Characterizations of regular local rings via syzygy modules of the residue field

Abstract

Let R be a commutative Noetherian local ring with residue field k. We show that if a finite direct sum of syzygy modules of k surjects onto 'a semidualizing module' or 'a non-zero maximal Cohen-Macaulay module of finite injective dimension', then R is regular. We also prove that R is regular if and only if some syzygy module of k has a non-zero direct summand of finite injective dimension.2010 Mathematics Subject Classification. Primary 13D02; Secondary 13D05, 13H05.

Complexity and rigidity of Ulrich modules, and some applications

“…Therefore, along with [30,Cor. 5.3], we get that M is a homomorphic image of a finite direct sum of some R-syzygy modules of k. So, by [19,Lem. 2.1], it follows that m ⊆ Soc(R) ⊆ ann R (M ).…”

“…(3) Similar characterizations of Gorenstein local rings are shown in Proposition 2. 19 in terms of certain consecutive vanishing of Ext or Tor involving an Ulrich module and a nonzero module of finite projective or injective dimension.…”

Auslander-Reiten conjecture and finite injective dimension of Hom

“…For example, R is regular if and only if its residue field k has finite injective dimension, see, e.g., [5, 3.1.26]. More generally, it is shown in [10,Thm. 3.7] that R is regular if and only if some syzygy Ω n R (k) (n 0) has a nonzero direct summand of finite injective dimension.…”

“…C]. (There is some overlapping among these eleven conditions, e.g., (1) is included in (2) as well as in (3); while (4), ( 5) and ( 8) are included in (10).) However the conjecture is widely open even for Gorenstein local rings.…”

Some characterizations of special rings by delta-invariant

“…Later on, Takahashi, in [21,Theorem 4.3], generalized the result in terms of the existence of a syzygy module of the residue field having a semidualizing module as its direct summand. Also Ghosh et.al, in [12,Theorem 3.7], have shown that the ring is regular if and only if a syzygy module of k has a non-zero direct summand of finite injective dimension. Now we investigate these notions by means of delta invariant.…”