On the volume of a tilting module |
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Authors: | L Hille |
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Institution: | 1. Mathematisches Seminar, Universit?t Hamburg, 20146, Hamburg, Germany
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Abstract: | We consider a finite dimensional k-algebraA and associate to each tilting module a cone in the Grothendieck groupK
0 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.
Dedicated to O. Riemenschneider on the occasion of his 65th birthday |
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Keywords: | 2000 Mathematics Subject Classification" target="_blank">2000 Mathematics Subject Classification Primary 16G30 Secondary 18G35 |
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