A method for constructing minimal projective resolutions over idempotent subrings

TL;DR

This paper develops a method to construct minimal projective resolutions over idempotent subrings of semiperfect noetherian rings and applies it to prove a conjecture about the self-orthogonality of certain simple modules.

Contribution

It introduces a new construction for minimal projective resolutions over idempotent subrings and proves a conjecture on the self-orthogonality of specific simple modules under finiteness conditions.

Findings

Established a construction for minimal projective resolutions over idempotent subrings.

Proved that certain simple modules are self-orthogonal when global dimension and projective dimension are finite.

Generalized results to sandwiched idempotent subrings with finite global dimension.

Abstract

We show how to obtain minimal projective resolutions of finitely generated modules over an idempotent subring $\Gamma_e := (1-e)R(1-e)$ of a semiperfect noetherian basic ring $R$ by a construction inside $\mathsf{mod} R$. This is then applied to investigate homological properties of idempotent subrings $\Gamma_e$ under the assumption of $R/\langle 1-e\rangle$ being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module $S_e := eR /\operatorname{rad} eR$ with $\operatorname{Ext}_R^1(S_e,S_e) = 0$ is self-orthogonal, that is $\operatorname{Ext}^k_R(S_e,S_e)$ vanishes for all $k \geq 1$, whenever $\operatorname{gl} R$ and $\operatorname{pdim} eR(1-e)_{\Gamma_e}$ are finite. Indeed, a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose $e \in R$ is an idempotent such that all idempotent subrings $\Gamma$ sandwiched between $\Gamma_e$ and $R$, that is $\Gamma_e \subset \Gamma \subset R$, have finite global dimension. Then the simple summands of $S_e$ can be numbered $S_1, \dots, S_n$ such that $\operatorname{Ext}_R^k(S_i, S_j) = 0$ for $1 \leq j \leq i \leq n$ and all $k > 0$.