Algebraic $K$-theory of the first Morava $K$-theory

Christian Ausoni; John Rognes

Displaying similar documents to “Algebraic $K$ -theory of the first Morava $K$ -theory”

Rational BV-algebra in string topology

Yves Félix, Jean-Claude Thomas (2008)

Bulletin de la Société Mathématique de France

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Let $M$ be a 1-connected closed manifold of dimension $m$ and $L M$ be the space of free loops on $M$ . M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of $L M$ , $H_{*} (L M; k)$ . When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology $H H^{*} (C^{*} (M); C^{*} (M))$ which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $H H^{*} (C^{*} (M); C^{*} (M))$ and the shifted homology $H_{* + m} (L M; k)$ . We also prove...

Annihilators of local homology modules

Shahram Rezaei (2019)

Czechoslovak Mathematical Journal

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Let $(R, 𝔪)$ be a local ring, $𝔞$ an ideal of $R$ and $M$ a nonzero Artinian $R$ -module of Noetherian dimension $n$ with $hd (𝔞, M) = n$ . We determine the annihilator of the top local homology module $H_{n}^{𝔞} (M)$ . In fact, we prove that ${Ann}_{R} (H_{n}^{𝔞} (M)) = {Ann}_{R} (N (𝔞, M)),$ where $N (𝔞, M)$ denotes the smallest submodule of $M$ such that $hd (𝔞, M / N (𝔞, M)) < n$ . As a consequence, it follows that for a complete local ring $(R, 𝔪)$ all associated primes of $H_{n}^{𝔞} (M)$ are minimal.

Taylor towers of symmetric and exterior powers

Brenda Johnson, Randy McCarthy (2008)

Fundamenta Mathematicae

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We study the Taylor towers of the nth symmetric and exterior power functors, Spⁿ and Λⁿ. We obtain a description of the layers of the Taylor towers, $D_{k} S p ⁿ$ and $D_{k} Λ ⁿ$ , in terms of the first terms in the Taylor towers of $S p^{t}$ and $Λ^{t}$ for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of $S p^{t}$ and $Λ^{t}$ . We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for $D_{k} S p ⁿ$ and $D_{k} Λ ⁿ$ .

Coxeter elements for vanishing cycles of types $A_{\frac{1}{2} \infty}$ and $D_{\frac{1}{2} \infty}$

Kyoji Saito (2011)

Annales de l’institut Fourier

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We introduce two entire functions $f_{A_{\frac{1}{2} \infty}}$ and $f_{D_{\frac{1}{2} \infty}}$ in two variables. Both of them have only two critical values $0$ and $1$ , and the associated maps $C^{2} \to C$ define topologically locally trivial fibrations over $C ∖ {0, 1}$ . All critical points in the singular fibers over $0$ and $1$ are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed infinite quivers of type $A_{\frac{1}{2} \infty}$ and $D_{\frac{1}{2} \infty}$ , respectively. Coxeter elements...

Topology of Fatou components for endomorphisms of $ℂ ℙ^{k}$ : linking with the Green’s current

Suzanne Lynch Hruska, Roland K. W. Roeder (2010)

Fundamenta Mathematicae

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Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f : ℂ ℙ^{k} \to ℂ ℙ^{k}$ , when k >1. Classical theory describes U(f) as the complement in $ℂ ℙ^{k}$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ ℙ^{k}$ , we give a definition of linking number between closed loops in $ℂ ℙ^{k} ∖ s u p p S$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H ₁ (ℂ ℙ^{k} ∖ s u p p S)$ . As an application, we use...

Realizable Galois module classes over the group ring for non abelian extensions

Nigel P. Byott, Bouchaïb Sodaïgui (2013)

Annales de l’institut Fourier

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Given an algebraic number field $k$ and a finite group $Γ$ , we write $ℛ (O_{k} [Γ])$ for the subset of the locally free classgroup $Cl (O_{k} [Γ])$ consisting of the classes of rings of integers $O_{N}$ in tame Galois extensions $N / k$ with $Gal (N / k) ≅ Γ$ . We determine $ℛ (O_{k} [Γ])$ , and show it is a subgroup of $Cl (O_{k} [Γ])$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $Γ = V ⋊ C$ , where $V$ is an elementary abelian group of order $p^{r}$ and $C$ is a cyclic group of order $m > 1$ acting faithfully...

On the unit group of a semisimple group algebra $𝔽_{q} S L (2, ℤ_{5})$

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

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We give the characterization of the unit group of $𝔽_{q} S L (2, ℤ_{5})$ , where $𝔽_{q}$ is a finite field with $q = p^{k}$ elements for prime $p > 5,$ and $S L (2, ℤ_{5})$ denotes the special linear group of $2 \times 2$ matrices having determinant $1$ over the cyclic group $ℤ_{5}$ .

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Matilde Marcolli, Gonçalo Tabuada (2016)

Journal of the European Mathematical Society

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In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum ${(k)}_{F}$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum ${(k)}_{F}$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined...

A countably cellular topological group all of whose countable subsets are closed need not be $ℝ$ -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

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We construct a Hausdorff topological group $G$ such that $ℵ_{1}$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$ -embedded in $G$ , but $G$ is not $ℝ$ -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

A Riemann-Roch theorem for dg algebras

François Petit (2013)

Bulletin de la Société Mathématique de France

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Given a smooth proper dg algebra $A$ , a perfect dg $A$ -module $M$ and an endomorphism $f$ of $M$ , we define the Hochschild class of the pair $(M, f)$ with values in the Hochschild homology of the algebra $A$ . Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

Hardness of embedding simplicial complexes in $ℝ^{d}$

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let ${𝙴𝙼𝙱𝙴𝙳}_{k \to d}$ be the following algorithmic problem: Given a ﬁnite simplicial complex $K$ of dimension at most $k$ , does there exist a (piecewise linear) embedding of $K$ into $ℝ^{d}$ ? Known results easily imply polynomiality of ${𝙴𝙼𝙱𝙴𝙳}_{k \to 2}$ ( $k = 1, 2$ ; the case $k = 1, d = 2$ is graph planarity) and of ${𝙴𝙼𝙱𝙴𝙳}_{k \to 2 k}$ for all $k \geq 3$ . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${𝙴𝙼𝙱𝙴𝙳}_{d \to d}$ and ${𝙴𝙼𝙱𝙴𝙳}_{(d - 1) \to d}$ are undecidable for each $_{d} \geq 5$ . Our main result is NP-hardness of ${𝙴𝙼𝙱𝙴𝙳}_{2 \to 4}$ and, more generally, of ${𝙴𝙼𝙱𝙴𝙳}_{k \to d}$ for all...

Algebraic and topological properties of some sets in ℓ₁

Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik (2012)

Colloquium Mathematicae

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For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series $\sum_{n = 1}^{\infty} x (n)$ . Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of $\sum_{n = 1}^{\infty} b (n)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable....

Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant

Gwénaël Massuyeau (2012)

Bulletin de la Société Mathématique de France

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Let $Σ$ be a compact connected oriented surface with one boundary component, and let $π$ be the fundamental group of $Σ$ . The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of $Σ$ , whose $k$ -th term consists of the self-homeomorphisms of $Σ$ that act trivially at the level of the $k$ -th nilpotent quotient of $π$ . Morita defined a homomorphism from the $k$ -th term of the Johnson filtration to the third homology group of the $k$ -th nilpotent quotient of $π$ . In this paper, we...

Bicrossed products of generalized Taft algebra and group algebras

Dingguo Wang, Xiangdong Cheng, Daowei Lu (2022)

Czechoslovak Mathematical Journal

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Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m, d} (q) ⋈ k [G]$ between the generalized Taft algebra $H_{m, d} (q)$ and group algebra $k [G]$ is actually the smash product $H_{m, d} (q) ♯ k [G]$ . Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$ . As an application, the classification of $H_{m, d} (q) ⋈ k [C_{n_{1}} \times C_{n_{2}}]$ is completely presented by generators and relations, where $C_{n}$ denotes the $n$ -cyclic group.

On $n$ -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

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A subset of a product of topological spaces is called $n$ -thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_{3}$ space $X$ without isolated points such that $X^{n}$ contains an $n$ -thin dense subset, but $X^{n + 1}$ does not contain any $n$ -thin dense subset. We also observe that part of the construction can be carried out under MA.

Displaying similar documents to “Algebraic K -theory of the first Morava K -theory”

Displaying similar documents to “Algebraic $K$ -theory of the first Morava $K$ -theory”