132 research outputs found
Algebraic K-theory of quasi-smooth blow-ups and cdh descent
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
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
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
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 -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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