Let be a regular local (noetherian) ring with maximal ideal and residue field . The purpose of this post is to construct an equivalence (in fact, a duality)

between the category of finite length -modules (i.e., finitely generated modules annihilated by a power of ) and its opposite. Such an anti-equivalence holds in fact for any noetherian local ring , 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 be any commutative ring, and let be the perfect derived category of . This is the derived category (or preferably, derived -category) of perfect complexes of -modules: that is, complexes containing a finite number of projectives. is the smallest stable subcategory of the derived category containing the complex in degree zero, and closed under retracts. It can also be characterized abstractly: consists of the compact objects in the derived category of . That is, a complex is quasi-isomorphic to something in if and only if the functor

commutes with homotopy colimits. (“Chain complexes” could replace “spaces.”) (more…)