The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings

Abstract

Let R be a Cohen-Macaulay local ring and Q be a parameter ideal of R. Due to M. Auslander, S. Ding, and Ø. Solberg, the Auslander-Reiten conjecture holds for R if and only if it holds for the residue ring R/Q. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring R/Q ℓ in connection with that for R and prove the equivalence of them for the case where R is Gorenstein and ℓ ≤ dim R.In the latter part, we study the existence of Ulrich ideals and generalize the result of maximal embedding dimension by J. Sally. Due to these two of our results, we finally show that the Auslander-Reiten conjecture holds if there is an Ulrich ideal whose residue ring is a complete intersection. Besides them, we also explore the Auslander-Reiten conjecture for determinantal rings.As is well known, non-zerodivisors preserve the Auslander-Reiten conjecture (Proposition 2.1). Hence, through Theorem 1.2, many Gorenstein rings which satisfy the Auslander-Reiten conjecture are given. Even if a given local ring is not Gorenstein, the conjecture still holds if the ring is Golod or almost Gorenstein ([21, Proposition 1.4.] and 2010 Mathematics Subject Classification. 13C40, 13D07, 13H10. Key words and phrases. Auslander-Reiten conjecture, Cohen-Macaulay ring, Gorenstein ring, complete intersection, power of parameter ideal, determinantal ring, Ulrich ideal.The author was partially supported by JSPS KAKENHI Grant Number JP19J10579.