Abstract
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.
Highlights
Let X be a scheme and i : Z → X a regular closed immersion
X, where the exceptional divisor is the projective bundle associated to the conormal sheaf NZ/X, which under the assumptions is locally free of rank n
Theorem A for algebraic K-theory was proven by Kerz–Strunk–Tamme [KST18], as part of their proof of Weibel’s conjecture on negative K-theory
Summary
Let X be a scheme and i : Z → X a regular closed immersion. This means that Z is, Zariski-locally on X, the zero-locus of some regular sequence of functions f1, . . . , fn ∈ Γ(X, OX). The canonical morphism i : Z → X is a quasi-smooth closed immersion In this case, Theorem A for algebraic K-theory was proven by Kerz–Strunk–Tamme [KST18] (where the blow-up BlZ/X was explicitly modelled as the derived fibred product X ×An Bl{0}/An), as part of their proof of Weibel’s conjecture on negative K-theory. — The presheaf of spectra S → KH(S) satisfies cdh descent on the site of quasi-compact quasi-separated algebraic spaces This was first proven by Haesemeyer [Hae04] for schemes over a field of characteristic zero, using resolution of singularities. Theorem E can be compared to a similar criterion due to Haesemeyer, implicit in [Hae04], which applies to Nisnevich sheaves of spectra on the category of schemes over a field k of characteristic zero It asserts that for such a sheaf, cdh descent is equivalent to descent for finite cdh squares and regularly immersed blow-up squares.
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