8,281 research outputs found
Equivariant sheaves for profinite groups
We develop the theory of equivariant sheaves over profinite spaces, where the
group is also taken to be profinite. We construct a good notion of equivariant
presheaves, with a suitable sheafification functor. Using these results on
equivariant presheaves, we give explicit constructions of products of
equivariant sheaves of R-modules. We introduce an equivariant analogue of
skyscraper sheaves, which allows us to show that the category of equivariant
sheaves of R-modules over a profinite space has enough injectives.
This paper also provides the basic theory for results by the authors on
giving an algebraic model for rational G-spectra in terms of equivariant
sheaves over profinite spaces. For those results, we need a notion of
Weyl-G-sheaves over the space of closed subgroups of G. We show that
Weyl-G-sheaves of R-modules form an abelian category, with enough injectives,
that is a full subcategory of equivariant sheaves of R-modules. Moreover, we
show that the inclusion functor has a right adjoint.Comment: 36 page
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
Loop Spaces and Connections
We examine the geometry of loop spaces in derived algebraic geometry and
extend in several directions the well known connection between rotation of
loops and the de Rham differential. Our main result, a categorification of the
geometric description of cyclic homology, relates S^1-equivariant quasicoherent
sheaves on the loop space of a smooth scheme or geometric stack X in
characteristic zero with sheaves on X with flat connection, or equivalently
D_X-modules. By deducing the Hodge filtration on de Rham modules from the
formality of cochains on the circle, we are able to recover D_X-modules
precisely rather than a periodic version. More generally, we consider the
rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega
S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary
connection, with curvature given by their Omega S^3-equivariance.Comment: Revised versio
Rational S^1-equivariant elliptic cohomology
For each elliptic curve A over the rational numbers we construct a 2-periodic
S^1-equivariant cohomology theory E whose cohomology ring is the sheaf
cohomology of A; the homology of the sphere of the representation z^n is the
cohomology of the divisor A(n) of points with order dividing n. The
construction proceeds by using the algebraic models of the author's AMS Memoir
``Rational S^1 equivariant homotopy theory.'' and is natural and explicit in
terms of sheaves of functions on A.
This is Version 5.2 of a paper of long genesis (this should be the final
version). The following additional topics were first added in the Fourth
Edition:
(a) periodicity and differentials treated
(b) dependence on coordinate
(c) relationship with Grojnowksi's construction and, most importantly,
(d) equivalence between a derived category of O_A-modules and a derived
category of EA-modules. The Fifth Edition included
(e) the Hasse square and
(f) explanation of how to calculate maps of EA-module spectra
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