## Algebraic $$K$$-theory of quasi-smooth blow-ups and cdh descent. (K-théorie algébrique des éclatements quasi-lisses et descente pour la topologie cdh.)(English. French summary)Zbl 07272950

Summary: 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.

### MSC:

 19E08 $$K$$-theory of schemes 14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 20C25 Projective representations and multipliers 14F42 Motivic cohomology; motivic homotopy theory 14A20 Generalizations (algebraic spaces, stacks) 14A30 Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.)
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### References:

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