arXiv subjects

Birgit Richter

Publications and source records attributed to Birgit Richter.

At least 37 records · Page 2Linked to original sources

arXiv2014

On the higher topological Hochschild homology of $\mathbb{F}_p$ and commutative $\mathbb{F}_p$-group algebras

We extend Torleif Veen's calculation of higher topological Hochschild homology ${\sf THH}^{[n]}_*(\mathbb{F}_p)$ from $n\leq 2p$ to $n\leq 2p+2$ for $p$ odd, and from $n=2$ to $n\leq 3$ for $p=2$. We calculate higher Hochschild homology ${\sf HH}_*^{[n]}(k[x])$ over $k$ for any integral domain $k$, and ${\sf HH}_*^{[n]}(\mathbb{F}_p[x]/x^{p^\ell})$ for all $n>0$. We use this and étale descent to calculate ${\sf HH}_*^{[n]}(\mathbb{F}_p[G])$ for all $n>0$ for any cyclic group $G$, and therefore also for any finitely generated abelian group $G$. We show a splitting result for higher ${\sf THH}$ of commutative $\mathbb{F}_p$-group algebras and use this technique to calculate higher topological Hochschild homology of such group algebras for as large an $n$ as ${\sf THH}^{[n]}_*(\mathbb{F}_p) $ is known for.

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arXiv2014

On the homology and homotopy of commutative shuffle algebras

For commutative algebras there are three important homology theories, Harrison homology, Andre-Quillen homology and Gamma-homology. In general these differ, unless one works with respect to a ground field of characteristic zero. We show that the analogues of these homology theories agree in the category of pointed commutative monoids in symmetric sequences and that Hochschild homology always possesses a Hodge decomposition in this setting. In addition we prove that the category of pointed differential graded commutative monoids in symmetric sequences has a model structure and that it is Quillen equivalent to the model category of pointed simplicial commutative monoids in symmetric sequences.

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arXiv2013

Some properties of the Thom spectrum over loop suspension of complex projective space

This note provides a reference for some properties of the Thom spectrum $Mξ$ over $ΩΣ\CPi$. Some of this material is used in recent work of Kitchloo and Morava. We determine the $Mξ$-cohomology of $\CPi$ and show that $Mξ^*(\CPi)$ injects into power series over the algebra of non-symmetric functions. We show that $Mξ$ gives rise to a commutative formal group law over the non-commutative ring $π_*Mξ$. We also discuss how $Mξ$ and some real and quaternionic analogues behave with respect to spectra that are related to these Thom spectra by splittings and by maps.

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arXiv2012

Brauer groups for commutative $S$-algebras

We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a K(n)-local variant of the notion of Brauer groups.

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arXiv2011

Some properties of Lubin-Tate cohomology for classifying spaces of finite groups

We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$ is either a Lubin-Tate spectrum $E_n$ or the related 2-periodic Morava K-theory $K_n$, and $G$ is a finite group. When $R$ is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a $G$-Galois extension in the sense of John Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ these extensions are always faithful in the $K_n$ local category. However, for a cyclic $p$-group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \to F({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it ramifies. As a consequence, it follows that the $E_n$-theory Eilenberg-Moore spectral sequence for $G$ and $BG$ does not always converge to its expected target.

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arXiv2011

Stable bundles over rig categories

The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku), the algebraic K-theory of topological K-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology. The main technical step is showing that for well-behaved small rig categories R (also known as bimonoidal categories) the algebraic K-theory space, K(HR), of the ring spectrum HR associated to R is equivalent to Z \times |BGL(R)|^+, where GL(R) is the monoidal category of weakly invertible matrices over R. If π_0R is a ring this is almost formal, and our approach is to replace R by a ring completed version provided by [BDRR1] whose π_0 is the ring completion of π_0R.

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arXiv2010

Ring completion of rig categories

We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category R (also known as a symmetric bimonoidal category), the additive group completion R' will be a commutative ring category. In an accompanying paper we show how this can be used to prove the conjecture from [BDR] that the algebraic K-theory of the connective topological K-theory spectrum ku is equivalent to the algebraic K-theory of the rig category V of complex vector spaces.

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arXiv2010

An interpretation of E_n-homology as functor homology

We prove that E_n-homology of non-unital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by natural maps, ranging from Hochschild homology and its higher order versions to Gamma homology.

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arXiv2009

A lower bound for coherences on the Brown-Peterson spectrum

We provide a lower bound for the coherence of the homotopy commutativity of the Brown-Peterson spectrum, BP, at a given prime p and prove that it is at least (2p^2 + 2p - 2)-homotopy commutative. We give a proof based on Dyer-Lashof operations that BP cannot be a Thom spectrum associated to n-fold loop maps to BSF for n=4 at 2 and n=2p+4 at odd primes. Other examples where we obtain estimates for coherence are the Johnson-Wilson spectra, localized away from the maximal ideal and unlocalized. We close with a negative result on Morava-K-theory.

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arXiv2008

Divided power structures and chain complexes

We interpret divided power structures on the homotopy groups of simplicial commutative rings as having a counterpart in divided power structures on chain complexes coming from a non-standard symmetric monoidal structure.

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arXiv2008

An involution on the K-theory of bimonoidal categories with anti-involution

We construct a combinatorially defined involution on the algebraic $K$-theory of the ring spectrum associated to a bimonoidal category with anti-involution. Particular examples of such are braided bimonoidal categories. We investigate examples such as algebraic K-theory of connective complex and real topological K-theory and Waldhausen's K-theory of spaces of the form BBG, for abelian groups G. We show that the involution agrees with the classical one for a bimonoidal category associated to a ring and prove that it is not trivial in the above mentioned examples.

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arXiv2008

Galois extensions of Lubin-Tate spectra

Let E_n be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E^{nr}_n whose coefficients are built from the coefficients of E_n and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E^{nr}_n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E^{nr}_n with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.

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arXiv2007

Quasisymmetric functions from a topological point of view

It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions, Symm. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as a topological model for the ring of quasisymmetric functions, QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on the loop space that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU. The canonical Thom spectrum over the loops on the suspension of BU(1) is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.

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arXiv2006

Uniqueness of $E_\infty$ structures for connective covers

We refine our earlier work on the existence and uniqueness of E-infinity structures on K-theoretic spectra to show that at each prime p, the connective Adams summand has an essentially unique structure as a commutative S-algebra. For the p-completion we show that the McClure-Staffeldt model for it is equivalent as an E-infinity ring spectrum to the connective cover of the periodic Adams summand. We establish Bousfield equivalence between the connective cover, c(E_n), of the Lubin-Tate spectrum E_n and BP and propose c(E_n) as an E-infinity approximation to the latter.

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arXiv2006

The collapse of the periodicity sequence in the stable range

The stabilization of Hochschild homology of commutative algebras is Gamma homology. We describe a cyclic variant of Gamma homology and prove that the associated analogue of Connes' periodicity sequence becomes almost trivial, because the cyclic version coincides with the ordinary version from homological degree two on. We offer an alternative explanation for this by proving that the B-operator followed by the stabilization map is trivial from degree one on.

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arXiv2005

Realizability of algebraic Galois extensions by strictly commutative ring spectra

We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central roles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the complex K-theory spectrum as a KO-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert's theorem 90 for the units associated with a Galois extension.

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arXiv2005

On the cooperation algebra of the connective Adams summand

The aim of this paper is to gain explicit information about the multiplicative structure of l_*l, where l is the connective Adams summand. Our approach differs from Kane's or Lellmann's because our main technical tool is the MU-based Kuenneth spectral sequence. We prove that the algebra structure on l_*l is inherited from the multiplication on a Koszul resolution of l_*BP.

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arXiv2005

$Γ$-cohomology of rings of numerical polynomials and $E_\infty$ structures on K-theory

We investigate Gamma-cohomology of some commutative cooperation algebras E_*E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Gamma-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E infinity structures. As a consequence we obtain an E infinity structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n > 0 we establish the existence of a unique E infinity structure for its I_n-adic completion.

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