15,040 research outputs found
On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings
We classify compactly generated co-t-structures on the derived category of a
commutative noetherian ring. In order to accomplish that, we develop a theory
for compactly generated Hom-orthogonal pairs (also known as torsion pairs in
the literature) in triangulated categories that resembles Bousfield
localization theory. Finally, we show that the category of perfect complexes
over a connected commutative noetherian ring admits only the trivial
co-t-structures and (de)suspensions of the canonical co-t-structure and use
this to describe all silting objects in the category.Comment: 34 pages. Version 2: minor corrections, references added and update
Algebraic K-theory of strict ring spectra
We view strict ring spectra as generalized rings. The study of their
algebraic K-theory is motivated by its applications to the automorphism groups
of compact manifolds. Partial calculations of algebraic K-theory for the sphere
spectrum are available at regular primes, but we seek more conceptual answers
in terms of localization and descent properties. Calculations for ring spectra
related to topological K-theory suggest the existence of a motivic cohomology
theory for strictly commutative ring spectra, and we present evidence for
arithmetic duality in this theory. To tie motivic cohomology to Galois
cohomology we wish to spectrally realize ramified extensions, which is only
possible after mild forms of localization. One such mild localization is
provided by the theory of logarithmic ring spectra, and we outline recent
developments in this area.Comment: Contribution to the proceedings of the ICM 2014 in Seou
Localization of algebras over coloured operads
We give sufficient conditions for homotopical localization functors to
preserve algebras over coloured operads in monoidal model categories. Our
approach encompasses a number of previous results about preservation of
structures under localizations, such as loop spaces or infinite loop spaces,
and provides new results of the same kind. For instance, under suitable
assumptions, homotopical localizations preserve ring spectra (in the strict
sense, not only up to homotopy), modules over ring spectra, and algebras over
commutative ring spectra, as well as ring maps, module maps, and algebra maps.
It is principally the treatment of module spectra and their maps that led us to
the use of coloured operads (also called enriched multicategories) in this
context.Comment: 34 page
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