Let {X} be an abelian variety over the algebraically closed field {k}. In the previous post, we studied the Picard scheme {\mathrm{Pic}_X}, or rather its connected component {\mathrm{Pic}^0_X} at the identity. The main result was that {\mathrm{Pic}^0_X} was itself an abelian variety (in particular, smooth) of the same dimension as {X}, which parametrizes precisely the translation-invariant line bundles on {X}.

We also saw how to construct isogenies between {X} and {\mathrm{Pic}^0_X}. Given an ample line bundle {\mathcal{L}} on {X}, the map

\displaystyle X \rightarrow \mathrm{Pic}^0_X, \quad x \mapsto t_x^* \mathcal{L} \otimes \mathcal{L}^{-1}

is an isogeny. Such maps were in fact the basic tool in proving the above result.

The goal of this post is to show that the contravariant functor

\displaystyle X \mapsto \mathrm{Pic}^0_X

from abelian varieties over {k} to abelian varieties over {k}, is a well-behaved duality theory. In particular, any abelian variety is canonically isomorphic to its bidual. (This explains why the double Picard functor on a general variety is the universal abelian variety generated by that variety, the so-called Albanese variety.) In fact, we won’t quite finish the proof in this post, but we will finish the most important step: the computation of the cohomology of the universal line bundle on X \times \mathrm{Pic}^0_X.

Motivated by this, we set the notation:

Definition 11 We write {\hat{X}} for {\mathrm{Pic}^0_X}.

The main reference for this post is Mumford’s Abelian varieties. (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…)