Generalized Matlis duality

Abstract

Abstract. Let R be a commutative noetherian ring and let E be the minimal injective cogenerator of the category of R-modules. A module M is said to be reflexive with respect to E if the natural evaluation map from M to Hom R (Hom R (M, E), E) is an isomorphism. We give a classification of modules which are reflexive with respect to E. A module M is reflexive with respect to E if and only if M has a finitely generated submodule S such that M/S is artinian and R/ ann(M ) is a complete semi-local ring.Matlis and Gabriel in [7] and [5] considered modules over a complete local ring R. They showed that if the dual of an R-module is taken with respect to E R (k) (the injective envelope of the residue field k of R), then finitely generated and artinian modules are reflexive.Various authors have considered related questions. For example, dropping the condition that R be complete or weakening local to semilocalIn this paper we let R be any commutative noetherian ring and let E be the minimal injective cogenerator of the category of R-modules. We give a classification of modules which are reflexive with respect to E. The result is that a module M is reflexive with respect to E if and only if M has a finitely generated submodule S such that M/S is artinian and if R/I is a complete semilocal ring where I = ann(M ).We denote by Ω the maximal spectrum of R, and we let E = m∈Ω E R (R/m) be the minimal injective cogenerator in the category of R-modules.∨∨ is an isomorphism we say that M is (Matlis) reflexive. We note that for any M , the map M → M ∨∨ is an injection. From this it is easy to conclude that ann(M ) = ann(M ∨ ). If S ⊂ R is a multiplicative set and the canonical mapIf R is a local ring we letR denote its completion. If M is finitely generated we note thatR ⊗ R M ∼ =M (the completion of M ). We writeR ⊗ R M ∼ =M = M to mean that M →R ⊗ R M ∼ =M is an isomorphism.We note that if m ∈ Ω and M is a finitely generated R-module M , then Hom R (M, E(R/m)) = 0 if and only if ann(M ) ⊂ m.

A change of rings result for Matlis reflexivity

“…We will make use of the main result of [1]: Proof. Since E R/I = Hom R (R/I, E R ), the result follows readily by Hom-tensor adjunction.…”

“…Then M is reflexive as an R-module if and only if M is reflexive as an R S -module. However, the proof given in [1] is incorrect (see Examples 3.1-3.3) and in fact the "if" part is false in general (cf. Proposition 3.4).…”

Artinian Local Cohomology Modules

“…Moreover it is closed under taking submodules, quotients and extensions, i.e., it is a Serre subcategory of the category of R-modules, [R, Z]. Obviously this class is strictly larger than the class of all finite modules and also Artinian modules, [BER,Theorem 12]. Keep in mind that a minimax R-module has only finitely many associated primes.…”

“…It is well known that an R-module M is reflexive (with respect to E) if and only if M is minimax and R/ Ann(M) is a complete semilocal ring, cf. [BER,Theorem 2]. Recall that if N is an arbitrary submodule of a module M, then M is reflexive if and only if both N and M/N are reflexive, cf.…”

“…Recall that if N is an arbitrary submodule of a module M, then M is reflexive if and only if both N and M/N are reflexive, cf. [BER,Lemma 5]. Consequently, a finite direct sum of modules is reflexive if and only if each direct summand is reflexive.…”