Tuesday, February 9, 14:00

Carlo Klapproth, Homological dimensions of idempotent subrings

Abstract:
We investigate homological dimensions of idempotent subrings (1-e)R(1-e) for R in a class of rings including artinian rings. We continue works by Ingalls and Paquette (see [IP15] and [IP17]) and Bravo and Paquette (see [BP20]). We establish a homological relationship of the rings R and (1-e)R(1-e), the semisimple R-module S:=(1-e)R/rad(1-e)R, the (1-e)R(1-e)-module M:=eR(1-e) and the graded Yoneda ring Y of S. In particular we show for R artinian

• how to construct minimal projective resolutions of (1-e)R(1-e)-modules only using homological properties of mod R,
• that Exti(S,S)=0 for i>0 if the global dimension of R and the projective dimension of M are finite and Y has uniform graded right Loewy length and
• that all sandwiched idempotent subrings (1-e)R(1-e) ⊂ (1-f)R(1-f) ⊂ R have finite global dimension iff gldim R < ∞ and all idempotent subrings of Y have finite global dimension.