commutative algebra

This post continues the series on local cohomology.

Let ${A}$ be a noetherian ring, ${\mathfrak{a} \subset A}$ an ideal. We are interested in the category ${\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))}$ of quasi-coherent sheaves on the complement of the closed subscheme cut out by ${\mathfrak{a}}$. When ${\mathfrak{a} = (f)}$ for ${f \in A}$, then

$\displaystyle \mathrm{Spec} A \setminus V(\mathfrak{a}) = \mathrm{Spec} A_f,$

and so ${\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))}$ is the category of modules over ${A_f}$. When ${\mathfrak{a}}$ is not principal, the open subschemes ${\mathrm{Spec} A \setminus V(\mathfrak{a})}$ 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 ${\mathbb{A}^{n+1} \setminus \left\{(0, 0, \dots, 0)\right\}}$. This is not affine once ${n > 0}$.

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

$\displaystyle X' = \mathrm{Spec} A \setminus V(\mathfrak{a}) , \quad X = \mathrm{Spec} A,$

and let ${i: X' \rightarrow X}$ be the open imbedding. This induces a functor

$\displaystyle i_* : \mathrm{QCoh}(X') \rightarrow \mathrm{QCoh}(X)$

which is right adjoint to the restriction functor ${i^* : \mathrm{QCoh}(X) \rightarrow \mathrm{QCoh}(X')}$. Since the composite ${i^* i_* }$ is the identity on ${\mathrm{QCoh}(X')}$, we find by a formal argument that ${i_*}$ 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 ${\mathcal{C} \rightarrow \mathcal{D}}$, one should imagine that ${\mathcal{C}}$ is obtained from ${\mathcal{D}}$ by inverting various morphisms, say a collection ${S}$. The category ${\mathcal{C}}$ sits inside ${\mathcal{D}}$ as the subcategory of ${S}$-local objects: in other words, those objects ${x}$ such that ${\hom(\cdot, x)}$ turns morphisms in ${S}$ into isomorphisms. (more…)

Let ${X \subset \mathbb{P}^r_{\mathbb{C}}}$ be a smooth projective variety, and let ${H}$ be a generic hyperplane. For generic enough ${H}$, the intersection ${X \cap H}$ is itself a smooth projective variety of dimension one less. The Lefschetz hyperplane theorem asserts that the map

$\displaystyle H \cap X \rightarrow X$

induces an isomorphism on ${\pi_1}$, if ${\dim X \geq 3}$.

We might be interested in analog over any field, possibly of characteristic ${p}$. Here ${\pi_1}$ has to be replaced with its étale analog, but otherwise it is a theorem of Grothendieck that ${H \cap X \rightarrow X}$ still induces an isomorphism on ${\pi_1}$, 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 ${Y \subset X}$ is a subvariety. To say that ${\pi_1(Y) \simeq \pi_1(X)}$ (always in the étale sense) is to say that there is an equivalence of categories

$\displaystyle \mathrm{Et}(X) \simeq \mathrm{Et}(Y)$

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

1. Compare ${\mathrm{Et}(X)}$ to ${\mathrm{Et}(U)}$, where ${U \supset Y}$ is a neighborhood of ${Y}$ in ${X}$.
2. Compare ${\mathrm{Et}(U)}$ to ${\mathrm{Et}(\hat{X}_Y)}$, where ${\hat{X}_Y}$ is the formal completion of ${X}$ along ${Y}$ (i.e., the inductive limit of the infinitesimal thickenings of ${Y}$).
3. Compare ${\mathrm{Et}( \hat{X})_Y)}$ and ${\mathrm{Et}(Y)}$.

In other words, to go from ${Y}$ to ${X}$, one first passes to the formal completion along ${Y}$, then to an open neighborhood, and then to all of ${Y}$. 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 ${Y}$ is not too small, the passage from ${U}$ to ${X}$ will involve adding only subvarieties of codimension ${\geq 2}$, 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 ${(A, \mathfrak{m})}$ be a regular local ring of dimension ${2}$, and let ${X = \mathrm{Spec} A}$. Then the map

$\displaystyle \mathrm{Et}(X) \rightarrow \mathrm{Et}(X \setminus \left\{\mathfrak{m}\right\})$

is an equivalence of categories.

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

Let ${(A, \mathfrak{m})}$ be a regular local (noetherian) ring of dimension ${d}$. In the previous post, we described loosely the local cohomology functors

$\displaystyle H^i_{\mathfrak{m}}: \mathrm{Mod}(A) \rightarrow \mathrm{Mod}(A)$

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

$\displaystyle H^i_{\mathfrak{m}}(M) \simeq \mathrm{Ext}^{d-i}(M, A)^{\vee}.$

Here ${\vee}$ is the Matlis duality functor ${\hom(\cdot, Q)}$, for ${Q}$ an injective envelope of the residue field ${A/\mathfrak{m}}$. 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 ${H^d_{\mathfrak{m}}(A) \simeq Q}$ (in fact, this could be taken as a definition of ${Q}$). For any ${M}$, there is a Yoneda pairing

$\displaystyle H^i_{\mathfrak{m}}(M) \times \mathrm{Ext}^{d-i}(M, A) \rightarrow H^d_{\mathfrak{m}}(A) \simeq Q,$

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

Example 1 Let ${k}$ be an algebraically closed field, and suppose that ${(A, \mathfrak{m})}$ is the local ring of a closed point ${p}$ on a smooth ${k}$-variety ${X}$. Then we can take for ${Q}$ the module

$\displaystyle \hom^{\mathrm{top}}_k(A, k) = \varinjlim \hom_k(A/\mathfrak{m}^i, k):$

in other words, the module of ${k}$-linear distributions (supported at that point). To see this, note that ${\hom_k(\cdot, k)}$ defines a duality functor on the category ${\mathrm{Mod}_{\mathrm{sm}}(A)}$ of finite length ${A}$-modules, and any such duality functor is unique. The associated representing object for this duality functor is precisely ${\hom^{\mathrm{top}}_k(A, k)}$.

In this case, we can think intuitively of ${H^i_{\mathfrak{m}}(A)}$ as the cohomology

$\displaystyle H^i(X, X \setminus \left\{p\right\}).$

These can be represented by meromorphic ${d}$-forms defined near ${p}$; any such ${\omega}$ defines a distribution by sending a function ${f}$ defined near ${p}$ to ${\mathrm{Res}_p(f \omega)}$. I’m not sure to what extent one can write an actual comparison theorem with the complex case. (more…)

Fix a noetherian local ring ${(A, \mathfrak{m})}$.

Let ${C \in D(A)}$ (for ${D(A)}$ the derived category of ${A}$, or preferably its higher-categorical analog). Let us define the local cohomology functor

$\displaystyle \Gamma_{\mathfrak{m}}: D(A) \rightarrow D(A), \quad \Gamma_{\mathfrak{m}}(C) = \varinjlim \mathbf{Hom}({A}/\mathfrak{m}^i, C).$

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

$\displaystyle \Gamma_{\mathfrak{m}}(M) = \varinjlim\hom(A/\mathfrak{m}^i, M)$

of ${\mathfrak{m}}$-power torsion elements. From this point of view, we can think of the cohomology groups

$\displaystyle H^i_{\mathfrak{m}}(M) \stackrel{\mathrm{def}}{=} H^i (\Gamma_{\mathfrak{m}}(M))$

as defining “cohomology with supports” for the pair ${(\mathrm{Spec} A, \mathrm{Spec} A \setminus \left\{\mathfrak{m}\right\})}$ with coefficients in the sheaf ${M}$. 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 ${\Gamma_{\mathfrak{m}}}$ in the regular case, in terms of dualizing objects. So, let’s suppose ${A}$ is regular local on, of dimension ${d = \dim A}$. In this case, each ${A/\mathfrak{m}^i}$ lives in the smaller perfect derived category ${\mathrm{D}_{\mathrm{perf}}(A)}$, and we will use the duality in that category.

Namely, recall that we have a functor ${D: \mathrm{D}_{\mathrm{perf}}(A) \rightarrow \mathrm{D}_{\mathrm{perf}}(A)^{op} }$ given by ${\mathbf{Hom}(\cdot, A)}$, which induces a duality on the perfect derived category of ${A}$, as we saw yesterday.

Let ${K = \varinjlim DA/\mathfrak{m}^i}$. We saw in the previous post that ${K}$ is cohomologically concentrated in the degree ${d}$, and it is a shift of the module ${Q = \varinjlim \mathrm{Ext}^d(A/\mathfrak{m}^i, A)}$: we saw that ${Q}$ was the injective envelope of ${k}$. The next result will reduce the computation of ${\Gamma_{\mathfrak{m}}}$ to an ${\mathrm{Ext}}$ computation.

Theorem 5 (Local duality) If ${C \in \mathrm{D}_{\mathrm{perf}}(A)}$ and ${A}$ is regular, then we have a canonical isomorphism in ${D(A)}$,

$\displaystyle \Gamma_{\mathfrak{m}}(C) \simeq \mathbf{Hom}( D C, K).$ (more…)

Let ${A}$ be a regular local (noetherian) ring with maximal ideal ${\mathfrak{m}}$ and residue field ${k}$. The purpose of this post is to construct an equivalence (in fact, a duality)

$\displaystyle \mathbb{D}: \mathrm{Mod}_{\mathrm{sm}}(A) \simeq \mathrm{Mod}_{\mathrm{sm}}(A)^{op}$

between the category ${\mathrm{Mod}_{\mathrm{sm}}(A)}$ of finite length ${A}$-modules (i.e., finitely generated modules annihilated by a power of ${\mathfrak{m}}$) and its opposite. Such an anti-equivalence holds in fact for any noetherian local ring ${A}$, 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 ${A}$ be any commutative ring, and let ${\mathrm{D}_{\mathrm{perf}}(A)}$ be the perfect derived category of ${A}$. This is the derived category (or preferably, derived ${\infty}$-category) of perfect complexes of ${A}$-modules: that is, complexes containing a finite number of projectives. ${\mathrm{D}_{\mathrm{perf}}(A)}$ is the smallest stable subcategory of the derived category containing the complex ${A}$ in degree zero, and closed under retracts. It can also be characterized abstractly: ${\mathrm{D}_{\mathrm{perf}}(A)}$ consists of the compact objects in the derived category of ${A}$. That is, a complex ${X}$ is quasi-isomorphic to something in ${\mathrm{D}_{\mathrm{perf}}(A)}$ if and only if the functor

$\displaystyle \hom(X, \cdot) : \mathrm{D}(A) \rightarrow \mathbf{Spaces}$

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

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 ${f: A \rightarrow B}$ is a finitely presented morphism of commutative rings, then ${\mathrm{Spec} B \rightarrow \mathrm{Spec} A}$ is an open immersion if and only if the restriction functor ${D^-(B) \rightarrow D^-(A)}$ 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 ${f: A \rightarrow B}$ (for instance, a localization ${A \rightarrow A_f}$), then the pull-back on derived categories is fully faithful: in other words, the composite of push-forward and pull-back is the identity. (more…)

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 ${(A, \mathfrak{m})}$ be a local noetherian ring with residue field ${k}$. Then a finitely generated ${A}$-module ${M}$ such that ${\mathrm{Tor}_i(M, k) = 0, i > 0}$ 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 ${D^-(A)}$ of ${A}$-modules: in other words, this is the category of bounded-below complexes of projectives and homotopy classes of maps. Any module ${M}$ can be identified with an object of ${D^-(A)}$ by choosing a projective resolution.

So, suppose ${M}$ satisfies ${\mathrm{Tor}_i(M, k) = 0, i > 0}$. Another way of saying this is that the derived tensor product

$\displaystyle M \stackrel{\mathbb{L}}{\otimes} k$

has no homology in negative degrees (it is ${M \otimes k}$ in degree zero). Choose a free ${A}$-module ${P}$ with a map ${P \rightarrow M}$ which induces an isomorphism ${P \otimes k \simeq M \otimes k}$. Then we have that

$\displaystyle P \stackrel{\mathbb{L}}{\otimes} k \simeq M \stackrel{\mathbb{L}}{\otimes} k$

by hypothesis. In particular, if ${C}$ is the cofiber (in ${D^-(A)}$) of ${P \rightarrow M}$, then ${C \stackrel{\mathbb{L}}{\otimes} k = 0}$.

We’d like to conclude from this that ${C}$ is actually zero, or that ${P \simeq M}$: this will imply the desired freeness. Here, we have:

Lemma 2 (Derived Nakayama) Let ${C \in D^-(A)}$ have finitely generated homology. Suppose ${C \stackrel{\mathbb{L}}{\otimes} k = 0}$. Then ${C = 0}$. (more…)

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