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COMPLETIONS OF GROTHENDIECK GROUPS PRAMOD N. ACHAR AND CATHARINA STROPPEL
 

Summary: COMPLETIONS OF GROTHENDIECK GROUPS
PRAMOD N. ACHAR AND CATHARINA STROPPEL
Abstract. For a certain class of abelian categories, we show how to make
sense of the "Euler characteristic" of an infinite projective resolution (or, more
generally, certain chain complexes that are only bounded above), by passing to
a suitable completion of the Grothendieck group. We also show that right-exact
functors (or their left-derived functors) induce continuous homomorphisms of
these completed Grothendieck groups, and we discuss examples and applica-
tions coming from categorification.
1. Introduction
Let A be a noetherian and artinian abelian category with enough projectives,
and let Db
(A) be its bounded derived category. The inclusion A Db
(A) gives
rise to a natural isomorphism of Grothendieck groups
(1.1) K(A)

K(Db
(A)).
When A has finite cohomological dimension, K(A) captures a great deal of infor-

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics