Abstract :
LetRbe a ring,Pbe a * -module, andS = End P. Inspired by the tilting theory, we investigate relations between the global dimensions gl dim Rand gl dim S, and between the structure of the Grothendieck groupsK0(mod-R) andK0(mod-S). We prove that gl dim S ≤ gl dim R + D, whereD = 1, e.g., forPalmost tilting, but the symmetry fails: for eachn < ω, there are almost tilting modules with gl dim R − gl dim S > n. IfRis right artinian andSright noetherian, then there is an isomorphismimage, whereimage. We also prove that ifRandR′ are right artinian rings such that there is a torsion theory counter equivalence between Mod-Rand Mod-R′, thenK0(mod-R) congruent with K0(mod-R′).
