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 Γ e ≔ ( 1 − e ) R ( 1 − e ) \Gamma _e ≔(1-e)R(1-e) of a semiperfect noetherian basic ring R R by a construction inside m o d R \mathsf {mod}\,R . This is then applied to investigate homological properties of idempotent subrings Γ e \Gamma _e under the assumption of R / ⟨ 1 − e ⟩ 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 ≔ e R / rad ⁡ e R S_e ≔eR /\operatorname {rad}eR with Ext R 1 ⁡ ( S e , S e ) = 0 \operatorname {Ext}_R^1(S_e,S_e) = 0 is self-orthogonal, that is Ext R k ⁡ ( S e , S e ) \operatorname {Ext}^k_R(S_e,S_e) vanishes for all k ≥ 1 k \geq 1 , whenever gldim ⁡ R \operatorname {gldim}R and pdim ⁡ e R ( 1 − e ) Γ e \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 ∈ R e \in R is an idempotent such that all idempotent subrings Γ \Gamma sandwiched between Γ e \Gamma _e and R R , that is Γ e ⊆ Γ ⊆ R \Gamma _e \subseteq \Gamma \subseteq R , have finite global dimension. Then the simple summands of S e S_e can be numbered S 1 , … , S n S_1, \dots , S_n such that Ext R k ⁡ ( S i , S j ) = 0 \operatorname {Ext}_R^k(S_i, S_j) = 0 for 1 ≤ j ≤ i ≤ n 1 \leq j \leq i \leq n and all k > 0 k > 0 .