On a generalized Auslander-Reiten conjecture

TL;DR

This paper investigates the symmetric Auslander condition (SAC) and its equivalences under various ring modifications, providing new results and classes of rings satisfying SAC, and exploring its relation to other invariants.

Contribution

It establishes equivalences of SAC under ring quotients and change of rings, and introduces new classes of rings satisfying SAC, including determinantal and numerical semigroup rings.

Findings

Proves equivalence of (SAC) for R and R/xR with non-zerodivisor x

Shows (SAC) for R implies (SAC) for S under finite flat dimension

Provides new classes of rings satisfying (SAC) such as determinantal and numerical semigroup rings

Abstract

It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten condition (ARC). In this paper we explore (SAC) along with the several canonical change of rings $R \to S$. First, we prove the equivalence of (SAC) for $R$ and $R/xR$, where $x$ is a non-zerodivisor on $R$, and the equivalence of (SAC) and (SACC) for rings with positive depth, where (SACC) is the symmetric Auslander condition for modules with constant rank. The latter assertion affirmatively answers a question posed by Celikbas and Takahashi. Secondly, for a ring homomorphism $R \to S$, we prove that if $S$ satisfies (SAC) (resp. (ARC)), then $R$ also satisfies (SAC) (resp. (ARC)) if the flat dimension of $S$ over $R$ is finite. We also prove that (SAC) for $R$ implies that (SAC) for $S$ when $R$ is Gorenstein and $S=R/Q^\ell$, where $Q$ is generated by a regular sequence of $R$ and the length of the sequence is at least $\ell$. This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to determinantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC). A relation between (SAC) and an invariant related to the finitistic extension degree is also explored.