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 $R/⟨1-e⟩$ being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module $S_e:=eR/\mathrm{rad}eR$ with ${\mathrm{Ext}}_R^1(S_e,S_e)=0$ is self-orthogonal, that is ${\mathrm{Ext}}_R^k(S_e,S_e)$ vanishes for all $k\ge 1$, whenever $\mathrm{gl}R$ and $\mathrm{pdim}eR(1-e{)}_{{\Gamma }_e}$ 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 ${\mathrm{Ext}}_R^k(S_i,S_j)=0$ for $1\le j\le i\le n$ and all $k>0$.

arXiv commentAccepted for publication at Proceedings of the American Mathematical Society