Math 764 -- Algebraic Geometry II -- Homeworks

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 $\displaystyle{ X }$; all sheaves and presheaves are sheaves on $\displaystyle{ X }$.

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

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

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

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

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

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

Homework 2

Due Friday, February 10th

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

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

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

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

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 $\displaystyle{ R_\alpha }$ be a finite collection of rings. Put $\displaystyle{ R=\prod_\alpha R_\alpha }$. Describe the topological space $\displaystyle{ {\mathop{\mathrm{Spec}}}(R) }$ in terms of $\displaystyle{ {\mathop{\mathrm{Spec}}}(R_\alpha) }$’s. What changes if the collection is infinite?
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 $\displaystyle{ R\to S }$ such that the image of a map $\displaystyle{ {\mathop{\mathrm{Spec}}}(S)\to{\mathop{\mathrm{Spec}}}(R) }$ is

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

(b) An infinite union of closed sets, but not constructible. (This part may be very hard.)

Contraction of a subvariety.

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

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

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

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

Homework 3

Due Friday, February 17th

1. (Gluing morphisms of sheaves) Let $\displaystyle{ F }$ and $\displaystyle{ G }$ be two sheaves on the same space $\displaystyle{ X }$. For any open set $\displaystyle{ U\subset X }$, consider the restriction sheaves $\displaystyle{ F|_U }$ and $\displaystyle{ G|_U }$, and let $\displaystyle{ Hom(F|_U,G|_U) }$ be the set of sheaf morphisms between them.

Prove that the presheaf on $\displaystyle{ X }$ given by the correspondence $\displaystyle{ U\mapsto Hom(F|_U,G|_U) }$ is in fact a sheaf.

2. (Gluing morphisms of ringed spaces) Let $\displaystyle{ X }$ and $\displaystyle{ Y }$ be ringed spaces. Denote by $\displaystyle{ \underline{Mor}(X,Y) }$ the following pre-sheaf on $\displaystyle{ X }$: its sections over an open subset $\displaystyle{ U\subset X }$ are morphisms of ringed spaces $\displaystyle{ U\to Y }$ where $\displaystyle{ U }$ is considered as a ringed space. (And the notion of restriction is the natural one.) Show that $\displaystyle{ \underline{Mor}(X,Y) }$ is in fact a sheaf.
3. (Affinization of a scheme) Let $\displaystyle{ X }$ be an arbitrary scheme. Prove that there exists an affine scheme $\displaystyle{ X_{aff} }$ and a morphism $\displaystyle{ X\to X_{aff} }$ that is universal in the following sense: any map form $\displaystyle{ X }$ to an affine scheme factors through it.
4. Let us consider direct and inverse limits of affine schemes. For simplicity, we will work with limits indexed by positive integers.

(a) Let $\displaystyle{ R_i }$ be a collection of rings ($\displaystyle{ i\gt 0 }$) together with homomorphisms $\displaystyle{ R_i\to R_{i+1} }$. Consider the direct limit $\displaystyle{ R:=\lim\limits_{\longrightarrow} R_i }$. Show that in the category of schemes, $\displaystyle{ {\mathop{\mathrm{Spec}}}(R)=\lim\limits_{\longleftarrow}{\mathop{\mathrm{Spec}}}R_i. }$

(b) Let $\displaystyle{ R_i }$ be a collection of rings ($\displaystyle{ i\gt 0 }$) together with homomorphisms $\displaystyle{ R_{i+1}\to R_i }$. Consider the inverse limit $\displaystyle{ R:=\lim\limits_{\longleftarrow} R_i }$. Show that generally speaking, in the category of schemes, $\displaystyle{ {\mathop{\mathrm{Spec}}}(R)\neq\lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i. }$

5. Here is an example of the situation from 4(b). Let $\displaystyle{ k }$ be a field, and let $\displaystyle{ R_i=k[t]/(t^i) }$, so that $\displaystyle{ \lim\limits_{\longleftarrow} R_i=k[[t]] }$. Describe the direct limit $\displaystyle{ \lim\limits_{\longrightarrow}{\mathop{\mathrm{Spec}}}R_i }$ in the category of ringed spaces. Is the direct limit a scheme?
6. Let $\displaystyle{ S }$ be a finite partially ordered set. Consider the following topology on $\displaystyle{ S }$: a subset $\displaystyle{ U\subset S }$ is open if and only if whenever $\displaystyle{ x\in U }$ and $\displaystyle{ y\gt x }$, it must be that $\displaystyle{ y\in U }$.

Construct a ring $\displaystyle{ R }$ such that $\displaystyle{ \mathop{\mathrm{Spec}}(R) }$ is homeomorphic to $\displaystyle{ S }$.

7. Show that any quasi-compact scheme has closed points. (It is not true that any scheme has closed points!)
8. 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

1. Show that the following two definitions of quasi-separated-ness of a scheme $\displaystyle{ S }$ are equivalent:
1. The intersection of any two quasi-compact open subsets of $\displaystyle{ S }$ is quasi-compact;
2. There is a cover of $\displaystyle{ S }$ by affine open subsets whose (pairwise) intersections are quasi-compact.
2. In class, we gave the following definition: a scheme $\displaystyle{ S }$ 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 non-empty open $\displaystyle{ U\subset S }$, $\displaystyle{ O_S(U) }$ is a domain.
3. Let us call a scheme $\displaystyle{ X }$ locally irreducible if every point has an irreducible neighborhood. (Since a non-empty 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.
4. Show that a locally Noetherian scheme is quasi-separated.
5. Show that the following two definitions of a Noetherian scheme $\displaystyle{ X }$ are equivalent:
1. $\displaystyle{ X }$ is a finite union of open affine sets, each of which is the spectrum of a Noetherian ring;
2. $\displaystyle{ X }$ is quasi-compact and locally Noetherian.
6. Show that any Noetherian scheme $\displaystyle{ X }$ is a disjoint union of finitely many connected open subsets (the connected components of $\displaystyle{ X }$.) (A problem from the last homework shows that things might go wrong if we do not assume that $\displaystyle{ X }$ is Noetherian.)
7. A locally closed subscheme $\displaystyle{ X\subset Y }$ is defined as a closed subscheme of an open subscheme of $\displaystyle{ Y }$. 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 $\displaystyle{ Y }$. 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

1. Fix a field $\displaystyle{ k }$, and put $\displaystyle{ X={\mathop{Spec}}k[x] }$ and $\displaystyle{ Y={\mathop{Spec}}k[y] }$. Consider the morphism $\displaystyle{ f:X\to Y }$ given by $\displaystyle{ y=x^2 }$. Describe the fiber product $\displaystyle{ X\times_YX }$ as explicitly as possible. (The answer may depend on $\displaystyle{ k }$.)
2. (The Frobenius morphism.) Let $\displaystyle{ X }$ be a scheme of characteristic $\displaystyle{ p }$: by definition, this means that $\displaystyle{ p=0 }$ in the structure sheaf of $\displaystyle{ X }$. Define the (absolute) Frobenius morphism $\displaystyle{ Fr_X:X\to X }$ as follows: it is the identity map on the underlying set, and the pullback $\displaystyle{ Fr_X^*(f) }$ equals $\displaystyle{ f^p }$ for any (local) function $\displaystyle{ f\in{\mathcal{O}}_X }$.

Verify that this defines an affine morphism of schemes. Assuming $\displaystyle{ X }$ is a scheme locally of finite type over a perfect field, verify that $\displaystyle{ Fr_X }$ is a morphism of finite type (it is in fact finite, if you know what it means).

3. (The relative Frobenius morphism.) Let $\displaystyle{ X\to Y }$ be a morphism of schemes of characteristic $\displaystyle{ p }$. Put $\displaystyle{ \overline X:=X\times_{Y,Fr_Y}Y, }$ where the notation means that $\displaystyle{ Y }$ is considered as a $\displaystyle{ Y }$-scheme via the Frobenius map.
1. Show that the Frobenius morphism $\displaystyle{ Fr_X }$ naturally factors as the composition $\displaystyle{ X\to\overline{X}\to X }$, where the first map $\displaystyle{ X\to\overline{X} }$ is naturally a morphism of schemes over $\displaystyle{ Y }$ (while the second map, generally speaking, is not). The map $\displaystyle{ X\to\overline{X} }$ is called the relative Frobenius morphism.
2. Suppose $\displaystyle{ Y={\mathop{Spec}}(\overline{\mathbb{F}}_p) }$, and $\displaystyle{ X }$ is an affine variety (that is, an affine reduced scheme of finite type) over $\displaystyle{ \overline{\mathbb{F}}_p }$. Describe $\displaystyle{ \overline X }$ and the relative Frobenius $\displaystyle{ X\to\overline{X} }$ explicitly in coordinates.
4. Let $\displaystyle{ X }$ be a scheme over $\displaystyle{ \mathbb{F}_p }$. In this case, the absolute Frobenius $\displaystyle{ Fr_X:X\to X }$ is a morphism of schemes over $\displaystyle{ \mathbb{F}_p }$ (and it coincides with the relative Frobenius of $\displaystyle{ X }$ over $\displaystyle{ \mathbb{F}_p }$.

Consider the extension of scalars $\displaystyle{ X'=X_{\overline{\mathbb{F}}_p}=X\otimes_{\mathbb{F}_p}\overline{\mathbb{F}}_p=X\times_{{\mathop{Spec}}(\mathbb{F}_p)}{\mathop{Spec}}(\overline{\mathbb{F}}_p). }$ Then $\displaystyle{ Fr_X }$ naturally extends to a morphism of $\displaystyle{ \overline{\mathbb{F}}_p }$-schemes $\displaystyle{ X'\to X' }$. Compare the map $\displaystyle{ X'\to X' }$ with the relative Frobenius of $\displaystyle{ X' }$ over $\displaystyle{ \overline{\mathbb{F}}_p }$.

5. 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 $\displaystyle{ f:X\to Z }$ is surjective and $\displaystyle{ g:Y\to Z }$ is arbitrary, then $\displaystyle{ X\times_ZY\to Y }$ is surjective.
6. (Normalization) A scheme is normal if all of its local rings are integrally closed domains. Let $\displaystyle{ X }$ be an integral scheme. Show that there exists a normal integral scheme $\displaystyle{ \tilde{X} }$ together with a morphism $\displaystyle{ \tilde{X}\to X }$ that is universal in the following sense: any dominant morphism $\displaystyle{ Y\to X }$ from a normal integral scheme to $\displaystyle{ X }$ factors through $\displaystyle{ \tilde{X} }$. (Just like in the case of varieties, a morphism is dominant if its image is dense.)
7. Let $\displaystyle{ X }$ be a scheme of finite type over a field $\displaystyle{ k }$. For every field extension $\displaystyle{ K\supset k }$, put $\displaystyle{ X_K:=X\otimes_kK={\mathop{Spec}}(K)\times_{{\mathop{Spec}}(k)}X. }$

Show that $\displaystyle{ X }$ is geometrically irreducible (that is, the morphism $\displaystyle{ X\to{\mathop{Spec}}(k) }$ has geometrically irreducible fibers) if and only if $\displaystyle{ X_K }$ is irreducible for all finite extensions $\displaystyle{ K\supset k }$.

Homework 6

Due Friday, March 10th

Sheaves of modules on ringed spaces.

Let $\displaystyle{ (X,{\mathcal{O}}_X) }$ be a ringed space, and let $\displaystyle{ {\mathcal{F}} }$ and $\displaystyle{ {\mathcal{G}} }$ be sheaves of $\displaystyle{ {\mathcal{O}}_X }$-modules. The tensor product of $\displaystyle{ {\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}} }$ is the sheafification of the presheaf $\displaystyle{ U\mapsto{\mathcal{F}}(U)\otimes_{{\mathcal{O}}_X(U)}{\mathcal{G}}(U). }$

1. Prove that the stalks of $\displaystyle{ {\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}} }$ are given by the tensor product: $\displaystyle{ ({\mathcal{F}}\otimes_{{\mathcal{O}}_X}{\mathcal{G}})_x={\mathcal{F}}_x\otimes_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x, }$ where $\displaystyle{ x\in X }$. Conclude that the tensor product is a right exact functor (in each of the two arguments).
2. Suppose that $\displaystyle{ {\mathcal{F}} }$ is locally free of finite rank. (That is to say, every point $\displaystyle{ x\in X }$ has a neighborhood $\displaystyle{ U }$ such that $\displaystyle{ {\mathcal{F}}|_U\simeq({\mathcal{O}}_U)^n }$. Prove that there exists a natural isomorphism $\displaystyle{ {{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{G}})={\mathcal{G}}\otimes{\mathcal{F}}^\vee. }$ Here $\displaystyle{ {\mathcal{F}}^\vee={\mathop{\mathcal{H}\mathit{om}}}_{{\mathcal{O}}_X}({\mathcal{F}},{\mathcal{O}}_X) }$ is the dual of the locally free sheaf $\displaystyle{ {\mathcal{F}} }$, and $\displaystyle{ {\mathop{\mathcal{H}\mathit{om}}} }$ is the sheaf of homomorphisms. (Note that $\displaystyle{ {\mathcal{G}} }$ is not assumed to be quasi-coherent.)
3. (Projection formula) Let $\displaystyle{ f:(X,{\mathcal{O}}_X)\to(Y,{\mathcal{O}}_Y) }$ be a morphism of ringed spaces. Suppose $\displaystyle{ {\mathcal{F}} }$ is an $\displaystyle{ {\mathcal{O}}_X }$-module and $\displaystyle{ {\mathcal{G}} }$ is a locally free $\displaystyle{ {\mathcal{O}}_Y }$-module of finite rank. Construct a natural isomorphism $\displaystyle{ f_*({\mathcal{F}}\otimes_{{\mathcal{O}}_X} f^*{\mathcal{G}})\simeq f_*({\mathcal{F}})\otimes_{{\mathcal{O}}_Y}{\mathcal{G}}. }$

Coherent sheaves on a noetherian scheme

4. Let $\displaystyle{ {\mathcal{F}} }$ be a coherent sheaf on a loclly noetherian scheme $\displaystyle{ X }$.

Show that $\displaystyle{ {\mathcal{F}} }$ is locally free if and only if its stalks $\displaystyle{ {\mathcal{F}}_x }$ are free $\displaystyle{ {\mathcal{O}}_{X,x} }$-modules for all $\displaystyle{ x\in X }$.

(b) Show that $\displaystyle{ {\mathcal{F}} }$ is locally free of rank one if and only if it is invertible: there exists a coherent sheaf $\displaystyle{ {\mathcal{G}} }$ such that $\displaystyle{ {\mathcal{F}}\otimes{\mathcal{G}}\simeq{\mathcal{O}}_X }$.

5. As in the previous problem, supposed $\displaystyle{ {\mathcal{F}} }$ be a coherent sheaf on a locally noetherian scheme $\displaystyle{ X }$. The fiber of $\displaystyle{ {\mathcal{F}} }$ at a point $\displaystyle{ x\in X }$ is the $\displaystyle{ k(x) }$-vector space $\displaystyle{ i^*{\mathcal{F}} }$ for the natural map $\displaystyle{ i:{\mathop{Spec}}(k(x))\to X }$ (where $\displaystyle{ k(x) }$ is the residue field of $\displaystyle{ x\in X }$). Denote by $\displaystyle{ \phi(x) }$ the dimension $\displaystyle{ \dim_{k(x)} i^*{\mathcal{F}} }$.

(a) Show that the function $\displaystyle{ \phi(x) }$ is upper semi-continuous: for every $\displaystyle{ n }$, the set $\displaystyle{ \{x\in X:\phi(x)\ge n\} }$ is closed.

(b) Suppose $\displaystyle{ X }$ is reduced. Show that $\displaystyle{ {\mathcal{F}} }$ is locally free if and only if $\displaystyle{ \phi(x) }$ is constant on each connected component of $\displaystyle{ X }$. (Do you see why we impose the assumption that $\displaystyle{ X }$ is reduced here?)

6. Let $\displaystyle{ X }$ be a locally noetherian scheme and let $\displaystyle{ U\subset X }$ be an open subset. Show that any coherent sheaf $\displaystyle{ {\mathcal{F}} }$ on $\displaystyle{ U }$ can be extended to a coherent sheaf on $\displaystyle{ \overline{{\mathcal{F}}} }$ on $\displaystyle{ X }$. (We say that $\displaystyle{ \overline{{\mathcal{F}}} }$ is an extension of $\displaystyle{ {\mathcal{F}} }$ if $\displaystyle{ \overline{{\mathcal{F}}}|_U\simeq{\mathcal{F}} }$.)

(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 $\displaystyle{ X }$ has a maximal closed reduced subscheme $\displaystyle{ X^{red} }$; the ideal sheaf of $\displaystyle{ X^{red} }$ is the nilradical (the sheaf of all nilpotents in $\displaystyle{ {\mathcal{O}}_X }$).

1. Let $\displaystyle{ f:X\to Y }$ be a morphism of schemes of finite type. Consider the induced map $\displaystyle{ f^{red}:X^{red}\to Y^{red} }$. Prove that $\displaystyle{ f }$ is separated (resp. proper) if and only if $\displaystyle{ f^{red} }$ is separated (resp. proper).

Vector bundles.

Fix an algebraically closed field $\displaystyle{ k }$. Any vector bundle on $\displaystyle{ {\mathbb{A}}^1_k={\mathop{Spec}}(k[t]) }$ is trivial, you can use this without proof. Let $\displaystyle{ X }$ be the ‘affine line with a doubled point’ obtained by gluing two copies of $\displaystyle{ {\mathbb{A}}^1_k }$ away from the origin.

2. Classify line bundles on $\displaystyle{ X }$ up to isomorphism.
3. (Could be hard) Prove that any vector bundle on $\displaystyle{ X }$ is a direct sum of several line bundles.

Tangent bundle.

4. Let $\displaystyle{ X }$ be an irreducible affine variety, not necessarily smooth. Let $\displaystyle{ M }$ be the $\displaystyle{ k[X] }$-module of $\displaystyle{ k }$-linear derivations $\displaystyle{ k[X]\to k[X] }$. (These are globally defined vector fields on $\displaystyle{ X }$, but keep in mind that $\displaystyle{ X }$ may be singular.) Consider its generic rank $\displaystyle{ r:=\dim_{k(X)}M\otimes_{k[X]}k(X) }$. Show that $\displaystyle{ r=\dim(X) }$.
5. Suppose now that $\displaystyle{ X }$ is smooth. Show that the module $\displaystyle{ M }$ is a locally free coherent module; the corresponding vector bundle is the tangent bundle $\displaystyle{ TX }$.
6. Let $\displaystyle{ f:X\to Y }$ be a morphism of algebraic varieties. Recall that a vector bundle $\displaystyle{ E }$ over $\displaystyle{ Y }$ gives a vector bundle $\displaystyle{ f^*E }$ on $\displaystyle{ X }$ whose total space is the fiber product $\displaystyle{ E\times_YX }$.
7. Suppose now that $\displaystyle{ X }$ and $\displaystyle{ Y }$ are affine and $\displaystyle{ Y }$ is smooth. Let $\displaystyle{ E=TY }$ be the tangent bundle to $\displaystyle{ Y }$. Show that the space of $\displaystyle{ k }$-linear derivations $\displaystyle{ k[Y]\to k[X] }$ (where $\displaystyle{ f }$ is used to equip $\displaystyle{ k[X] }$ with the structure of a $\displaystyle{ k[Y] }$-module) is identified with $\displaystyle{ \Gamma(X,f^*(TY)) }$.
8. Let $\displaystyle{ X }$ be a smooth affine variety. Let $\displaystyle{ I_\Delta\subset k[X\times X] }$ be the ideal sheaf of the diagonal $\displaystyle{ \Delta\subset X\times X }$. Prove that there is a bijection $\displaystyle{ I_\Delta/I_\Delta^2=\Gamma(X,\Omega^1_X), }$ where $\displaystyle{ \Omega^1_X }$ is the sheaf of differential 1-forms (that is, the sheaf of sections of the cotangent bundle $\displaystyle{ T^\vee X }$, which is the dual vector bundle of $\displaystyle{ TX }$).

Homework 8

Due Friday, April 7th

1. (Hartshorne, II.4.4) Fix a Noetherian scheme $\displaystyle{ S }$, let $\displaystyle{ X }$ and $\displaystyle{ Y }$ be schemes of finite type and separated over $\displaystyle{ S }$, and let $\displaystyle{ f:X\to Y }$ be a morphism of $\displaystyle{ S }$-schemes. Suppose that $\displaystyle{ Z\subset X }$ be a closed subscheme that is proper over $\displaystyle{ S }$. Show that $\displaystyle{ f(Z)\subset Y }$ is closed.
2. In the setting of the previous problem, show that if we consider $\displaystyle{ f(Z) }$ as a closed subscheme (its ideal of functions consists of all functions whose composition with $\displaystyle{ f }$ is zero), then $\displaystyle{ f }$ induces a proper map fro $\displaystyle{ Z }$ to $\displaystyle{ f(Z) }$.

(Galois descent, inspired by Hartshorne II.4.7) Let $\displaystyle{ F/k }$ be a finite Galois extension of fields. The Galois group $\displaystyle{ G:=Gal(F/k) }$ acts on the scheme $\displaystyle{ {\mathop{Spec}}(F) }$. Given any $\displaystyle{ k }$-scheme $\displaystyle{ X }$, we let $\displaystyle{ X_F:={\mathop{Spec}}(F)\times_{{\mathop{Spec}}(k)}X }$ be its extension of scalars; the group $\displaystyle{ G }$ acts on $\displaystyle{ X_F }$ in a way compatible with its action on $\displaystyle{ {\mathop{Spec}}(F) }$ (i.e., this action is ‘semilinear’).

3. Show that $\displaystyle{ X }$ is affine if and only if $\displaystyle{ X_F }$ is affine.
4. Prove that this operation gives a fully faithful functor from the category of $\displaystyle{ k }$-schemes into the category of $\displaystyle{ F }$-schemes with a semi-linear action of $\displaystyle{ G }$.
5. Suppose that $\displaystyle{ Y }$ is a separated $\displaystyle{ F }$-scheme such that any finite subset of $\displaystyle{ Y }$ is contained in an affine open chart (this holds, for instance, if $\displaystyle{ Y }$ is quasi-projective). Then for any semi-linear action of $\displaystyle{ G }$ on $\displaystyle{ Y }$, there exists a $\displaystyle{ k }$-scheme $\displaystyle{ X }$ and an isomorphism $\displaystyle{ X_F\simeq Y }$ that agrees with an action of $\displaystyle{ G }$. (That is, the action of $\displaystyle{ G }$ gives a $\displaystyle{ k }$-structure on the scheme $\displaystyle{ Y }$.)
6. Suppose $\displaystyle{ X }$ is an $\displaystyle{ {\mathbb{R}} }$-scheme such that $\displaystyle{ X_{\mathbb{C}}\simeq{\mathbb{A}}_{\mathbb{C}}^1 }$. Show that $\displaystyle{ X\simeq{\mathbb{A}}^1_{\mathbb{R}} }$.
7. Suppose $\displaystyle{ X }$ is an $\displaystyle{ {\mathbb{R}} }$-scheme such that $\displaystyle{ X_{\mathbb{C}}\simeq{\mathbb{P}}_{\mathbb{C}}^1 }$. Show that there are two possibilities for the isomorphism class of $\displaystyle{ X }$.

Homework 9

Due Friday, April 21st

1. Let $\displaystyle{ X }$ be a singular cubic in $\displaystyle{ {\mathbb{P}}^2 }$, given (in non-homogeneous coordinates) either by $\displaystyle{ y^2=x^3+x^2 }$ (nodal cubic) or by $\displaystyle{ y^2=x^3 }$ (cuspidal cubic). Compute the class group of Cartier divisors on $\displaystyle{ X }$.
2. Let $\displaystyle{ X }$ and $\displaystyle{ Y }$ be schemes over some base scheme $\displaystyle{ S }$. For any map $\displaystyle{ f:X\to Y }$, use the functoriality of the module of Kähler differentials to construct a morphism $\displaystyle{ f^*\Omega_{Y/S}\to\Omega_{X/S} }$ and verify that $\displaystyle{ \Omega_{X/Y}=\mathrm{coker}(f^*\Omega_{Y/S}\to\Omega_{X/S}) }$.
3. Suppose now that $\displaystyle{ X }$ and $\displaystyle{ Y }$ be schemes over an algebraically closed field $\displaystyle{ k }$. A morphism $\displaystyle{ f:X\to Y }$ is unramified if $\displaystyle{ \Omega_{X/Y}=0 }$. Show that this is equivalent to the following condition: given $\displaystyle{ D={\mathop{Spec}}k[\epsilon]/{\epsilon^2} }$, the map $\displaystyle{ f }$ induces an injection $\displaystyle{ \mathrm{Maps}(D,X)\to\mathrm{Maps}(D,Y) }$.
4. Let us compute the algebraic de Rham cohomology of the affine space. Put $\displaystyle{ X={\mathop{Spec}}R }$, $\displaystyle{ R=k[t_1,\dots,t_n] }$. Since $\displaystyle{ X }$ is a smooth $\displaystyle{ k }$-scheme, $\displaystyle{ \Omega^1_R=\Omega_{R/k} }$ is a locally free $\displaystyle{ R }$-module. Denote by $\displaystyle{ \Omega^\bullet_R }$ the exterior algebra of $\displaystyle{ \Omega^1_R }$, so that $\displaystyle{ \Omega^i_R=\bigwedge^i\Omega^1_R }$. Define the de Rham differential $\displaystyle{ d:\Omega^i_R\to\Omega^{i+1}_R }$ by starting with the Kähler differential $\displaystyle{ d:R\to\Omega^1_R }$ and then extending it by the graded Leibniz rule: $\displaystyle{ d(\omega_1\wedge\omega_2)=(d\omega_1)\wedge\omega_2+(-1)^i\omega_1\wedge d(\omega_2),\qquad \omega_1\in\Omega^i_R. }$

Compute the cohomology of the complex $\displaystyle{ \Omega^\bullet_R }$ equipped with the differential $\displaystyle{ d }$. The answer will depend on the characteristic of $\displaystyle{ k }$.

5. Let $\displaystyle{ X }$ be a Noetherian scheme.Let $\displaystyle{ K(X) }$ be the $\displaystyle{ K }$-group of $\displaystyle{ X }$: it is generated by elements $\displaystyle{ [F] }$ for each coherent sheaf $\displaystyle{ F }$ with relations $\displaystyle{ [F]=[F_1]+[F_2] }$ whenever there is a short exact sequence $\displaystyle{ 0\to F_1\to F_2\to F_3\to 0. }$ Prove that $\displaystyle{ K({\mathbb{A}}^n)={\mathbb{Z}} }$. (This is much easier if you know Hilbert’s Syzygy Theorem.)
6. Let $\displaystyle{ X }$ be a smooth curve over an algebraically closed field. Show that $\displaystyle{ K(X) }$ is generated by $\displaystyle{ [L] }$ for line bundles $\displaystyle{ L }$.
7. Let $\displaystyle{ X }$ be a smooth curve over an algebraically closed field. Show that $\displaystyle{ K(X) }$ is isomorphic to $\displaystyle{ {\mathbb{Z}}\oplus \mathrm{Pic}(X) }$. (If this problem is too hard, look at Hartshorne’s II.6.11 for a step-by-step approach.)