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…)