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

Khan, Adeel A.

Annales Henri Lebesgue, Tome 3 (2020), pp. 1091-1116.

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.

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.

Classification : 20C25, 14J99, 14A20
Mots clés : derived algebraic geometry, semi-orthogonal decompositions, algebraic K-theory, cdh descent

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