A method for constructing minimal projective resolutions over idempotent subrings

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 $k > 0$.