On the $K$-theory of pullbacks

Land, Markus; Tamme, Georg

doi:10.4007/annals.2019.190.3.4

Cited by 43 publications

(42 citation statements)

References 32 publications

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“…Remark 2.18. In the above statement, the p-completion R p can be taken to be either derived or ordinary p-completion; it doesn't matter for the statement, as the (mod p) K-theory of Z[1/p]algebras is nil-invariant and truncating in the sense of [LT19].…”

Section: Complements Combining With the Main Results Of [Cmm18]mentioning

confidence: 99%

Remarks on K(1)-local K-theory

Bhatt

Clausen

Mathew

2020

Sel. Math. New Ser.

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We prove two basic structural properties of the algebraic K-theory of rings after K(1)-localization at an implicit prime p. Our first result (also recently obtained by Land-Meier-Tamme by different methods) states that L K(1) K(R) is insensitive to inverting p on R; we deduce this from recent advances in prismatic cohomology and TC. Our second result yields a Künneth formula in K(1)-local K-theory for adding p-power roots of unity to R.

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Section: Complements Combining With the Main Results Of [Cmm18]mentioning

confidence: 99%

Remarks on K(1)-local K-theory

Bhatt

Clausen

Mathew

2020

Sel. Math. New Ser.

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“…Remark 5.10. -Let E be a localizing invariant and suppose that it is moreover truncating in the sense of [LT18]. That is, if R is a connective E 1 -ring spectrum and Remark 5.11.…”

Section: A Cdh Descent Criterionmentioning

confidence: 99%

Algebraic K-theory of quasi-smooth blow-ups and cdh descent

Khan¹

2020

Annales Henri Lebesgue

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We construct a semi-orthogonal decomposition on the category of perfect complexes on the blow-up of a derived Artin stack in a quasi-smooth centre. This gives a generalization of Thomason's blow-up formula in algebraic K-theory to derived stacks. We also provide a new criterion for descent in Voevodsky's cdh topology, which we use to give a direct proof of Cisinski's theorem that Weibel's homotopy invariant K-theory satisfies cdh descent. Résumé.-Nous construisons une décomposition semi-orthogonale sur la catégorie des complexes parfaits sur l'éclaté d'un champ d'Artin dérivé le long d'un centre quasi-lisse. Ceci conduit à une généralisation de la formule d'éclatement de Thomason en K-théorie algébrique à une situation dérivée. Nous établissons aussi un nouveau critère de descente pour la topologie cdh de Voevodsky, que nous utilisons pour donner une démonstration directe d'un thèorème de Cisinski qui affirme que la K-théorie invariante par homotopie de Weibel satisfait la descente cdh.

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“…is a Milnor square of commutative rings, i.e. a pull-back diagram of rings with g surjective, see [Bas68, Chapter IX, §5] for an early account and [LT18] for a current development. If, in addition, a finite group G acts on eq.…”

Section: Excisionmentioning

confidence: 99%