420 research outputs found
From Sheaf Cohomology to the Algebraic de Rham Theorem
Let X be a smooth complex algebraic variety with the Zariski topology, and
let Y be the underlying complex manifold with the complex topology.
Grothendieck's algebraic de Rham theorem asserts that the singular cohomology
of Y with complex coefficients can be computed from the complex of sheaves of
algebraic differential forms on X. This article gives an elementary proof of
Grothendieck's algebraic de Rham theorem, elementary in the sense that we use
only tools from standard textbooks as well as Serre's FAC and GAGA papers.Comment: 53 pages; this version replaces an earlier version submitted in
August 2013. Some misprints have been correcte
Geometricity of the Hodge filtration on the -stack of perfect complexes over
We construct a locally geometric -stack of perfect
complexes with -connection structure on a smooth projective variety
. This maps to , so it can be considered as the Hodge filtration
of its fiber over 1 which is , parametrizing complexes of
-modules which are -perfect. We apply the result of Toen-Vaquie that
is locally geometric. The proof of geometricity of the map
uses a Hochschild-like notion of weak complexes
of modules over a sheaf of rings of differential operators. We prove a
strictification result for these weak complexes, and also a strictification
result for complexes of sheaves of -modules over the big crystalline site
Residue Complexes over Noncommutative Rings
Residue complexes were introduced by Grothendieck in algebraic geometry.
These are canonical complexes of injective modules that enjoy remarkable
functorial properties (traces).
In this paper we study residue complexes over noncommutative rings. These
objects are even more complicated than in the commutative case, since they are
complexes of bimodules. We develop methods to prove uniqueness, existence and
functoriality of residue complexes.
For a noetherian affine PI algebra over a field (admitting a noetherian
connected filtration) we prove existence of the residue complex and describe
its structure in detail.Comment: 37 pages, AMSLaTeX with XYpic figures; some changes in Section 3;
final version, to appear in Algebr. Represent. Theor
Two-dimensional Id\`eles with Cycle Module Coefficients
We give a theory of id\`eles with coefficients for smooth surfaces over a
field. It is an analogue of Beilinson/Huber's theory of higher ad\`eles, but
handling cycle module sheaves instead of quasi-coherent ones. We prove that
they give a flasque resolution of the cycle module sheaves in the Zariski
topology. As a technical ingredient we show the Gersten property for cycle
modules on equicharacteristic complete regular local rings, which might be of
independent interest.Comment: major change in exposition, streamlined, removed incorrect claim
about product map (many thanks to S. Gorchinskiy for pointing this out to
me), bibliography update
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