A method for constructing minimal projective resolutions over idempotent subrings

Abstract

We show how to obtain a minimal projective resolution of finitely generated modules over an idempotent subring Γe := (1−e)R(1−e) of a semiperfect noetherian basic ring R by a construction inside modR. This is then applied to investigate homological properties of idempotent subrings Γe 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 Se := eR/ rad eR with Ext 1 R (Se, Se) = 0 is self-orthogonal, that is Ext k R (Se, Se) vanishes for all k ≥ 1, whenever gldim R and pdim eR(1 − e) Γe are finite (see [IP16, conjecture 4.13]). Indeed a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose e ∈ R is an idempotent such that all idempotent subrings Γ sandwiched between Γe and R, that is Γe ⊆ Γ ⊆ R, have finite global dimension. Then the simple summands of Se can be numbered S1, . . . , Sn such that Ext k R (Si, Sj) = 0 for 1 ≤ j ≤ i ≤ n and all k > 0.