Deformation bicomplex of module algebras - Yau, Donald
The deformation bicomplex of a module algebra over a bialgebra is constructed. It
is then applied to study algebraic deformations in which both the module
structure and the algebra structure are deformed. The cases of module
coalgebras, comodule (co)algebras, and (co)module bialgebras are also
On affine morphisms of Hopf algebroids - Powell, Geoffrey M.L.
The paper considers the FPQC stacks which are associated to affine groupoid
schemes. Using a formulation of a descent datum in terms of morphisms of affine
groupoid schemes, explicit arguments are given which avoid appeal to the general
principle of faithfully flat descent. This theory is applied to consider the
notion of affine morphism.
The Euler characteristic of a category as the sum of a divergent series - Berger, Clemens; Leinster, Tom
The Euler characteristic of a cell complex is often thought of as the alternating
sum of the number of cells of each dimension. When the complex is infinite, the
sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this
is possible when the complex is the nerve of a finite category. This provides an
alternative definition of the Euler characteristic of a category, which is in
many cases equivalent to the original one.
Excision for $K$-theory of connective ring spectra - Dundas, Bjørn Ian; Kittang, Harald Øyen
We extend Geisser and Hesselholt's result on ``bi-relative K-theory'' from
discrete rings to connective ring spectra. That is, if $\cal A$ is a homotopy
cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then
the $(n+1)$-cube induced by the cyclotomic trace
¶ is A) \to TC(\cal A)$$ is homotopy cartesian after profinite completion. In other
words, the fiber of the profinitely completed cyclotomic trace satisfies
Splittings in the Burnside ring and in $SF_G$ - French, Christopher P.
Let $G$ be a finite $p$-group, $p \neq 2$. We construct a map from the space
$J_G$, defined as the fiber of $\psi^k-1: B_G O \to B_G Spin$, to the space
$(SF_G)_p$, defined as the 1-component of the zeroth space of the equivariant
$p$-complete sphere spectrum. Our map produces the same splitting of the
$G$-connected cover of $(SF_G)_p$ as we have described in previous work, but it
also induces a natural splitting of the $p$-completions of the component groups
of fixed point subspaces.
Hopf-Hochschild (co)homology of module algebras - Kaygun, Atabey
We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show that this (co)homology, called Hopf-Hochschild (co)homology, can also be defined as a derived functor on the category of representations of an equivariant analogue of the enveloping algebra of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf-Hochschild (co)homology.
Homotopy types of truncated projective resolutions - Mannan, W. H.
We work over an arbitrary ring R. Given two truncated projective resolutions of equal length for the same module, we consider their underlying chain complexes. We show they may be stabilized by projective modules to obtain a pair of complexes of the same homotopy type.
Model structure on operads in orthogonal spectra - Kro, Tore August
We generalize Berger and Moerdijk's results on axiomatic homotopy theory for operads to the setting of enriched symmetric monoidal model categories, and show how this theory applies to orthogonal spectra. In particular, we provide a symmetric fibrant replacement functor for the positive stable model structure.
Beyond the hit problem: Minimal presentations of odd-primary Steenrod modules, with application to CP(∞) and BU - Pengelley, David J.; Williams, Frank
We describe a minimal unstable module presentation over the Steenrod algebra for the odd-primary cohomology of infinite-dimensional complex projective space and apply it to obtain a minimal algebra presentation for the cohomology of the classifying space of the infinite unitary group. We also show that there is a unique Steenrod module structure on any unstable cyclic module that has dimension one in each complex degree (half the topological degree) with a fixed alpha-number (sum of `digits') and is zero in other degrees.
A statistical approach to persistent homology - Bubenik, Peter; Kim, Peter T.
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach to this problem. We assume that the data is randomly sampled from an unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using statistical estimators for samples from certain families of distributions, we show that we can recover the persistent homology of the underlying...
Thick subcategories of the derived category of a hereditary algebra - Brüning, Kristian
We classify thick subcategories of the bounded derived category of a hereditary abelian category A in terms of subcategories of A. The proof can be applied to characterize the localizing subcategories of the full derived category of A. As an application we prove an algebraic analog of the telescope conjecture for the derived category of a representation finite hereditary artin algebra.