6,047 research outputs found
Enhanced adic formalism and perverse t-structures for higher Artin stacks
In this sequel of arXiv:1211.5294 and arXiv:1211.5948, we develop an adic
formalism for \'etale cohomology of Artin stacks and prove several desired
properties including the base change theorem. In addition, we define perverse
t-structures on Artin stacks for general perversity, extending Gabber's work on
schemes. Our results generalize results of Laszlo and Olsson on adic formalism
and middle perversity. We continue to work in the world of -categories
in the sense of Lurie, by enhancing all the derived categories, functors, and
natural transformations to the level of -categories.Comment: 53 pages. v2: reformulatio
Hochschild (co)homology of the second kind I
We define and study the Hochschild (co)homology of the second kind (known
also as the Borel-Moore Hochschild homology and the compactly supported
Hochschild cohomology) for curved DG-categories. An isomorphism between the
Hochschild (co)homology of the second kind of a CDG-category B and the same of
the DG-category C of right CDG-modules over B, projective and finitely
generated as graded B-modules, is constructed. Sufficient conditions for an
isomorphism of the two kinds of Hochschild (co)homology of a DG-category are
formulated in terms of the two kinds of derived categories of DG-modules over
it. In particular, a kind of "resolution of the diagonal" condition for the
diagonal CDG-bimodule B over a CDG-category B guarantees an isomorphism of the
two kinds of Hochschild (co)homology of the corresponding DG-category C.
Several classes of examples are discussed.Comment: LaTeX 2e, 67 pages. v.2: The case of matrix factorizations discussed
in detail in the new subsections 4.8 and 4.1
Periodic twisted cohomology and T-duality
The initial motivation of this work was to give a topological interpretation
of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary
coefficients. To this end we develop a sheaf theory in the context of locally
compact topological stacks with emphasis on the construction of the sheaf
theory operations in unbounded derived categories, elements of Verdier duality
and integration. The main result is the construction of a functorial
periodization functor associated to a U(1)-gerbe. As applications we verify the
-duality isomorphism in periodic twisted cohomology and in periodic twisted
orbispace cohomology.Comment: 128 pages; v2: small corrections (e.g. of typos), version to appear
in Asterisqu
Representations of Spaces
We explain how the notion of homotopy colimits gives rise to that of mapping
spaces, even in categories which are not simplicial. We apply the technique of
model approximations and use elementary properties of the category of spaces to
be able to construct resolutions. We prove that the homotopy category of any
monoidal model category is always a central algebra over the homotopy category
of Spaces.Comment: Final version, almost as it will appear in "Algebraic and Geometric
Topology"; 30 page
Riemann-Hilbert correspondence for unit -crystals on embeddable algebraic varieties
For a separated scheme of finite type over a perfect field of
characteristic which admits an immersion into a proper smooth scheme over
the truncated Witt ring , we define the bounded derived category of
locally finitely generated unit -crystals with finite Tor-dimension on
over , independently of the choice of the immersion. Then we prove the
anti-equivalence of this category with the bounded derived category of
constructible \'etale sheaves of -modules with
finite Tor dimension. We also discuss the relationship of -structures on
these derived categories when . Our result is a generalization of the
Riemann-Hilbert correspondence for unit -crystals due to Emerton-Kisin to
the case of (possibly singular) embeddable algebraic varieties in
characteristic .Comment: This is the final version, to appear in Annales de l'Institut Fourie
- …