Dimension estimates for representable equivalences of module categories

[摘要] Let R be a ring, P be a *-module, and S = End P. Inspired by the tilting theory, we investigate relations between the global dimensions gl dim R and gl dim S, and between the structure of the Grothendieck groups K-0(mod-R) and K-0(mod-S). We prove that gl dim S less than or equal to gl dim R + D, where D = 1, e.g., for P almost tilting, but the symmetry fails: for each n < w, there are almost tilting modules with gl dim R - gl dim S > n. If R is right artinian and S right noetherian, then there is an isomorphism K-0(mod-S)congruent to K-0(mod-(R) over bar), where (R) over bar = R/Ann(R)(P). We also prove that if R and R' are right artinian rings such that there is a torsion theory counter equivalence between Mod-R and Mod-R', then K-0(mod-R)congruent to K-0(mod-R'). (C) 1997 Academic Press.

[发布日期] 1997-07-15