Independence of the Total Reflexivity Conditions for Modules

Abstract

We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative Noetherian local ring R and a reflexive R-module M such that Ext i R (M, R) = 0 for all i > 0, but Ext i R (M * , R) = 0 for all i > 0. (2000): 13D07. Mathematics Subject Classification

Algebras that satisfy Auslander’s condition on vanishing of cohomology

“…By another example of Jorgensen andŞega [30], also this question has a negative answer, even for commutative local finite dimensional k-algebras. The following partial answer is part of 4.4.…”

On Reflexive Modules Over Commutative Rings

“…By [23], there exist finitely generated R-reflexive modules which are not in G(R) over a commutative Noetherian local ring R. Now we get the following corollary.…”

Acyclic complexes of finitely generated free modules over local rings

“…Indeed, Jorgensen andŞega in [14] construct minimal acyclic complexes of finitely generated free R-modules C with the property that Hi(C * ) = 0 if and only if i ≥ 1. We thus have the following diagram of implications for complexes of finitely generated free R-modules (with the right implication also following directly from the definitions):…”

“…The properties and uses of complete resolutions have been studied extensively. More recently, the failure of acyclic complexes of free modules to be totally acyclic is studied by Jorgensen andŞega in [14], and, in a more general setting, by Iyengar and Krause in [13]. Structure theorems for acyclic complexes of finitely generated free modules and the rings which admit non-trivial such complexes are given by Christensen and Veliche [11], in the case ᒊ 3 = 0.…”

“…These complexes are precisely those acyclic complexes of finitely generated free modules A satisfying Hi(A * ) = 0 for all i 0 (see Lemma 2.3). Totally acyclic complexes are thus sesquiacyclic, and the examples in [14] show that the converse does not hold. As noted in [11], it was previously not known whether every minimal acyclic complex of finitely generated free modules is sesqui-acyclic.…”