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.”)

Now, {\mathrm{D}_{\mathrm{perf}}(A)} is a symmetric monoidal {\infty}-category under (derived) tensor product, and as it happens here we have dual objects. There is a duality

\displaystyle D: \mathrm{D}_{\mathrm{perf}}(A) \simeq \mathrm{D}_{\mathrm{perf}}(A)^{op}

sending a complex {X} to its dual {\mathbf{Hom}(X, A)} (where the bolded {\mathbf{Hom}} means the internal mapping object: that is, the mapping chain complex). Given a symmetric monoidal stable {\infty}-category, this sort of duality on compact objects is a frequent occurrence. For instance, in the case of spectra (i.e., modules over the sphere spectrum), the compact objects are the finite spectra, and the relevant duality is Spanier-Whitehead duality. The analog in the 1-categorical case might be duality for finitely generated projective modules over a ring.

The rough idea is that there is a natural map

\displaystyle X \rightarrow \mathbf{Hom}(\mathbf{Hom}(X, A)) ,

for {X} an arbitrary complex, adjoint to the evaluation map {X \otimes \mathbf{Hom}(X, A) \rightarrow A}. This is an equivalence for {X = A}, and thus on the smallest {\infty}-category containing {A} closed under finite limits and colimits and retracts: that is, on the perfect derived category.

2.The duality functor as an Ext

Now let’s return to commutative algebra. We saw in the previous section that there is an equivalence

\displaystyle \mathrm{D}_{\mathrm{perf}}(A) \simeq \mathrm{D}_{\mathrm{perf}}(A)^{op}

for any commutative ring {A}. We want to turn this into a statement about the ordinary 1-categories {\mathrm{Mod}_{\mathrm{sm}}(A)}, when {A} is a regular local ring.

So suppose that {A} is regular local. Then every {A}-module determines an object in the derived category of {A}, and a finitely generated {A}-module in fact lives in {\mathrm{D}_{\mathrm{perf}}(A)}. This is not formal: it follows from the fact that {A} has finite global dimension. In particular, we have a (fully faithful) inclusion of categories

\displaystyle \mathrm{Mod}_{\mathrm{sm}}(A) \subset \mathrm{D}_{\mathrm{perf}}(A).

A complex {X_\bullet} belongs to {\mathrm{Mod}_{\mathrm{sm}}(A)} if and only if it is homologically concentrated in degree zero and the group {\pi_0 X_\bullet} is annihilated by a power of {\mathfrak{m} }.

The point of the present post is that the functor

\displaystyle \mathrm{Mod}_{\mathrm{sm}}(A) \subset \mathrm{D}_{\mathrm{perf}}(A) \stackrel{D}{\rightarrow} \mathrm{D}_{\mathrm{perf}}(A)^{op}

has its image contained in a shift of {\mathrm{Mod}_{\mathrm{sm}}(A)}. In fact, let {M \in \mathrm{Mod}_{\mathrm{sm}}(A)}; we have to show that the complex {\mathbf{Hom}(M, A)} is homologically concentrated in one degree and is {\mathfrak{m}}-power torsion there. However, we observe that

\displaystyle \pi_i( \mathbf{Hom}(M, A)) = \pi_0 \hom ( M[i], A) = \mathrm{Ext}^{-i}(M, A),

for all {i}.

We now invoke:

Proposition 1 Let {A} be regular local of dimension {d}. Then if {M \in \mathrm{Mod}_{\mathrm{sm}}(A)}, we have

\displaystyle \mathrm{Ext}^i( M, A) = 0 \quad \text{if} \ i \neq d.

Moreover, {\mathrm{Ext}^d(M, A) \in \mathrm{Mod}_{\mathrm{sm}}(A)}.

When {i < d}, this is a consequence of the fact that {A} has depth {d}, and the description of depth in terms of vanishing of {\mathrm{Ext}} groups. When {i > d}, the vanishing is a consequence of the fact that {A} has global dimension {d}. The fact that {\mathrm{Ext}^d(M, A)} is annihilated by a power of {\mathfrak{m}} follows from the fact that {M} is.

It follows that the duality functor {\mathbf{Hom}(\cdot, A)} sends {\mathrm{Mod}_{\mathrm{sm}}(A)} into the category {\mathrm{Mod}_{\mathrm{sm}}(A) [-d] \subset \mathrm{D}_{\mathrm{perf}}(A)}.

Combining these observations, we get a fully faithful duality functor {\mathbb{D}} fitting into the above diagram

Concretely, it is given by

\displaystyle \mathbb{D}: M \mapsto \mathrm{Ext}^d(M, A).

The last thing we need to check is that {\mathbb{D}} actually induces an equivalence. To see this, we observe that its inverse is again {\mathbb{D}}: this follows from the fact that {\mathbf{Hom}(\cdot, A)} was a duality functor.

We can conclude:

Theorem 2 The duality functor {\mathbb{D}: M \mapsto \mathrm{Ext}^d(M, A)} induces an equivalence (in fact, a duality) {\mathrm{Mod}_{\mathrm{sm}}(A) \simeq \mathrm{Mod}_{\mathrm{sm}}(A)^{op}}.

Our next goal is to describe this functor in another way.

3. The dualizing module

Observe that this functor {\mathbb{D}} is actually ind-representable, because it is exact (being an equivalence). In other words, there is an {\mathrm{Ind} }-object {Q} of {\mathrm{Mod}_{\mathrm{sm}}(A)} (that is, an {\mathfrak{m}}-torsion module which is not necessarily finitely generated) such that

\displaystyle \mathbb{D} M = \hom(M, Q) , \quad M \in \mathrm{Mod}_{\mathrm{sm}}(A).

This follows from a suitable version of the adjoint functor theorem and {\mathrm{Ind}}-ization. We can actually write down this module very explicitly. Namely, we can take

\displaystyle Q = \varinjlim \mathbb{D} A/\mathfrak{m}^i.

This is an {\mathfrak{m}}-torsion module. Once one knows that {\mathbb{D}} is {\mathrm{Ind}}-representable, it is straightforward that the representing object must be as written.

Observe that it satisfies {\hom_A(k, Q) = k}, because {\hom_A(k, Q) = \mathbb{D} k} must be a simple object of {\mathrm{Mod}_{\mathrm{sm}}(A)} (i.e., {k}). Below, we will give a characterization of {Q} in terms of {k}.

We will need:

Proposition 3 {Q} is an injective {A}-module: in fact, it is the injective hull of {k}.

Proof: We know that {\mathbb{D}} is exact on the category {\mathrm{Mod}_{\mathrm{sm}}(A)}, from which we can see that {Q} is an injective in the category {\mathrm{Ind}(\mathrm{Mod}_{\mathrm{sm}}(A))} of {\mathfrak{m}}-torsion {A}-modules. In fact, injectivity of {Q} will follow from the following result: the functor

\displaystyle \mathrm{Ind}(\mathrm{Mod}_{\mathrm{sm}}(A)) \rightarrow \mathrm{Mod}(A)

preserves injective objects. That is, an {\mathfrak{m}}-torsion module which has the relevant extension property for monomorphisms in {\mathrm{Mod}_{\mathrm{sm}}(A)} is actually an injective {A}-module.

This is not (as far as I can tell) completely formal, and I’ll just sketch the proof. Given a module {I} which is injective in {\mathrm{Ind}(\mathrm{Mod}_{\mathrm{sm}}(A))}, and given an injection of finitely generated {A}-modules {E \hookrightarrow F}, any map

\displaystyle E \rightarrow I

factors through {E/(\mathfrak{m}^t F \cap E) \rightarrow I} for some {t \gg 0}, in view of the Artin-Rees lemma. Now we can extend the map

\displaystyle E/(\mathfrak{m}^t F \cap E) \rightarrow I

to a map {F/\mathfrak{m}^t F \rightarrow I}, by hypothesis. This proves injectivity.

Next we need to see that {Q} is actually the injective hull of {k}. In other words, we have to see that the map {k \rightarrow Q} (which is well-defined up to scaling) is essential: that is, any nonzero submodule of {Q} intersects {k} nontrivially. We can reduce to the finitely generated (hence small) case, so we need to show that for an injection {W \rightarrow \varinjlim \mathbb{D} A/\mathfrak{m}^i}, the fibered product

is nonempty. Suppose {W \rightarrow Q} factors through some {\mathbb{D} A/\mathfrak{m}^N}, {N \gg 0}. Then we might as well take the fibered product

which is dual, under the anti-equivalence {\mathbb{D}}, to the push-out square

Now, by duality, {A/\mathfrak{m}^i \rightarrow \mathbb{D} W} is a surjection, and by defintion the map {A/\mathfrak{m}^i \rightarrow \mathbb{D} k = k} is the reduction map. It follows that the push-out is a quotient of {A/\mathfrak{m}^i} by an ideal contained in {\mathfrak{m}}, so that {\mathbb{D}(W \cap k)} is not zero (in fact, it is {k}). This means {W \cap k \neq 0}, as desired. \Box

So this means that we’ve gotten a fairly concrete result. Given a regular local ring {A}, we can consider the injective envelope {Q} of the residue field; then {\hom(\cdot, Q)} determines an anti-equivalence between {\mathrm{Mod}_{\mathrm{sm}}(A) } and itself. We could have, in fact, proved this directly, and in fact:

Proposition 4 Let {A} be a noetherian local ring (not necessarily regular) and let {Q} be the injective envelope of the residue field {k}. Then {\hom(\cdot, Q)} defines a duality {\mathrm{Mod}_{\mathrm{sm}}(A) \simeq \mathrm{Mod}_{\mathrm{sm}}(A)^{op}}.

In fact, we need to check that the canonical biduality map

\displaystyle t_M: M \rightarrow \hom(\hom(M, Q), Q)

is an isomorphism if {M \in \mathrm{Mod}_{\mathrm{sm}}(A)}. Both functors are exact, as {Q} is injective, and the collection of {M} such that {t_M} is an isomorphism is stable under extensions. Consequently it suffices to check that {t_M} is an isomorphism when {M = k}. Then it follows from the definition of an injective hull: {\hom(k, Q) \simeq k}. (The same sort of argument can be used to show that \mathbb{D} actually takes values in finite length modules.)

In the next post, we’ll apply this to a calculation of the local cohomology groups of {A}.