Geodesic
The "fundamental theorem" for the higher algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings
Documenta mathematica, Tome 26 (2021), pp. 1557-1599 Cet article a éte moissonné depuis la source EMS Press

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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 NKq±​. 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 actions are trivial in the classical case). The analogues of the groups NKq±​ are identified with the reduced K-theory of homotopy nilpotent twisted endomorphisms, and appropriate versions of Mayer-Vietoris and localisation sequences are established.
DOI : 10.4171/dm/849
Classification : 16E20, 18G35, 19D35, 19D50
Mots-clés : higher algebraic K-theory, fundamental theorem, strongly Z-graded ring, twisted endomorphism, nil term, projective line 
@article{10_4171_dm_849,
     author = {Thomas H\"uttemann},
     title = {The "fundamental theorem" for the higher algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings},
     journal = {Documenta mathematica},
     pages = {1557--1599},
     year = {2021},
     volume = {26},
     doi = {10.4171/dm/849},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/849/}
}
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Thomas Hüttemann. The "fundamental theorem" for the higher algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings. Documenta mathematica, Tome 26 (2021), pp. 1557-1599. doi: 10.4171/dm/849

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