Geodesic
The ``fundamental theorem'' for the higher algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings
Documenta mathematica, Tome 26 (2021), pp. 1557-1599.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 groups $\text{NK}^{\pm}_q $. It is shown here that a modified version of this result generalises to strongly $ \mathbb{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 actions are trivial in the classical case). The analogues of the groups $\text{NK}^{\pm}_q$ are identified with the reduced $K$-theory of homotopy nilpotent twisted endomorphisms, and appropriate versions of Mayer-Vietoris and localisation sequences are established.
Classification : 19D50, 19D35, 16E20, 18G35
Keywords: fundamental theorem, higher algebraic \(K\)-theory, strongly \(\mathbb{Z}\)-graded ring, twisted endomorphism, nil term, projective line 
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     title = {The ``fundamental theorem'' for the higher algebraic {\(K\)-theory} of strongly {\(\mathbb{Z}\)-graded} rings},
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Hüttemann, Thomas. The ``fundamental theorem'' for the higher algebraic \(K\)-theory of strongly \(\mathbb{Z}\)-graded rings. Documenta mathematica, Tome 26 (2021), pp. 1557-1599. http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a15/