 
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 rightexact
functors (or their leftderived 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
