15 research outputs found
Representability of derived stacks
Lurie's representability theorem gives necessary and sufficient conditions
for a functor to be an almost finitely presented derived geometric stack. We
establish several variants of Lurie's theorem, making the hypotheses easier to
verify for many applications. Provided a derived analogue of Schlessinger's
condition holds, the theorem reduces to verifying conditions on the underived
part and on cohomology groups. Another simplification is that functors need
only be defined on nilpotent extensions of discrete rings. Finally, there is a
pre-representability theorem, which can be applied to associate explicit
geometric stacks to dg-manifolds and related objects.Comment: 28 pages; v2 Lemma 1.16 added; v3 final version, to appear in JKT; v4
refs update
Unifying derived deformation theories
AbstractWe develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L∞-algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toën–Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as André–Quillen cohomology
The de Rham homotopy theory and differential graded category
This paper is a generalization of arXiv:0810.0808. We develop the de Rham
homotopy theory of not necessarily nilpotent spaces, using closed dg-categories
and equivariant dg-algebras. We see these two algebraic objects correspond in a
certain way. We prove an equivalence between the homotopy category of schematic
homotopy types and a homotopy category of closed dg-categories. We give a
description of homotopy invariants of spaces in terms of minimal models. The
minimal model in this context behaves much like the Sullivan's minimal model.
We also provide some examples. We prove an equivalence between fiberwise
rationalizations and closed dg-categories with subsidiary data.Comment: 47 pages. final version. The final publication is available at
http://www.springerlink.co
A K-theoretic interpretation of real Deligne cohomology
We show that real Deligne cohomology of a complex manifold arises locally as
a topological vector space completion of the analytic Lie groupoid of
holomorphic vector bundles. Thus Beilinson's regulator arises naturally as a
comparison map between -theory groups of different types.Comment: 24 pages; v5 even more small changes and corrections, final version
to appear in Adv. Mat