Gorenstein-projective modules over short local algebras

TL;DR

This paper investigates the structure of Gorenstein-projective modules over short local algebras, establishing conditions under which these algebras are self-injective and proving the Auslander-Reiten conjecture for this class.

Contribution

It characterizes Gorenstein-projective modules over short local algebras and proves the Auslander-Reiten conjecture in this context, extending known results from commutative to non-commutative cases.

Findings

If a short local algebra has a non-projective Gorenstein-projective module, then it is either self-injective with small radical square or satisfies specific size relations.

Existence of non-projective reflexive modules imposes additional restrictions on the algebra's radical structure.

Any non-projective semi-Gorenstein-projective module has non-zero self-Ext, leading to the proof of the Auslander-Reiten conjecture for these algebras.

Abstract

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra $A$ with radical $J$ will be said to be short provided $J^3 = 0$. As in the commutative case, we show: if a short local algebra $A$ has an indecomposable non-projective Gorenstein-projective module $M$, then either $A$ is self-injective (so that all modules are Gorenstein-projective) and then $|J^2| \le 1$, or else $|J^2| = |J/J^2| - 1$ and $|JM| = |J^2||M/JM|$. More generally, we focus the attention to semi-Gorenstein-projective and $\infty$-torsionfree modules, even to $\mho$-paths of length 2, 3 and 4. In particular, we show that the existence of a non-projective reflexive module implies that $|J^2| < |J/J^2|$ and further restrictions. In addition, we consider exact complexes of projective modules with a non-projective image. Again, as in the commutative case, we see that if such a complex exists, then $A$ is self-injective or satisfies the condition $|J^2| = |J/J^2| - 1.$ Also, we show that any non-projective semi-Gorenstein-projective module $M$ satisfies $Ext^1(M,M) \neq 0$. In this way, we prove the Auslander-Reiten conjecture (one of the classical homological conjectures) for arbitrary short local algebras. Many arguments used in the commutative case actually work in general, but there are interesting differences and some of our results may be new also in the commutative case.