Completions of Grothendieck groups

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 bounded only 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 applications coming from categorification.