37 research outputs found
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
Towers and fibered products of model categories
Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization
Stratifying derived categories of cochains on certain spaces
In recent years, Benson, Iyengar and Krause have developed a theory of
stratification for compactly generated triangulated categories with an action
of a graded commutative Noetherian ring. Stratification implies a
classification of localizing and thick subcategories in terms of subsets of the
prime ideal spectrum of the given ring. In this paper two stratification
results are presented: one for the derived category of a commutative
ring-spectrum with polynomial homotopy and another for the derived category of
cochains on certain spaces. We also give the stratification of cochains on a
space a topological content.Comment: 27 page
Hermitian K-theory and 2-regularity for totally real number fields
We completely determine the 2-primary torsion subgroups of the hermitian
K-groups of rings of 2-integers in totally real 2-regular number fields. The
result is almost periodic with period 8. We also identify the homotopy fibers
of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory.
The result is then exactly periodic of period 8. In both the orthogonal and
symplectic cases, we prove the 2-primary hermitian Quillen-Lichtenbaum
conjecture.Comment: To appear in Mathematische Annale