132 research outputs found

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

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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.Comment: 24 pages; to appear in Annales Henri Lebesgu

    The cdh-local motivic homotopy category

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    We construct a cdh-local motivic homotopy category SH_cdh(S) over an arbitrary base scheme S, and show that there is a canonical equivalence between SH_cdh(S) and SH(S). We learned this result from D.-C. Cisinski.Comment: 6 pages, posted on author's webpage in 2019; to appear in JPA

    The derived homogeneous Fourier transform

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    We study a derived version of Laumon's homogeneous Fourier transform, which exchanges G_m-equivariant sheaves on a derived vector bundle and its dual. In this context, the Fourier transform exhibits a duality between derived and stacky phenomena. This is the first in a series of papers on derived microlocal sheaf theory.Comment: 29 page

    Modularity of higher theta series II: Chow group of the generic fiber

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    Higher theta series on moduli spaces of Hermitian shtukas were constructed by Feng--Yun--Zhang and conjectured to be modular, parallel to classical conjectures in the Kudla program. In this paper we prove the modularity of higher theta series after restriction to the generic locus. The proof is an upgrade, using motivic homotopy theory, of earlier work of Feng--Yun--Zhang which established generic modularity of â„“\ell-adic realizations. In the process, we develop some general tools of broader utility. One such is the "motivic sheaf-cycle correspondence", a categorical trace formalism for extracting computations in the Chow group from computations in Voevodsky's derived category of motives. Another new tool is the "derived homogeneous Fourier transform", which we use to implement a form of Fourier analysis for motives.Comment: Corrected some statements in section 7.5. This paper absorbs arXiv:2311.13270 as section
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