math>M</math> is a locally free coherent module; the corresponding vector bundle is the ''tangent bundle'' <math>TX</math>.  
+  # Let <math>f:X\to Y</math> be a morphism of algebraic varieties. Recall that a vector bundle <math>E</math> over <math>Y</math> gives a vector bundle <math>f^*E</math> on <math>X</math> whose total space is the fiber product <math>E\times_YX</math>.  
+  # Suppose now that <math>X</math> and <math>Y</math> are affine and <math>Y</math> is smooth. Let <math>E=TY</math> be the tangent bundle to <math>Y</math>. Show that the space of <math>k</math>linear derivations <math>k[Y]\to k[X]</math> (where <math>f</math> is used to equip <math>k[X]</math> with the structure of a <math>k[Y]</math>module) is identified with <math>\Gamma(X,f^*(TY))</math>.  
+  # Let <math>X</math> be a smooth affine variety. Let <math>I_\Delta\subset k[X\times X]</math> be the ideal sheaf of the diagonal <math>\Delta\subset X\times X</math>. Prove that there is a bijection <math>I_\Delta/I_\Delta^2=\Gamma(X,\Omega^1_X),</math> where <math>\Omega^1_X</math> is the sheaf of differential 1forms (that is, the sheaf of sections of the cotangent bundle <math>T^\vee X</math>, which is the dual vector bundle of <math>TX</math>).  
+  
+  ===Homework 8===  
+  
+  Due Friday, April 7th  
+  
+  # (Hartshorne, II.4.4) Fix a Noetherian scheme <math>S</math>, let <math>X</math> and <math>Y</math> be schemes of finite type and separated over <math>S</math>, and let <math>f:X\to Y</math> be a morphism of <math>S</math>schemes. Suppose that <math>Z\subset X</math> be a closed subscheme that is proper over <math>S</math>. Show that <math>f(Z)\subset Y</math> is closed.  
+  # In the setting of the previous problem, show that if we consider <math>f(Z)</math> as a closed subscheme (its ideal of functions consists of all functions whose composition with <math>f</math> is zero), then <math>f</math> induces a proper map fro <math>Z</math> to <math>f(Z)</math>. <p>(Galois descent, inspired by Hartshorne II.4.7) Let <math>F/k</math> be a finite Galois extension of fields. The Galois group <math>G:=Gal(F/k)</math> acts on the scheme <math>{\mathop{Spec}}(F)</math>. Given any <math>k</math>scheme <math>X</math>, we let <math>X_F:={\mathop{Spec}}(F)\times_{{\mathop{Spec}}(k)}X</math> be its extension of scalars; the group <math>G</math> acts on <math>X_F</math> in a way compatible with its action on <math>{\mathop{Spec}}(F)</math> (i.e., this action is ‘semilinear’).</p>  
+  # Show that <math>X</math> is affine if and only if <math>X_F</math> is affine.  
+  # Prove that this operation gives a fully faithful functor from the category of <math>k</math>schemes into the category of <math>F</math>schemes with a semilinear action of <math>G</math>.  
+  # Suppose that <math>Y</math> is a separated <math>F</math>scheme such that any finite subset of <math>Y</math> is contained in an affine open chart (this holds, for instance, if <math>Y</math> is quasiprojective). Then for any semilinear action of <math>G</math> on <math>Y</math>, there exists a <math>k</math>scheme <math>X</math> and an isomorphism <math>X_F\simeq Y</math> that agrees with an action of <math>G</math>. (That is, the action of <math>G</math> gives a <math>k</math>structure on the scheme <math>Y</math>.)  
+  # Suppose <math>X</math> is an <math>{\mathbb{R}}</math>scheme such that <math>X_{\mathbb{C}}\simeq{\mathbb{A}}_{\mathbb{C}}^1</math>. Show that <math>X\simeq{\mathbb{A}}^1_{\mathbb{R}}</math>.  
+  # Suppose <math>X</math> is an <math>{\mathbb{R}}</math>scheme such that <math>X_{\mathbb{C}}\simeq{\mathbb{P}}_{\mathbb{C}}^1</math>. Show that there are two possibilities for the isomorphism class of <math>X</math>.  
+  
+  
+  ===Homework 9===  
+  
+  Due Friday, April 21st  
+  
+  # Let <math>X</math> be a singular cubic in <math>{\mathbb{P}}^2</math>, given (in nonhomogeneous coordinates) either by <math>y^2=x^3+x^2</math> (nodal cubic) or by <math>y^2=x^3</math> (cuspidal cubic). Compute the class group of Cartier divisors on <math>X</math>.  
+  # Let <math>X</math> and <math>Y</math> be schemes over some base scheme <math>S</math>. For any map <math>f:X\to Y</math>, use the functoriality of the module of Kähler differentials to construct a morphism <math>f^*\Omega_{Y/S}\to\Omega_{X/S}</math> and verify that <math>\Omega_{X/Y}=\mathrm{coker}(f^*\Omega_{Y/S}\to\Omega_{X/S})</math>.  
+  # Suppose now that <math>X</math> and <math>Y</math> be schemes over an algebraically closed field <math>k</math>. A morphism <math>f:X\to Y</math> is ''unramified'' if <math>\Omega_{X/Y}=0</math>. Show that this is equivalent to the following condition: given <math>D={\mathop{Spec}}k[\epsilon]/{\epsilon^2}</math>, the map <math>f</math> induces an injection <math>\mathrm{Maps}(D,X)\to\mathrm{Maps}(D,Y)</math>.  
+  # Let us compute the algebraic de Rham cohomology of the affine space. Put <math>X={\mathop{Spec}}R</math>, <math>R=k[t_1,\dots,t_n]</math>. Since <math>X</math> is a smooth <math>k</math>scheme, <math>\Omega^1_R=\Omega_{R/k}</math> is a locally free <math>R</math>module. Denote by <math>\Omega^\bullet_R</math> the exterior algebra of <math>\Omega^1_R</math>, so that <math>\Omega^i_R=\bigwedge^i\Omega^1_R</math>. Define the de Rham differential <math>d:\Omega^i_R\to\Omega^{i+1}_R</math> by starting with the Kähler differential <math>d:R\to\Omega^1_R</math> and then extending it by the graded Leibniz rule: <math>d(\omega_1\wedge\omega_2)=(d\omega_1)\wedge\omega_2+(1)^i\omega_1\wedge d(\omega_2),\qquad \omega_1\in\Omega^i_R.</math><p> Compute the cohomology of the complex <math>\Omega^\bullet_R</math> equipped with the differential <math>d</math>. The answer will depend on the characteristic of <math>k</math>.</p>  
+  # Let <math>X</math> be a Noetherian scheme.Let <math>K(X)</math> be the ''<math>K</math>group of <math>X</math>'': it is generated by elements <math>[F]</math> for each coherent sheaf <math>F</math> with relations <math>[F]=[F_1]+[F_2]</math> whenever there is a short exact sequence <math>0\to F_1\to F_2\to F_3\to 0.</math> Prove that <math>K({\mathbb{A}}^n)={\mathbb{Z}}</math>. (This is much easier if you know Hilbert’s Syzygy Theorem.)  
+  # Let <math>X</math> be a smooth curve over an algebraically closed field. Show that <math>K(X)</math> is generated by <math>[L]</math> for line bundles <math>L</math>.  
+  # Let <math>X</math> be a smooth curve over an algebraically closed field. Show that <math>K(X)</math> is isomorphic to <math>{\mathbb{Z}}\oplus \mathrm{Pic}(X)</math>. (If this problem is too hard, look at Hartshorne’s II.6.11 for a stepbystep approach.) 
Latest revision as of 12:27, 15 April 2017
Contents
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 3 (PDF), due February 17th.
 Homework 4 (PDF), due February 24th.
 Homework 5 (PDF), due March 3rd.
 Homework 6 (PDF), due March 10th.
 Homework 7 (PDF), due March 31th.
 Homework 8 (PDF), due April 7th.
 Homework 9 (PDF), due April 21st.
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]?

Locally constant sheaves:
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].
[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.
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].)
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].)
 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]
(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].)
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 [math]R_\alpha[/math] be a finite collection of rings. Put [math]R=\prod_\alpha R_\alpha[/math]. Describe the topological space [math]{\mathop{\mathrm{Spec}}}(R)[/math] in terms of [math]{\mathop{\mathrm{Spec}}}(R_\alpha)[/math]’s. What changes if the collection is infinite?
 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
(a) An infinite intersection of open sets, but not constructible.
(b) An infinite union of closed sets, but not constructible. (This part may be very hard.)
Contraction of a subvariety.
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:
 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].)
In each of the following examples, determine whether the quotient [math]X/Y[/math] is an algebraic variety; if it is, describe it explicitly.
 [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).
Homework 3
Due Friday, February 17th
 (Gluing morphisms of sheaves) Let [math]F[/math] and [math]G[/math] be two sheaves on the same space [math]X[/math]. For any open set [math]U\subset X[/math], consider the restriction sheaves [math]F_U[/math] and [math]G_U[/math], and let [math]Hom(F_U,G_U)[/math] be the set of sheaf morphisms between them.
Prove that the presheaf on [math]X[/math] given by the correspondence [math]U\mapsto Hom(F_U,G_U)[/math] is in fact a sheaf.
 (Gluing morphisms of ringed spaces) Let [math]X[/math] and [math]Y[/math] be ringed spaces. Denote by [math]\underline{Mor}(X,Y)[/math] the following presheaf on [math]X[/math]: its sections over an open subset [math]U\subset X[/math] are morphisms of ringed spaces [math]U\to Y[/math] where [math]U[/math] is considered as a ringed space. (And the notion of restriction is the natural one.) Show that [math]\underline{Mor}(X,Y)[/math] is in fact a sheaf.
 (Affinization of a scheme) Let [math]X[/math] be an arbitrary scheme. Prove that there exists an affine scheme [math]X_{aff}[/math] and a morphism [math]X\to X_{aff}[/math] that is universal in the following sense: any map form [math]X[/math] to an affine scheme factors through it.
 Let us consider direct and inverse limits of affine schemes. For simplicity, we will work with limits indexed by positive integers.
(a) Let [math]R_i[/math] be a collection of rings ([math]i\gt 0[/math]) together with homomorphisms [math]R_i\to R_{i+1}[/math]. Consider the direct limit [math]R:=\lim\limits_{\longrightarrow} R_i[/math]. Show that in the category of schemes, [math]{\mathop{\mathrm{Spec}}}(R)=\lim\limits_{\longleftarrow}{\mathop{\mathrm{Spec}}}R_i.[/math]
(b) Let [math]R_i[/math] be a collection of rings ([math]i\gt 0[/math]) together with homomorphisms [math]R_{i+1}\to R_i[/math]. Consider the inverse limit [math]R:=\lim\limits_{\longleftarrow} R_i[/math]. Show that generally speaking, in the category of schemes, [math]{\mathop{\mathrm{Spec}}}(R)\neq\lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i.[/math]
 Here is an example of the situation from 4(b). Let [math]k[/math] be a field, and let [math]R_i=k[t]/(t^i)[/math], so that [math]\lim\limits_{\longleftarrow} R_i=k[[t]][/math]. Describe the direct limit [math]\lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i[/math] in the category of ringed spaces. Is the direct limit a scheme?
 Let [math]S[/math] be a finite partially ordered set. Consider the following topology on [math]S[/math]: a subset [math]U\subset S[/math] is open if and only if whenever [math]x\in U[/math] and [math]y\gt x[/math], it must be that [math]y\in U[/math].
Construct a ring [math]R[/math] such that [math]\mathop{\mathrm{Spec}}(R)[/math] is homeomorphic to [math]S[/math].
 Show that any quasicompact scheme has closed points. (It is not true that any scheme has closed points!)
 Give an example of a scheme that has no open connected subsets. In particular, such a scheme is not locally connected. Of course, my convention here is that the empty set is not connected...
Homework 4
Due Friday, February 24th
 Show that the following two definitions of quasiseparatedness of a scheme [math]S[/math] are equivalent:
 The intersection of any two quasicompact open subsets of [math]S[/math] is quasicompact;
 There is a cover of [math]S[/math] by affine open subsets whose (pairwise) intersections are quasicompact.
 In class, we gave the following definition: a scheme [math]S[/math] is integral if it is irreducible and reduced. Show that this is equivalent to the definition from Vakil’s notes: a scheme is integral if for any nonempty open [math]U\subset S[/math], [math]O_S(U)[/math] is a domain.
 Let us call a scheme [math]X[/math] locally irreducible if every point has an irreducible neighborhood. (Since a nonempty open subset of an irreducible space is irreducible, this implies that all smaller neighborhoods of this point are irreducible as well.) Prove or disprove the following claim: a scheme is irreducible if and only if it is connected and locally irreducible.
 Show that a locally Noetherian scheme is quasiseparated.
 Show that the following two definitions of a Noetherian scheme [math]X[/math] are equivalent:
 [math]X[/math] is a finite union of open affine sets, each of which is the spectrum of a Noetherian ring;
 [math]X[/math] is quasicompact and locally Noetherian.
 Show that any Noetherian scheme [math]X[/math] is a disjoint union of finitely many connected open subsets (the connected components of [math]X[/math].) (A problem from the last homework shows that things might go wrong if we do not assume that [math]X[/math] is Noetherian.)
 A locally closed subscheme [math]X\subset Y[/math] is defined as a closed subscheme of an open subscheme of [math]Y[/math]. Accordingly, a locally closed embedding is a composition of a closed embedding followed by an open embedding (in this order). In principle, one can try to reverse the order, and consider open subschemes of closed subschemes of [math]Y[/math]. Does this yield an equivalent definition?
Remark. The difficulty of such questions (and, sometimes, the answer to them) depends on the class of schemes one works with: often, very mild assumptions (such as, say, quasicompactness) would make the question easy. A complete answer to this problem would include both the mild assumptions that would make the two versions equivalent, and a description of what happens for general schemes.
Homework 5
Due Friday, March 3rd
 Fix a field [math]k[/math], and put [math]X={\mathop{Spec}}k[x][/math] and [math]Y={\mathop{Spec}}k[y][/math]. Consider the morphism [math]f:X\to Y[/math] given by [math]y=x^2[/math]. Describe the fiber product [math]X\times_YX[/math] as explicitly as possible. (The answer may depend on [math]k[/math].)
 (The Frobenius morphism.) Let [math]X[/math] be a scheme of characteristic [math]p[/math]: by definition, this means that [math]p=0[/math] in the structure sheaf of [math]X[/math]. Define the (absolute) Frobenius morphism [math]Fr_X:X\to X[/math] as follows: it is the identity map on the underlying set, and the pullback [math]Fr_X^*(f)[/math] equals [math]f^p[/math] for any (local) function [math]f\in{\mathcal{O}}_X[/math].
Verify that this defines an affine morphism of schemes. Assuming [math]X[/math] is a scheme locally of finite type over a perfect field, verify that [math]Fr_X[/math] is a morphism of finite type (it is in fact finite, if you know what it means).
 (The relative Frobenius morphism.) Let [math]X\to Y[/math] be a morphism of schemes of characteristic [math]p[/math]. Put [math]\overline X:=X\times_{Y,Fr_Y}Y,[/math] where the notation means that [math]Y[/math] is considered as a [math]Y[/math]scheme via the Frobenius map.
 Show that the Frobenius morphism [math]Fr_X[/math] naturally factors as the composition [math]X\to\overline{X}\to X[/math], where the first map [math]X\to\overline{X}[/math] is naturally a morphism of schemes over [math]Y[/math] (while the second map, generally speaking, is not). The map [math]X\to\overline{X}[/math] is called the relative Frobenius morphism.
 Suppose [math]Y={\mathop{Spec}}(\overline{\mathbb{F}}_p)[/math], and [math]X[/math] is an affine variety (that is, an affine reduced scheme of finite type) over [math]\overline{\mathbb{F}}_p[/math]. Describe [math]\overline X[/math] and the relative Frobenius [math]X\to\overline{X}[/math] explicitly in coordinates.
 Let [math]X[/math] be a scheme over [math]\mathbb{F}_p[/math]. In this case, the absolute Frobenius [math]Fr_X:X\to X[/math] is a morphism of schemes over [math]\mathbb{F}_p[/math] (and it coincides with the relative Frobenius of [math]X[/math] over [math]\mathbb{F}_p[/math].
Consider the extension of scalars [math]X'=X_{\overline{\mathbb{F}}_p}=X\otimes_{\mathbb{F}_p}\overline{\mathbb{F}}_p=X\times_{{\mathop{Spec}}(\mathbb{F}_p)}{\mathop{Spec}}(\overline{\mathbb{F}}_p).[/math] Then [math]Fr_X[/math] naturally extends to a morphism of [math]\overline{\mathbb{F}}_p[/math]schemes [math]X'\to X'[/math]. Compare the map [math]X'\to X'[/math] with the relative Frobenius of [math]X'[/math] over [math]\overline{\mathbb{F}}_p[/math].
 A morphism of schemes is surjective if it is surjective as a morphism of sets. Show that surjectivity is preserved by base changes. That is, if [math]f:X\to Z[/math] is surjective and [math]g:Y\to Z[/math] is arbitrary, then [math]X\times_ZY\to Y[/math] is surjective.
 (Normalization) A scheme is normal if all of its local rings are integrally closed domains. Let [math]X[/math] be an integral scheme. Show that there exists a normal integral scheme [math]\tilde{X}[/math] together with a morphism [math]\tilde{X}\to X[/math] that is universal in the following sense: any dominant morphism [math]Y\to X[/math] from a normal integral scheme to [math]X[/math] factors through [math]\tilde{X}[/math]. (Just like in the case of varieties, a morphism is dominant if its image is dense.)
 Let [math]X[/math] be a scheme of finite type over a field [math]k[/math]. For every field extension [math]K\supset k[/math], put [math]X_K:=X\otimes_kK={\mathop{Spec}}(K)\times_{{\mathop{Spec}}(k)}X.[/math] </p>
Show that [math]X[/math] is geometrically irreducible (that is, the morphism [math]X\to{\mathop{Spec}}(k)[/math] has geometrically irreducible fibers) if and only if [math]X_K[/math] is irreducible for all finite extensions [math]K\supset k[/math].
Homework 6
Due Friday, March 10th
Sheaves of modules on ringed spaces.
Let [math](X,{\mathcal{O}}_X)[/math] be a ringed space, and let [math]{\mathcal{F}}[/math] and [math]{\mathcal{G}}[/math] be sheaves of [math]{\mathcal{O}}_X[/math]modules. The tensor product of [math]{\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}}[/math] is the sheafification of the presheaf [math]U\mapsto{\mathcal{F}}(U)\otimes_{{\mathcal{O}}_X(U)}{\mathcal{G}}(U).[/math]
 Prove that the stalks of [math]{\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}}[/math] are given by the tensor product: [math]({\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}})_x={\mathcal{F}}_x\otimes_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x,[/math] where [math]x\in X[/math]. Conclude that the tensor product is a right exact functor (in each of the two arguments).
 Suppose that [math]{\mathcal{F}}[/math] is locally free of finite rank. (That is to say, every point [math]x\in X[/math] has a neighborhood [math]U[/math] such that [math]{\mathcal{F}}_U\simeq({\mathcal{O}}_U)^n[/math]. Prove that there exists a natural isomorphism [math]{{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{G}})={\mathcal{G}}\otimes{\mathcal{F}}^\vee.[/math] Here [math]{\mathcal{F}}^\vee={\mathop{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{O}}_X)[/math] is the dual of the locally free sheaf [math]{\mathcal{F}}[/math], and [math]{\mathop{\mathcal{H}\mathit{om}}}[/math] is the sheaf of homomorphisms. (Note that [math]{\mathcal{G}}[/math] is not assumed to be quasicoherent.)
 (Projection formula) Let [math]f:(X,{\mathcal{O}}_X)\to(Y,{\mathcal{O}}_Y)[/math] be a morphism of ringed spaces. Suppose [math]{\mathcal{F}}[/math] is an [math]{\mathcal{O}}_X[/math]module and [math]{\mathcal{G}}[/math] is a locally free [math]{\mathcal{O}}_Y[/math]module of finite rank. Construct a natural isomorphism [math]f_*({\mathcal{F}}\otimes_{{\mathcal{O}}_X} f^*{\mathcal{G}})\simeq f_*({\mathcal{F}})\otimes_{{\mathcal{O}}_Y}{\mathcal{G}}.[/math]
Coherent sheaves on a noetherian scheme
 Let [math]{\mathcal{F}}[/math] be a coherent sheaf on a loclly noetherian scheme [math]X[/math].
Show that [math]{\mathcal{F}}[/math] is locally free if and only if its stalks [math]{\mathcal{F}}_x[/math] are free [math]{\mathcal{O}}_{X,x}[/math]modules for all [math]x\in X[/math].
(b) Show that [math]{\mathcal{F}}[/math] is locally free of rank one if and only if it is invertible: there exists a coherent sheaf [math]{\mathcal{G}}[/math] such that [math]{\mathcal{F}}\otimes{\mathcal{G}}\simeq{\mathcal{O}}_X[/math].
 As in the previous problem, supposed [math]{\mathcal{F}}[/math] be a coherent sheaf on a locally noetherian scheme [math]X[/math]. The fiber of [math]{\mathcal{F}}[/math] at a point [math]x\in X[/math] is the [math]k(x)[/math]vector space [math]i^*{\mathcal{F}}[/math] for the natural map [math]i:{\mathop{Spec}}(k(x))\to X[/math] (where [math]k(x)[/math] is the residue field of [math]x\in X[/math]). Denote by [math]\phi(x)[/math] the dimension [math]\dim_{k(x)} i^*{\mathcal{F}}[/math].
(a) Show that the function [math]\phi(x)[/math] is upper semicontinuous: for every [math]n[/math], the set [math]\{x\in X:\phi(x)\ge n\}[/math] is closed.
(b) Suppose [math]X[/math] is reduced. Show that [math]{\mathcal{F}}[/math] is locally free if and only if [math]\phi(x)[/math] is constant on each connected component of [math]X[/math]. (Do you see why we impose the assumption that [math]X[/math] is reduced here?)
 Let [math]X[/math] be a locally noetherian scheme and let [math]U\subset X[/math] be an open subset. Show that any coherent sheaf [math]{\mathcal{F}}[/math] on [math]U[/math] can be extended to a coherent sheaf on [math]\overline{{\mathcal{F}}}[/math] on [math]X[/math]. (We say that [math]\overline{{\mathcal{F}}}[/math] is an extension of [math]{\mathcal{F}}[/math] if [math]\overline{{\mathcal{F}}}_U\simeq{\mathcal{F}}[/math].)
(If you need a hint for this problem, look at Problem II.5.15 in Hartshorne.)
Homework 7
Due Friday, March 31st
Proper and separated morphisms.
Each scheme [math]X[/math] has a maximal closed reduced subscheme [math]X^{red}[/math]; the ideal sheaf of [math]X^{red}[/math] is the nilradical (the sheaf of all nilpotents in [math]{\mathcal{O}}_X[/math]).
 Let [math]f:X\to Y[/math] be a morphism of schemes of finite type. Consider the induced map [math]f^{red}:X^{red}\to Y^{red}[/math]. Prove that [math]f[/math] is separated (resp. proper) if and only if [math]f^{red}[/math] is separated (resp. proper).
Vector bundles.
Fix an algebraically closed field [math]k[/math]. Any vector bundle on [math]{\mathbb{A}}^1_k={\mathop{Spec}}(k[t])[/math] is trivial, you can use this without proof. Let [math]X[/math] be the ‘affine line with a doubled point’ obtained by gluing two copies of [math]{\mathbb{A}}^1_k[/math] away from the origin.
 Classify line bundles on [math]X[/math] up to isomorphism.
 (Could be hard) Prove that any vector bundle on [math]X[/math] is a direct sum of several line bundles.
Tangent bundle.
 Let [math]X[/math] be an irreducible affine variety, not necessarily smooth. Let [math]M[/math] be the [math]k[X][/math]module of [math]k[/math]linear derivations [math]k[X]\to k[X][/math]. (These are globally defined vector fields on [math]X[/math], but keep in mind that [math]X[/math] may be singular.) Consider its generic rank [math]r:=\dim_{k(X)}M\otimes_{k[X]}k(X)[/math]. Show that [math]r=\dim(X)[/math].
 Suppose now that [math]X[/math] is smooth. Show that the module [math]M[/math] is a locally free coherent module; the corresponding vector bundle is the tangent bundle [math]TX[/math].
 Let [math]f:X\to Y[/math] be a morphism of algebraic varieties. Recall that a vector bundle [math]E[/math] over [math]Y[/math] gives a vector bundle [math]f^*E[/math] on [math]X[/math] whose total space is the fiber product [math]E\times_YX[/math].
 Suppose now that [math]X[/math] and [math]Y[/math] are affine and [math]Y[/math] is smooth. Let [math]E=TY[/math] be the tangent bundle to [math]Y[/math]. Show that the space of [math]k[/math]linear derivations [math]k[Y]\to k[X][/math] (where [math]f[/math] is used to equip [math]k[X][/math] with the structure of a [math]k[Y][/math]module) is identified with [math]\Gamma(X,f^*(TY))[/math].
 Let [math]X[/math] be a smooth affine variety. Let [math]I_\Delta\subset k[X\times X][/math] be the ideal sheaf of the diagonal [math]\Delta\subset X\times X[/math]. Prove that there is a bijection [math]I_\Delta/I_\Delta^2=\Gamma(X,\Omega^1_X),[/math] where [math]\Omega^1_X[/math] is the sheaf of differential 1forms (that is, the sheaf of sections of the cotangent bundle [math]T^\vee X[/math], which is the dual vector bundle of [math]TX[/math]).
Homework 8
Due Friday, April 7th
 (Hartshorne, II.4.4) Fix a Noetherian scheme [math]S[/math], let [math]X[/math] and [math]Y[/math] be schemes of finite type and separated over [math]S[/math], and let [math]f:X\to Y[/math] be a morphism of [math]S[/math]schemes. Suppose that [math]Z\subset X[/math] be a closed subscheme that is proper over [math]S[/math]. Show that [math]f(Z)\subset Y[/math] is closed.
 In the setting of the previous problem, show that if we consider [math]f(Z)[/math] as a closed subscheme (its ideal of functions consists of all functions whose composition with [math]f[/math] is zero), then [math]f[/math] induces a proper map fro [math]Z[/math] to [math]f(Z)[/math].
(Galois descent, inspired by Hartshorne II.4.7) Let [math]F/k[/math] be a finite Galois extension of fields. The Galois group [math]G:=Gal(F/k)[/math] acts on the scheme [math]{\mathop{Spec}}(F)[/math]. Given any [math]k[/math]scheme [math]X[/math], we let [math]X_F:={\mathop{Spec}}(F)\times_{{\mathop{Spec}}(k)}X[/math] be its extension of scalars; the group [math]G[/math] acts on [math]X_F[/math] in a way compatible with its action on [math]{\mathop{Spec}}(F)[/math] (i.e., this action is ‘semilinear’).
 Show that [math]X[/math] is affine if and only if [math]X_F[/math] is affine.
 Prove that this operation gives a fully faithful functor from the category of [math]k[/math]schemes into the category of [math]F[/math]schemes with a semilinear action of [math]G[/math].
 Suppose that [math]Y[/math] is a separated [math]F[/math]scheme such that any finite subset of [math]Y[/math] is contained in an affine open chart (this holds, for instance, if [math]Y[/math] is quasiprojective). Then for any semilinear action of [math]G[/math] on [math]Y[/math], there exists a [math]k[/math]scheme [math]X[/math] and an isomorphism [math]X_F\simeq Y[/math] that agrees with an action of [math]G[/math]. (That is, the action of [math]G[/math] gives a [math]k[/math]structure on the scheme [math]Y[/math].)
 Suppose [math]X[/math] is an [math]{\mathbb{R}}[/math]scheme such that [math]X_{\mathbb{C}}\simeq{\mathbb{A}}_{\mathbb{C}}^1[/math]. Show that [math]X\simeq{\mathbb{A}}^1_{\mathbb{R}}[/math].
 Suppose [math]X[/math] is an [math]{\mathbb{R}}[/math]scheme such that [math]X_{\mathbb{C}}\simeq{\mathbb{P}}_{\mathbb{C}}^1[/math]. Show that there are two possibilities for the isomorphism class of [math]X[/math].
Homework 9
Due Friday, April 21st
 Let [math]X[/math] be a singular cubic in [math]{\mathbb{P}}^2[/math], given (in nonhomogeneous coordinates) either by [math]y^2=x^3+x^2[/math] (nodal cubic) or by [math]y^2=x^3[/math] (cuspidal cubic). Compute the class group of Cartier divisors on [math]X[/math].
 Let [math]X[/math] and [math]Y[/math] be schemes over some base scheme [math]S[/math]. For any map [math]f:X\to Y[/math], use the functoriality of the module of Kähler differentials to construct a morphism [math]f^*\Omega_{Y/S}\to\Omega_{X/S}[/math] and verify that [math]\Omega_{X/Y}=\mathrm{coker}(f^*\Omega_{Y/S}\to\Omega_{X/S})[/math].
 Suppose now that [math]X[/math] and [math]Y[/math] be schemes over an algebraically closed field [math]k[/math]. A morphism [math]f:X\to Y[/math] is unramified if [math]\Omega_{X/Y}=0[/math]. Show that this is equivalent to the following condition: given [math]D={\mathop{Spec}}k[\epsilon]/{\epsilon^2}[/math], the map [math]f[/math] induces an injection [math]\mathrm{Maps}(D,X)\to\mathrm{Maps}(D,Y)[/math].
 Let us compute the algebraic de Rham cohomology of the affine space. Put [math]X={\mathop{Spec}}R[/math], [math]R=k[t_1,\dots,t_n][/math]. Since [math]X[/math] is a smooth [math]k[/math]scheme, [math]\Omega^1_R=\Omega_{R/k}[/math] is a locally free [math]R[/math]module. Denote by [math]\Omega^\bullet_R[/math] the exterior algebra of [math]\Omega^1_R[/math], so that [math]\Omega^i_R=\bigwedge^i\Omega^1_R[/math]. Define the de Rham differential [math]d:\Omega^i_R\to\Omega^{i+1}_R[/math] by starting with the Kähler differential [math]d:R\to\Omega^1_R[/math] and then extending it by the graded Leibniz rule: [math]d(\omega_1\wedge\omega_2)=(d\omega_1)\wedge\omega_2+(1)^i\omega_1\wedge d(\omega_2),\qquad \omega_1\in\Omega^i_R.[/math]
Compute the cohomology of the complex [math]\Omega^\bullet_R[/math] equipped with the differential [math]d[/math]. The answer will depend on the characteristic of [math]k[/math].
 Let [math]X[/math] be a Noetherian scheme.Let [math]K(X)[/math] be the [math]K[/math]group of [math]X[/math]: it is generated by elements [math][F][/math] for each coherent sheaf [math]F[/math] with relations [math][F]=[F_1]+[F_2][/math] whenever there is a short exact sequence [math]0\to F_1\to F_2\to F_3\to 0.[/math] Prove that [math]K({\mathbb{A}}^n)={\mathbb{Z}}[/math]. (This is much easier if you know Hilbert’s Syzygy Theorem.)
 Let [math]X[/math] be a smooth curve over an algebraically closed field. Show that [math]K(X)[/math] is generated by [math][L][/math] for line bundles [math]L[/math].
 Let [math]X[/math] be a smooth curve over an algebraically closed field. Show that [math]K(X)[/math] is isomorphic to [math]{\mathbb{Z}}\oplus \mathrm{Pic}(X)[/math]. (If this problem is too hard, look at Hartshorne’s II.6.11 for a stepbystep approach.)