The "fundamental theorem" for the algebraic K-theory of strongly ℤ-graded rings

2020/03/03 by Huettemann, Thomas
#FOS: Mathematics #K-Theory and Homology (math.KT) #Primary 19D50 #Rings and Algebras (math.RA) #Secondary 19D35 16E20 18G35

paper · doi:10.48550/arxiv.2003.01506

Abstract

The "fundamental theorem" for algebraic K-theory expresses the K-groups of a Laurent polynomial ring L[t,t^-1] as a direct sum of two copies of the K-groups of L (with a degree shift in one copy), and certain "nil" groups of L. It is shown here that a modified version of this result generalises to strongly ℤ-graded rings; rather than the algebraic K-groups of L, the splitting involves groups related to the shift actions on the category of L-modules coming from the graded structure. (These action are trivial in the classical case). The nil groups are identified with the reduced K-theory of homotopy nilpotent twisted endomorphisms, and analogues of Mayer-Vietoris and localisation sequences are established.

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