409 research outputs found
Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces
We construct a new spectrum of units for a commutative symmetric ring
spectrum that detects the difference between a periodic ring spectrum and its
connective cover. It is augmented over the sphere spectrum. The homotopy
cofiber of its augmentation map is a non-connected delooping of the usual
spectrum of units whose bottom homotopy group detects periodicity.
Our approach builds on the graded variant of E-infinity spaces introduced in
joint work with Christian Schlichtkrull. We construct a group completion model
structure for graded E-infinity spaces and use it to exhibit our spectrum of
units functor as right adjoint on the level of homotopy categories. The
resulting group completion functor is an essential tool for studying ring
spectra with graded logarithmic structures.Comment: v3: 33 pages; exposition improved, accepted for publication in
Algebraic and Geometric Topolog
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
A new proof for the decidability of D0L ultimate periodicity
We give a new proof for the decidability of the D0L ultimate periodicity
problem based on the decidability of p-periodicity of morphic words adapted to
the approach of Harju and Linna.Comment: In Proceedings WORDS 2011, arXiv:1108.341
The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)
We show that there is a homotopy cofiber sequence of spectra relating
Carlsson's deformation K-theory of a group G to its "deformation representation
ring," analogous to the Bott periodicity sequence relating connective K-theory
to ordinary homology. We then apply this to study simultaneous similarity of
unitary matrices
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