1,533 research outputs found
Two-dimensional categorified Hall algebras
In the present paper, we introduce two-dimensional categorified Hall algebras
of smooth curves and smooth surfaces. A categorified Hall algebra is an
associative monoidal structure on the stable -category
of complexes of sheaves with
bounded coherent cohomology on a derived moduli stack .
In the surface case, is a suitable derived enhancement
of the moduli stack of coherent sheaves on the surface. This
construction categorifies the K-theoretical and cohomological Hall algebras of
coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case,
we define three categorified Hall algebras associated with suitable derived
enhancements of the moduli stack of Higgs sheaves on a curve , the moduli
stack of vector bundles with flat connections on , and the moduli stack of
finite-dimensional local systems on , respectively. In the Higgs sheaves
case we obtain a categorification of the K-theoretical and cohomological Hall
algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in
the other two cases our construction yields, by passing to , new
K-theoretical Hall algebras, and by passing to ,
new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and
the non-abelian Hodge correspondences can be lifted to the level of our
categorified Hall algebras of a curve.Comment: 54 page
GAGA problems for the Brauer group via derived geometry
In this paper we prove that the Brauer group of any (derived) scheme ,
proper over the spectrum of a quasi-excellent Henselian ring, injects into the
Brauer group of the Henselization of along the base, generalizing a
classical result of Grothendieck. We offer two proofs of this fact, one based
on a formal GAGA-type theorem for smooth and proper stable -categories
enriched over the -category of quasi-coherent
-modules, and a second one based on a GAGA-type theorem for
perfect complexes on -gerbes.Comment: 33 page
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