750 research outputs found
Calculating obstruction groups for E-infinity ring spectra
We describe a special instance of the Goerss-Hopkins obstruction theory, due
to Senger, for calculating the moduli of ring spectra with given
mod- homology. In particular, for the -primary Brown-Peterson spectrum we
give a chain complex that calculates the first obstruction groups, locate the
first potential genuine obstructions, and discuss how some of the obstruction
classes can be interpreted in terms of secondary operations.Comment: 27 pages. Comments welcom
The Shimura curve of discriminant 15 and topological automorphic forms
We find defining equations for the Shimura curve of discriminant 15 over
Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic
integers, as well as the higher cohomology. We apply this to calculate the
homotopy groups of a spectrum of "topological automorphic forms" associated to
this curve, as well as one associated to a quotient by an Atkin-Lehner
involution.Comment: 36 pages, 5 figures, updated with corrections and new introduction
(this version corrects image issues in the previous
Localization of enriched categories and cubical sets
The invertibility hypothesis for a monoidal model category S asks that
localizing an S-enriched category with respect to an equivalence results in an
weakly equivalent enriched category. This is the most technical among the
axioms for S to be an excellent model category in the sense of Lurie, who
showed that the category of S-enriched categories then has a model structure
with characterizable fibrant objects. We use a universal property of cubical
sets, as a monoidal model category, to show that the invertibility hypothesis
is consequence of the other axioms
Structured ring spectra and displays
We combine Lurie's generalization of the Hopkins-Miller theorem with work of
Zink-Lau on displays to give a functorial construction of even-periodic
commutative ring spectra, concentrated in chromatic layers 2 and above,
associated to certain n by n invertible matrices with coefficients in Witt
rings. This is applied to examples related to Lubin-Tate and Johnson-Wilson
spectra. We also give a Hopf algebroid presentation of the moduli of
p-divisible groups of height greater than or equal to 2.Comment: 18 page
The product formula in unitary deformation K-theory
We prove that "unitary deformation K-theory" takes products of finitely
generated groups to coproducts of algebra spectra over ku, the connective
K-theory spectrum. Additionally, we give spectral sequences for computing the
homotopy groups of the unitary deformation K-theory of a group G and the
cofiber of the Bott map in terms of PU(n)-equivariant K-theory and homology of
spaces of G-representations.Comment: 36 pages (revised version
Secondary power operations and the Brown-Peterson spectrum at the prime 2
The dual Steenrod algebra has a canonical subalgebra isomorphic to the
homology of the Brown-Peterson spectrum. We will construct a secondary
operation in mod-2 homology and show that this canonical subalgebra is not
closed under it. This allows us to conclude that the 2-primary Brown-Peterson
spectrum does not admit the structure of an E_n-algebra for any n greater than
or equal to 12, answering a question of May in the negative.Comment: Revised version with some expanded details. Comments welcom
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
Regularity of structured ring spectra and localization in K-theory
We identify a regularity property for structured ring spectra, and with it we
prove a natural analogue of Quillen's localization theorem for algebraic
K-theory in this setting.Comment: 9 pages. Corrected references and aspects of the expositio
Brauer groups and Galois cohomology of commutative ring spectra
In this paper we develop methods for classifying Baker-Richter-Szymik's
Azumaya algebras over a commutative ring spectrum, especially in the largely
inaccessible case where the ring is nonconnective. We give
obstruction-theoretic tools, constructing and classifying these algebras and
their automorphisms with Goerss-Hopkins obstruction theory, and give
descent-theoretic tools, applying Lurie's work on -categories to show
that a finite Galois extension of rings in the sense of Rognes becomes a
homotopy fixed-point equivalence on Brauer spaces.
For even-periodic ring spectra , we find that the "algebraic" Azumaya
algebras whose coefficient ring is projective are governed by the Brauer-Wall
group of , recovering a result of Baker-Richter-Szymik. This allows
us to calculate many examples. For example, we find that the algebraic Azumaya
algebras over Lubin-Tate spectra have either 4 or 2 Morita equivalence classes
depending on whether the prime is odd or even, that all algebraic Azumaya
algebras over the complex K-theory spectrum are Morita trivial, and that
the group of the Morita classes of algebraic Azumaya algebras over the
localization is .
Using our descent results and an obstruction theory spectral sequence, we
also study Azumaya algebras over the real K-theory spectrum which become
Morita-trivial -algebras. We show that there exist exactly two Morita
equivalence classes of these. The nontrivial Morita equivalence class is
realized by an "exotic" -algebra with the same coefficient ring as
. This requires a careful analysis of what happens in the
homotopy fixed-point spectral sequence for the Picard space of , previously
studied by Mathew and Stojanoska.Comment: 51 pages, 3 figures. Comments welcom
Commutativity conditions for truncated Brown-Peterson spectra of height 2
An algebraic criterion, in terms of closure under power operations, is
determined for the existence and uniqueness of generalized trun- cated
Brown-Peterson spectra of height 2 as E_\infty-ring spectra. The criterion is
checked for an example at the prime 2 derived from the universal elliptic curve
equipped with a level \Gamma_1(3)-structure.Comment: minor revision, essentially in final form, to appear in Journal of
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