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 (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.
Due Friday, February 3rd
In all these problems, we fix a topological space ; all sheaves and presheaves are sheaves on .
- Example: Let be the unit circle, and let be the sheaf of -functions on . Find the (sheaf) image and the kernel of the morphism Here is the polar coordinate on the circle.
- Sheaf operations: Let and be sheaves of sets. Recall that a morphism is a (categorical) monomorphism if and only if for any sheaf and any two morphisms , the equality implies . Show that is a monomorphism if and only if it induces injective maps on all stalks.
- Let and be sheaves of sets. Recall that a morphism is a (categorical) epimorphism if and only if for any sheaf and any two morphisms , the equality implies . Show that is a epimorphism if and only if it induces surjective maps on all stalks.
- Show that any morphism of sheaves can be written as a composition of an epimorphism and a monomorphism. (You should know what order of composition I mean here.)
- Let be a sheaf, and let be a sub-presheaf of (thus, for every open set , is a subset of and the restriction maps for and agree). Show that the sheafification of is naturally identified with a subsheaf of .
- Let be a family of sheaves of abelian groups on indexed by a set (not necessarily finite). Show that the direct sum and direct product of this family exists in the category of sheaves of abelian groups. (E.g., a direct sum would be a sheaf of abelian groups together with a universal family of homomorphisms .) Do these operations agree with (a) taking stalks at a point (b) taking sections over an open subset ?
Locally constant sheaves:
Definition. A sheaf is constant over an open set if there is a subset such that the map (the germ of at ) gives a bijection between and for all .
is locally constant (on ) if every point of has a neighborhood on which is constant.
Recall that a covering space is a continuous map of topological spaces such that every has a neighborhood whose preimage is homeomorphic to for some discrete topological space . ( may depend on ; also, the homeomorphism is required to respect the projection to .)
Show that if is a covering space, its sheaf of sections is locally constant. Moreover, prove that this correspondence is an equivalence between the category of covering spaces and the category of locally constant sheaves. (If is pathwise connected, both categories are equivalent to the category of sets with an action of the fundamental group of .)
- Sheafification: (This problem may be hard, but it is still a good idea to try it) Prove or disprove the following statement (contained in the lecture notes). Let be a presheaf on , and let be its sheafification. Then every section can be represented as (the equivalence class of) the following gluing data: an open cover and a family of sections such that .
Due Friday, February 10th
Extension of a sheaf by zero. Let be a topological space, let be an open subset, and let be a sheaf of abelian groups on .
The extension by zero of (here is the embedding ) is the sheaf on that can be defined as the sheafification of the presheaf such that
- Is the sheafication necessary in this definition? (Or maybe is a sheaf automatically?)
- Describe the stalks of over all points of and the espace étalé of .
- Verify that is the left adjoint of the restriction functor from to : that is, for any sheaf on , there exists a natural isomorphism
(The restriction of a sheaf from to an open set is defined by for .)
Side question (not part of the homework): What changes if we consider the version of extension by zero for sheaves of sets (‘the extension by empty set’)?
Examples of affine schemes.
- Let be a finite collection of rings. Put . Describe the topological space in terms of ’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 such that the image of a map is
(a) An infinite intersection of open sets, but not constructible.
(b) An infinite union of closed sets, but not constructible. (This part may be very hard.)
Contraction of a subvariety.
Let be a variety (over an algebraically closed field ) and let be a closed subvariety. Our goal is to construct a -ringed space that is in some sense the result of ‘gluing’ together the points of . While can be described by a universal property, we prefer an explicit construction:
- The topological space is the ‘quotient-space’ : as a set, ; a subset is open if and only if is open. Here the natural projection is identity on and sends all of to the ‘center’ .
- The structure sheaf is defined as follows: for any open subset , is the algebra of functions such that the composition is a regular function that is constant along . (The last condition is imposed only if , in which case .)
In each of the following examples, determine whether the quotient is an algebraic variety; if it is, describe it explicitly.
- , (embedded as a line in ).
- , .
- , is a two-point set (if you want a more challenging version, let be any finite set).
Due Friday, February 17th
- (Gluing morphisms of sheaves) Let and be two sheaves on the same space . For any open set , consider the restriction sheaves and , and let be the set of sheaf morphisms between them.
Prove that the presheaf on given by the correspondence is in fact a sheaf.
- (Gluing morphisms of ringed spaces) Let and be ringed spaces. Denote by the following pre-sheaf on : its sections over an open subset are morphisms of ringed spaces where is considered as a ringed space. (And the notion of restriction is the natural one.) Show that is in fact a sheaf.
- (Affinization of a scheme) Let be an arbitrary scheme. Prove that there exists an affine scheme and a morphism that is universal in the following sense: any map form 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 be a collection of rings () together with homomorphisms . Consider the direct limit . Show that in the category of schemes,
(b) Let be a collection of rings () together with homomorphisms . Consider the inverse limit . Show that generally speaking, in the category of schemes,
- Here is an example of the situation from 4(b). Let be a field, and let , so that . Describe the direct limit in the category of ringed spaces. Is the direct limit a scheme?
- Let be a finite partially ordered set. Consider the following topology on : a subset is open if and only if whenever and , it must be that .
Construct a ring such that is homeomorphic to .
- Show that any quasi-compact 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...
Due Friday, February 24th
- Show that the following two definitions of quasi-separated-ness of a scheme are equivalent:
- The intersection of any two quasi-compact open subsets of is quasi-compact;
- There is a cover of by affine open subsets whose (pairwise) intersections are quasi-compact.
- In class, we gave the following definition: a scheme 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 , is a domain.
- Let us call a scheme 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.
- Show that a locally Noetherian scheme is quasi-separated.
- Show that the following two definitions of a Noetherian scheme are equivalent:
- is a finite union of open affine sets, each of which is the spectrum of a Noetherian ring;
- is quasi-compact and locally Noetherian.
- Show that any Noetherian scheme is a disjoint union of finitely many connected open subsets (the connected components of .) (A problem from the last homework shows that things might go wrong if we do not assume that is Noetherian.)
- A locally closed subscheme is defined as a closed subscheme of an open subscheme of . 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 . 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.
Due Friday, March 3rd
- Fix a field , and put and . Consider the morphism given by . Describe the fiber product as explicitly as possible. (The answer may depend on .)
- (The Frobenius morphism.) Let be a scheme of characteristic : by definition, this means that in the structure sheaf of . Define the (absolute) Frobenius morphism as follows: it is the identity map on the underlying set, and the pullback equals for any (local) function .
Verify that this defines an affine morphism of schemes. Assuming is a scheme locally of finite type over a perfect field, verify that is a morphism of finite type (it is in fact finite, if you know what it means).
- (The relative Frobenius morphism.) Let be a morphism of schemes of characteristic . Put where the notation means that is considered as a -scheme via the Frobenius map.
- Show that the Frobenius morphism naturally factors as the composition , where the first map is naturally a morphism of schemes over (while the second map, generally speaking, is not). The map is called the relative Frobenius morphism.
- Suppose , and is an affine variety (that is, an affine reduced scheme of finite type) over . Describe and the relative Frobenius explicitly in coordinates.
- Let be a scheme over . In this case, the absolute Frobenius is a morphism of schemes over (and it coincides with the relative Frobenius of over .
Consider the extension of scalars Then naturally extends to a morphism of -schemes . Compare the map with the relative Frobenius of over .
- 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 is surjective and is arbitrary, then is surjective.
- (Normalization) A scheme is normal if all of its local rings are integrally closed domains. Let be an integral scheme. Show that there exists a normal integral scheme together with a morphism that is universal in the following sense: any dominant morphism from a normal integral scheme to factors through . (Just like in the case of varieties, a morphism is dominant if its image is dense.)
- Let be a scheme of finite type over a field . For every field extension , put </p>
Show that is geometrically irreducible (that is, the morphism has geometrically irreducible fibers) if and only if is irreducible for all finite extensions .
Due Friday, March 10th
Sheaves of modules on ringed spaces.
Let be a ringed space, and let and be sheaves of -modules. The tensor product of is the sheafification of the presheaf
- Prove that the stalks of are given by the tensor product: where . Conclude that the tensor product is a right exact functor (in each of the two arguments).
- Suppose that is locally free of finite rank. (That is to say, every point has a neighborhood such that . Prove that there exists a natural isomorphism Here is the dual of the locally free sheaf , and is the sheaf of homomorphisms. (Note that is not assumed to be quasi-coherent.)
- (Projection formula) Let be a morphism of ringed spaces. Suppose is an -module and is a locally free -module of finite rank. Construct a natural isomorphism
Coherent sheaves on a noetherian scheme
- Let be a coherent sheaf on a loclly noetherian scheme .
Show that is locally free if and only if its stalks are free -modules for all .
(b) Show that is locally free of rank one if and only if it is invertible: there exists a coherent sheaf such that .
- As in the previous problem, supposed be a coherent sheaf on a locally noetherian scheme . The fiber of at a point is the -vector space for the natural map (where is the residue field of ). Denote by the dimension .
(a) Show that the function is upper semi-continuous: for every , the set is closed.
(b) Suppose is reduced. Show that is locally free if and only if is constant on each connected component of . (Do you see why we impose the assumption that is reduced here?)
- Let be a locally noetherian scheme and let be an open subset. Show that any coherent sheaf on can be extended to a coherent sheaf on on . (We say that is an extension of if .)
(If you need a hint for this problem, look at Problem II.5.15 in Hartshorne.)
Due Friday, March 31st
Proper and separated morphisms.
Each scheme has a maximal closed reduced subscheme ; the ideal sheaf of is the nilradical (the sheaf of all nilpotents in ).
- Let be a morphism of schemes of finite type. Consider the induced map . Prove that is separated (resp. proper) if and only if is separated (resp. proper).
Fix an algebraically closed field . Any vector bundle on is trivial, you can use this without proof. Let be the ‘affine line with a doubled point’ obtained by gluing two copies of away from the origin.
- Classify line bundles on up to isomorphism.
- (Could be hard) Prove that any vector bundle on is a direct sum of several line bundles.
- Let be an irreducible affine variety, not necessarily smooth. Let be the -module of -linear derivations . (These are globally defined vector fields on , but keep in mind that may be singular.) Consider its generic rank . Show that .
- Suppose now that is smooth. Show that the module is a locally free coherent module; the corresponding vector bundle is the tangent bundle .
- Let be a morphism of algebraic varieties. Recall that a vector bundle over gives a vector bundle on whose total space is the fiber product .
- Suppose now that and are affine and is smooth. Let be the tangent bundle to . Show that the space of -linear derivations (where is used to equip with the structure of a -module) is identified with .
- Let be a smooth affine variety. Let be the ideal sheaf of the diagonal . Prove that there is a bijection where is the sheaf of differential 1-forms (that is, the sheaf of sections of the cotangent bundle , which is the dual vector bundle of ).
Due Friday, April 7th
- (Hartshorne, II.4.4) Fix a Noetherian scheme , let and be schemes of finite type and separated over , and let be a morphism of -schemes. Suppose that be a closed subscheme that is proper over . Show that is closed.
- In the setting of the previous problem, show that if we consider as a closed subscheme (its ideal of functions consists of all functions whose composition with is zero), then induces a proper map fro to .
(Galois descent, inspired by Hartshorne II.4.7) Let be a finite Galois extension of fields. The Galois group acts on the scheme . Given any -scheme , we let be its extension of scalars; the group acts on in a way compatible with its action on (i.e., this action is ‘semilinear’).
- Show that is affine if and only if is affine.
- Prove that this operation gives a fully faithful functor from the category of -schemes into the category of -schemes with a semi-linear action of .
- Suppose that is a separated -scheme such that any finite subset of is contained in an affine open chart (this holds, for instance, if is quasi-projective). Then for any semi-linear action of on , there exists a -scheme and an isomorphism that agrees with an action of . (That is, the action of gives a -structure on the scheme .)
- Suppose is an -scheme such that . Show that .
- Suppose is an -scheme such that . Show that there are two possibilities for the isomorphism class of .
Due Friday, April 21st
- Let be a singular cubic in , given (in non-homogeneous coordinates) either by (nodal cubic) or by (cuspidal cubic). Compute the class group of Cartier divisors on .
- Let and be schemes over some base scheme . For any map , use the functoriality of the module of Kähler differentials to construct a morphism and verify that .
- Suppose now that and be schemes over an algebraically closed field . A morphism is unramified if . Show that this is equivalent to the following condition: given , the map induces an injection .
- Let us compute the algebraic de Rham cohomology of the affine space. Put , . Since is a smooth -scheme, is a locally free -module. Denote by the exterior algebra of , so that . Define the de Rham differential by starting with the Kähler differential and then extending it by the graded Leibniz rule:
Compute the cohomology of the complex equipped with the differential . The answer will depend on the characteristic of .
- Let be a Noetherian scheme.Let be the -group of : it is generated by elements for each coherent sheaf with relations whenever there is a short exact sequence Prove that . (This is much easier if you know Hilbert’s Syzygy Theorem.)
- Let be a smooth curve over an algebraically closed field. Show that is generated by for line bundles .
- Let be a smooth curve over an algebraically closed field. Show that is isomorphic to . (If this problem is too hard, look at Hartshorne’s II.6.11 for a step-by-step approach.)