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 R (M, R) = 0 for all i > 0, but Ext R (M, R) 6= 0 for all i > 0. introduction Let R be a commutative Noetherian ring. For any R-module M we set M = HomR(M,R). An R-module M is said to be reflexive if it is finite and the canonical map M → M is bijective. A finite R-module M is said to be totally reflexive if it satisfies the following conditions: (i) M is reflexive (ii) ExtiR(M,R) = 0 for all i > 0 (iii) ExtiR(M , R) = 0 for all i > 0. This notion is due to Auslander and Bridger [1]: the totally reflexive modules are precisely the modules of G-dimension zero. The G-dimension of a module is one of the best studied non-classical homological dimensions, and is defined in terms of the length of a resolution of the module by totally reflexive modules. Given any homological dimension, a serious concern is whether its defining conditions can be verified effectively. For example, the projective dimension of a finite R-module M is zero if and only if ExtR(M,N) = 0 for all finite R-modules N . However, when R is local with maximal ideal m, one only needs to check vanishing for N = R/m. In the same spirit, it is natural to ask whether the set of conditions defining total reflexivity is overdetermined (cf. [4, §2]) and in particular, whether total reflexivity for a module can be established by verifying vanishing of only finitely many Ext modules. When R is a local Gorenstein ring, (ii) implies the other two conditions above, and it is equivalent to M being maximal Cohen-Macaulay. Recently, Yoshino [9] studied other situations when (ii) alone implies total reflexivity, and raised the question whether this is always the case. In the present paper we give an example of a local Artinian ring R which admits modules whose total reflexivity conditions are independent, in that (ii) implies neither (i) nor (iii); (i) and (ii) do not imply (iii), equivalently, (i) and (iii) do not imply (ii). More precisely, we prove the following result as Theorem 1.7: Theorem. There exists a local Artinian ring R, and a family {Ms}s>1 of reflexive R-modules such that Date: April 7, 2008. 1 2 D. A. JORGENSEN AND L. M. ŞEGA (1) ExtiR(Ms, R) = 0 if and only if 1 ≤ i ≤ s− 1; (2) ExtiR(M ∗ s , R) = 0 for all i > 0. Moreover, there exists a non-reflexive R-module L such that (1) ExtiR(L,R) = 0 for all i > 0; (2) ExtiR(L , R) 6= 0 for all i > 0. By taking Ns = M ∗ s for all s ≥ 1, we get a statement dual to that of the first part above: there exists a family {Ns}s>1 of reflexive R-modules such that (1) ExtiR(N ∗ s , R) = 0 if and only if 1 ≤ i ≤ s− 1; (2) Ext i R(Ns, R) = 0 for all i > 0. This theorem shows that in order to check whether or not a module M is totally reflexive — even for a local Artinian ring — one needs to check vanishing of ExtiR(M,R) and Ext i R(M , R) for infinitely many values of i. In Section 2, however, we point out that when R is standard graded, in the sense that R = ⊕∞ i=0 Ri with R0 = k, a field, and R = R0[R1], one may skip checking finitely many values of i of the same parity. In our example, R is a standard graded Koszul algebra and has Hilbert series ∑ i≥0 rankk(Ri)t i = 1 + 4t + 3t. The ring R is thus local, and its maximal ideal m satisfies m = 0. Note that our example is minimal in the following sense: if m 2 = 0, then any finite R-module M which satisfies ExtiR(M,R) = 0 for some i > 1 is totally reflexive, hence (ii) alone implies total reflexivity. (See 1.1 below.) Our construction involves a minimal acyclic complex C of finite free R-modules such that the sequence {rankR(Ci)}i>0 is strictly increasing and has exponential growth, while the sequence {rankR(C−i)}i>0 is constant. In the last section we raise several related questions.

Cite this paper

@inproceedings{Jorgensen2007IndependenceOT, title={Independence of the Total Reflexivity Conditions for Modules}, author={David A. Jorgensen and LIANA M. ŞEGA}, year={2007} }