Completions of Grothendieck groups

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 applications coming from categorification.

13 pages; corrected a mistake in the previous version

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• Derived equivalences for symmetric groups and sl\_2-categorification
• A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products
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• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
• Canonical bases and higher representation theory
• Categorifying fractional Euler characteristics, Jones-Wenzl projector and $3j$-symbols
• An approach to categorification of Verma modules
• A Note on the Grothendieck Group of an Additive Category
• Categorification of tensor powers of the vector representation of $U_q(\mathfrak{gl}(1|1))$
• Constructing categorical idempotents
• Asymptotic Grothendieck groups and c.b.l.f. positive dg-algebras