Local cohomology III

This post continues the series on local cohomology.

Let be a noetherian ring, an ideal. We are interested in the category of quasi-coherent sheaves on the complement of the closed subscheme cut out by . When for , then

and so is the category of modules over . When is not principal, the open subschemes are generally no longer affine, but understanding quasi-coherent sheaves on them is still of interest. For instance, we might be interested in studying vector bundles on projective space, which pull back to vector bundles on the complement . This is not affine once .

In order to do this, let’s adopt the notation

and let be the open imbedding. This induces a functor

which is right adjoint to the restriction functor . Since the composite is the identity on , we find by a formal argument that is fully faithful.

Fully faithful right adjoint functors have a name in category theory: they are localization functors. In other words, when one sees a fully faithful right adjoint , one should imagine that is obtained from by inverting various morphisms, say a collection . The category sits inside as the subcategory of -local objects: in other words, those objects such that turns morphisms in into isomorphisms.

 

Purity in dimension two

Let be a smooth projective variety, and let be a generic hyperplane. For generic enough , the intersection is itself a smooth projective variety of dimension one less. The Lefschetz hyperplane theorem asserts that the map

induces an isomorphism on , if .

We might be interested in analog over any field, possibly of characteristic . Here has to be replaced with its étale analog, but otherwise it is a theorem of Grothendieck that still induces an isomorphism on , under the same hypotheses. This is one of the main results of SGA 2, and it uses the local cohomology machinery developed there. One of my goals in the next few posts is to understand some of the ideas that go into Grothendieck’s argument.

More generally, suppose is a subvariety. To say that (always in the étale sense) is to say that there is an equivalence of categories

between étale covers of and étale covers of . How might one prove such a result? Grothendieck’s strategy is to attack this problem in three stages:

1. Compare to , where is a neighborhood of in .
2. Compare to , where is the formal completion of along (i.e., the inductive limit of the infinitesimal thickenings of ).
3. Compare and .

In other words, to go from to , one first passes to the formal completion along , then to an open neighborhood, and then to all of . The third step is the easiest: it is the topological invariance of the étale site. The second step is technical. In this post, we’ll only say something about the first step.

The idea behind the first step is that, if is not too small, the passage from to will involve adding only subvarieties of codimension , and these will unaffect the category of étale covers. There are various “purity” theorems to this effect.

The goal of  this post is to sketch Grothendieck’s proof of the following result of Zariski and Nagata.

Theorem 10 (Purity in dimension two) Let be a regular local ring of dimension , and let . Then the map

is an equivalence of categories.

In other words, “puncturing” the spectrum of a regular local ring does not affect the fundamental group.

 

Local cohomology II

Let be a regular local (noetherian) ring of dimension . In the previous post, we described loosely the local cohomology functors

(in fact, described them in three different ways), and proved a fundamental duality theorem

Here is the Matlis duality functor , for an injective envelope of the residue field . This was stated initially as a result in the derived category, but we are going to use the above form.

The duality can be rewritten in a manner analogous to Serre duality. We have that (in fact, this could be taken as a definition of ). For any , there is a Yoneda pairing

and the local duality theorem states that it is a perfect pairing.

Example 1 Let be an algebraically closed field, and suppose that is the local ring of a closed point on a smooth -variety . Then we can take for the module

in other words, the module of -linear distributions (supported at that point). To see this, note that defines a duality functor on the category of finite length -modules, and any such duality functor is unique. The associated representing object for this duality functor is precisely .

In this case, we can think intuitively of as the cohomology

These can be represented by meromorphic -forms defined near ; any such defines a distribution by sending a function defined near to . I’m not sure to what extent one can write an actual comparison theorem with the complex case.

 

Local cohomology I

Fix a noetherian local ring .

Let (for the derived category of , or preferably its higher-categorical analog). Let us define the local cohomology functor

We can think of this at a number of levels: for instance, it is the (derived) functor of the ordinary functor on -modules which sends an -module to its submodule

of -power torsion elements. From this point of view, we can think of the cohomology groups

as defining “cohomology with supports” for the pair with coefficients in the sheaf . I’ll try to elaborate more on this point of view later.

Notation: The derived categories in this post will use cohomological grading conventions, for simplicity.

Our first goal here is to describe the calculation (which is now quite formal) of in the regular case, in terms of dualizing objects. So, let’s suppose is regular local on, of dimension . In this case, each lives in the smaller perfect derived category , and we will use the duality in that category.

Namely, recall that we have a functor given by , which induces a duality on the perfect derived category of , as we saw yesterday.

Let . We saw in the previous post that is cohomologically concentrated in the degree , and it is a shift of the module : we saw that was the injective envelope of . The next result will reduce the computation of to an computation.

Theorem 5 (Local duality) If and is regular, then we have a canonical isomorphism in ,

 

Local duality

Let be a regular local (noetherian) ring with maximal ideal and residue field . The purpose of this post is to construct an equivalence (in fact, a duality)

between the category of finite length -modules (i.e., finitely generated modules annihilated by a power of ) and its opposite. Such an anti-equivalence holds in fact for any noetherian local ring , but in this post we will mostly stick to the regular case. In the next post, we’ll use this duality to give a description of the local cohomology groups of a noetherian local ring. Most of this material can be found in the first couple of sections of SGA 2 or in Hartshorne’s Local Cohomology.

1. Duality in the derived category

Let be any commutative ring, and let be the perfect derived category of . This is the derived category (or preferably, derived -category) of perfect complexes of -modules: that is, complexes containing a finite number of projectives. is the smallest stable subcategory of the derived category containing the complex in degree zero, and closed under retracts. It can also be characterized abstractly: consists of the compact objects in the derived category of . That is, a complex is quasi-isomorphic to something in if and only if the functor

commutes with homotopy colimits. (“Chain complexes” could replace “spaces.”)

 

A derived characterization of open immersions

I’d like to discuss today a category-theoretic characterization of Zariski open immersions of rings, which I learned from Toen-Vezzosi’s article.

Theorem 1 If is a finitely presented morphism of commutative rings, then is an open immersion if and only if the restriction functor between derived categories is fully faithful.

Toen and Vezzosi use this to define a Zariski open immersion in the derived context, but I’d like to work out carefully what this means in the classical sense. If one has an open immersion (for instance, a localization ), then the pull-back on derived categories is fully faithful: in other words, the composite of push-forward and pull-back is the identity.

 

Re-learning homological algebra

I’ve been trying to re-understand some of the proofs in commutative and homological algebra. I never really had a good feeling for spectral sequences, but they seemed to crop up in purely theoretical proofs quite frequently. (Of course, they crop up in computations quite frequently, too.) After learning about derived categories it became possible to re-interpret many of these proofs. That’s what I’d like to do in this post.

Here is a toy example of a result, which does not use spectral sequences in its usual proof, but which can be interpreted in terms of the derived category.

Proposition 1 Let be a local noetherian ring with residue field . Then a finitely generated -module such that is free.

Let’s try to understand the usual proof in terms of the derived category. Throughout, this will mean the bounded-below derived category of -modules: in other words, this is the category of bounded-below complexes of projectives and homotopy classes of maps. Any module can be identified with an object of by choosing a projective resolution.

So, suppose satisfies . Another way of saying this is that the derived tensor product

has no homology in negative degrees (it is in degree zero). Choose a free -module with a map which induces an isomorphism . Then we have that

by hypothesis. In particular, if is the cofiber (in ) of , then .

We’d like to conclude from this that is actually zero, or that : this will imply the desired freeness. Here, we have:

Lemma 2 (Derived Nakayama) Let have finitely generated homology. Suppose . Then .

 

Topological invariance of the etale site via category theory

Let be a ring. An -algebra is said to be étale if it is finitely presented and for every -algebra and every nilpotent ideal (or ideal consisting of nilpotents), we have an isomorphism

In other words, given a homomorphism of -algebras , we can lift it uniquely to the “nilpotent thickening” .

The algebras étale over form a category, ; this is the étale site of . For example, for a field, it consists of the category of all finite separable extensions. \’Etaleness is preserved under base-change, so for any morphism , there is a functor

A basic property of étale morphisms is the following:

Theorem 1 Let be a ring and a square zero (or nilpotent) ideal. Then there is an equivalence of categories

given by tensoring with .

This result is often proved using the local structure theory for étale morphisms, but this is fairly difficult: as far as I know, the local structure theory requires Zariski’s Main Theorem for its proof. (Correction: as observed below, one only needs a portion of the local structure theory to make the argument, and that portion does not require ZMT.) Here is a more elementary argument.

 

The Quillen-Suslin theorem

[Apologies in the delay in posts on the Segal paper — there are a couple of things I’m confused on that are preventing me from proceeding.]

A classical problem (posed by Serre) was to determine whether there were any nontrivial algebraic vector bundles over affine space , for an algebraically closed field. In other words, it was to determine whether a finitely generated projective module over the ring is necessarily free. The topological analog, whether (topological) vector bundles on are trivial is easy because is contractible. The algebraic case is harder.

The problem was solved affirmatively by Quillen and Suslin. In this post, I would like to describe an elementary proof, due to Vaserstein, of the Quillen-Suslin theorem.

 

The cotangent complex I: group objects in categories of algebras

I’d now like to begin a series of posts on the cotangent complex, following Daniel Quillen’s paper “Homology of Commutative Rings.” (There are also two very nice articles from a 2004 summer school on “Homotopy and algebra” on the subject, those by Goerss-Schemmerhorn and Iyengar, that discuss the topic.) While the cotangent complex can be defined quite cleanly once one has the appropriate categorical setting, it will be useful to spend a brief period formulating that.

1. Generalities

Let be a commutative ring. Ultimately, we are going to think of the cotangent complex of an -algebra as a “linearization” or “abelianization.” Viewed more precisely, the cotangent complex will be the derived functor of abelianization (this is the general means of defining “homology” in a model category). It will turn out that abelianization will correspond to taking the module of Kähler differentials, so that the cotangent complex will also be a derived functor of those.

The problem is the category of -algebras does not exactly admit a nice abelianization functor. Recall:

Definition 1 If is a category with finite products, then an abelian monoid object in is an object together with a multiplication morphism and a unit (where is the terminal object). These are required to satisfy the usual commutativity and associativity constraints. For instance,

should be the identity.

The terminal object in the category is the zero ring, and it cannot map on any nonzero ring. So there are no nontrivial abelian group objects in this category!