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…)