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
`Ext`

for^{i}(S,S)=0`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.