Defines algebraic K-theory for unbounded chain complexes via Abel summation, proves it non-trivial, and shows the usual K-theory of R maps into it with an infinite cyclic subgroup in the image.
An Euler Characteristic for Unbounded Chain Complexes
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abstract
We propose a definition of an Euler characteristic for unbounded chain complexes by taking the (usual) Euler characteristics of successively longer parts of the complex, weighted inversely proportional to the length, and passing to the limit. This amounts to taking the limit of the sequence of ranks of homology modules with alternating signs in the sense of the H\"older summation method. We establish the structure of a category with cofibrations and weak equivalences on unbounded complexes for which the infinite Euler characteristic is defined, and show that its Grothendieck group is unusually large (viz., uncountable).
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math.KT 1years
2026 1verdicts
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The Abel Summation Method and Infinite Euler Characteristic
Defines algebraic K-theory for unbounded chain complexes via Abel summation, proves it non-trivial, and shows the usual K-theory of R maps into it with an infinite cyclic subgroup in the image.