论文标题
在广义的奥斯兰德 - 里列顿猜想上
On a generalized Auslander-Reiten conjecture
论文作者
论文摘要
众所周知，广义的Auslander-Reiten条件（GARC）和对称的Auslander条件（SAC）是等效的，（GARC）意味着Auslander-Reiten条件（ARC）。在本文中，我们将探索（SAC），以及$ r \ to s $的几个规范更改。首先，我们证明了（SAC）对于$ r $和$ r/xr $的等效性，其中$ x $是$ r $上的非嵌合式，以及（sac）和（SAC）和（SACC）的戒指的等效性，对于具有正等级的模块的对称的Auslander条件（SACC）是模块的对称的Auslander条件。后者的断言肯定回答了Celikbas和Takahashi提出的问题。 其次，对于戒指同态$ r \ to s $，我们证明，如果$ s $满足（sac）（sac）（resp。（arc）），则$ r $也可以满足（sac）（sac）（resp。（arc）），如果$ s $的平面尺寸超过$ r $是有限的。我们还证明，$ r $的（sac）意味着（$ r $是gorenstein和$ s = r/q^\ ell $）的（sac），其中$ q $是由常规序列$ r $生成的，序列的长度至少为$ \ ell $。这是本文证明的关于Ulrich理想的更一般结果的结果。将这些结果应用于确定环和数值半群环，我们提供了满足的新类环（SAC）。还探讨了（SAC）与与有限扩展学位有关的不变性之间的关系。
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.
