Algebra / Topology Seminar

Title: Rigid and Test Modules

Speaker: Olgur Celikbas

Abstract: Let R be a commutative Noetherian local ring with unique maximal ideal m. All
R-modules are assumed to be finitely generated. We define: 
([1]) M is said to be Tor-rigid if, for all R-modules N , one has:
Tor^R_n (M, N ) = 0 for some n ≥ 0 =⇒ Tor_{n+1}^R(M,N ) = 0.
([2]) M is said to be a test module if, for all R-modules N , one has:
pd(N ) = ∞ =⇒ Tor^R_n(M, N ) = 0 for infinitely many integers n.
([3]) M is said to be a rigid-test module if it is a Tor-rigid test module.
Test modules are abundant in the literature: for example, over singular hyper-
surface rings, each module of infinite projective dimension is a test module. A
classical example of a rigid-test module is the maximal ideal m. In fact it was
established in that each integrally closed m-primary ideal is a rigid-test module.
In this talk I will discuss some of the tools used to establish the following the-
orem that gives a characterization of Gorenstein rings in terms of the homological
dimensions of rigid and test modules:
Theorem: The following conditions are equivalent:
(i) R is Gorenstein.
(ii) R admits a nonzero test module of finite Gorenstein dimension.
(iii) R admits a nonzero Tor-rigid module of finite injective dimension.
(iv) R is Cohen-Macaulay and R admits a nonzero rigid-test module of finite
Gorenstein injective dimension.
A consequence of the theorem is that, if the Gorenstein injective dimension of
the maximal ideal m is finite, then R is Gorenstein, i.e., in case m is considered, one
does not need to assume R is Cohen-Macaulay, but in general it is an open question
whether or not the Cohen-Macaulay assumption in the theorem (iv) is essential.
The talk is based on the joint works with Dao and Takahashi , Gheibi, Zargar
and Sadeghi, and Wagstaff .