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…)