On the volume of a tilting module

Abstract

We consider a finite dimensional k-algebraA and associate to each tilting module a cone in the Grothendieck groupK0 of finitely generated A-modules. We prove that the set of cones associated to tilting modules of projective dimension at most one defines a, not necessarily finite, fan Σ(A). IfA is of finite global dimension, the fan Σ(A) is smooth. Moreover, using the cone of a tilting module, we can associate a volume to each tilting module. Using the fan and the volume, we obtain new proofs for several classical results; we obtain certain convergent sums naturally associated to the algebraA and obtain criteria for the completeness of a list of tilting modules. Finally, we consider several examples.