$L^2$-cohomology

J. Eichhorn

Displaying similar documents to “ $L^{2}$ -cohomology”

Overconvergent de Rham-Witt cohomology

Christopher Davis, Andreas Langer, Thomas Zink (2011)

Annales scientifiques de l'École Normale Supérieure

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The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p > 0$ , an overconvergent de Rham-Witt complex $W^{†} Ω_{X / k}$ as a suitable subcomplex of the de Rham-Witt complex of Deligne-Illusie. This complex, which is functorial in $X$ , is a complex of étale sheaves and a differential graded algebra over the ring $W^{†} (𝒪_{X})$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent...

Chen–Ruan Cohomology of $ℳ_{1, n}$ and ${\bar{ℳ}}_{1, n}$

Nicola Pagani (2013)

Annales de l’institut Fourier

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In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$ -pointed curves of genus $1$ . In the first part of the paper we study and describe stack theoretically the twisted sectors of $ℳ_{1, n}$ and ${\bar{ℳ}}_{1, n}$ . In the second part, we study the orbifold intersection theory of ${\bar{ℳ}}_{1, n}$ . We suggest a definition for an orbifold tautological ring in genus $1$ , which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

Batalin-Vilkovisky algebra structures on Hochschild cohomology

Luc Menichi (2009)

Bulletin de la Société Mathématique de France

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Let $M$ be any compact simply-connected oriented $d$ -dimensional smooth manifold and let $𝔽$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$ , $H H^{*} (S^{*} (M), S^{*} (M))$ , extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$ , $H_{* + d} (L M)$ introduced by Chas and Sullivan. We also show that the negative cyclic...

Bounded cohomology and isometry groups of hyperbolic spaces

Ursula Hamenstädt (2008)

Journal of the European Mathematical Society

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Let $X$ be an arbitrary hyperbolic geodesic metric space and let $Γ$ be a countable subgroup of the isometry group $Iso (X)$ of $X$ . We show that if $Γ$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_{b}^{2} (Γ, ℝ)$ , $H_{b}^{2} (Γ, ℓ^{p} (Γ))$ $(1 < p < \infty)$ are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually splits as a direct...

Noetherian loop spaces

Natàlia Castellana, Juan Crespo, Jérôme Scherer (2011)

Journal of the European Mathematical Society

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The class of loop spaces of which the mod $p$ cohomology is Noetherian is much larger than the class of $p$ -compact groups (for which the mod $p$ cohomology is required to be finite). It contains Eilenberg–Mac Lane spaces such as $ℂ P^{\infty}$ and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space $B X$ of such an object and prove it is as small as expected, that is, comparable to that of $B ℂ P^{\infty}$ . We also show that $B$ X differs basically from the classifying space of a $p$ -compact...

Cohomological dimension filtration and annihilators of top local cohomology modules

Ali Atazadeh, Monireh Sedghi, Reza Naghipour (2015)

Colloquium Mathematicae

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Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration $= {M_{i}}_{i = 0}^{c}$ , where c = cd(,M) and $M_{i}$ denotes the largest submodule of M such that $c d (, M_{i}) \leq i$ . Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module $H_{}^{c} (M)$ , namely $A n n_{R} (H_{}^{c} (M)) = A n n_{R} (M / M_{c - 1})$ . As a consequence, there exists an ideal of R such that $A n n_{R} (H_{}^{c} (M)) = A n n_{R} (M / H ⁰_{} (M))$ . This generalizes the...

$1$ -cocycles on the group of contactomorphisms on the supercircle $S^{1 | 3}$ generalizing the Schwarzian derivative

Boujemaa Agrebaoui, Raja Hattab (2016)

Czechoslovak Mathematical Journal

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The relative cohomology $H_{diff}^{1} (𝕂 (1 | 3), 𝔬𝔰𝔭 (2, 3); 𝒟_{λ, μ} (S^{1 | 3}))$ of the contact Lie superalgebra $𝕂 (1 | 3)$ with coefficients in the space of differential operators $𝒟_{λ, μ} (S^{1 | 3})$ acting on tensor densities on $S^{1 | 3}$ , is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$ -cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$ -cocycle $s (X_{f}) = D_{1} D_{2} D_{3} (f) α_{3}^{1 / 2}$ , $X_{f} \in 𝕂 (1 | 3)$ which is invariant with respect to the conformal subsuperalgebra $𝔬𝔰𝔭 (2, 3)$ of $𝕂 (1 | 3)$ . In this work we study the supergroup case. We give an explicit construction of $1$ -cocycles...

On $1$ -cocycles induced by a positive definite function on a locally compact abelian group

Jordan Franks, Alain Valette (2014)

Annales mathématiques Blaise Pascal

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For $ϕ$ a normalized positive definite function on a locally compact abelian group $G$ , let $π_{ϕ}$ be the unitary representation associated to $ϕ$ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^{1} (G, π_{ϕ})$ and reduced 1-cohomology ${\bar{H}}^{1} (G, π_{ϕ})$ . For example, ${\bar{H}}^{1} (G, π_{ϕ}) = 0$ if and only if either $Hom (G, ℂ) = 0$ or $μ_{ϕ} (1_{G}) = 0$ , where $1_{G}$ is the trivial character of $G$ and $μ_{ϕ}$ is the probability measure on the Pontryagin dual $\hat{G}$ associated to $ϕ$ by Bochner’s Theorem. This streamlines an argument of Guichardet...

Some bounds for the annihilators of local cohomology and Ext modules

Ali Fathi (2022)

Czechoslovak Mathematical Journal

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Let $𝔞$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a nonnegative integer. Let $M$ and $N$ be two finitely generated $R$ -modules. In certain cases, we give some bounds under inclusion for the annihilators of ${Ext}_{R}^{t} (M, N)$ and $H_{𝔞}^{t} (M)$ in terms of minimal primary decomposition of the zero submodule of $M$ , which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.

Invariants, torsion indices and oriented cohomology of complete flags

Baptiste Calmès, Viktor Petrov, Kirill Zainoulline (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$ . Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$ -theory, Chow groups, Grothendieck’s $K_{0}$ , etc.) with formal group law $F$ . We construct a ring from $F$ and the characters of $T$ , that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁 (G / B)$ where $G / B$ is the variety of Borel subgroups of $G$ . Our main result says that when the...

Poincaré inequalities and rigidity for actions on Banach spaces

Piotr Nowak (2015)

Journal of the European Mathematical Society

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The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that $H^{1} (G, π) = 0$ for every isometric representation $π$ of $G$ on $X$ . The condition is expressed in terms of $p$ -Poincaré constants and we...

Fredholm spectrum and growth of cohomology groups

Jörg Eschmeier (2008)

Studia Mathematica

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Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let $σ_{F} (T) = σ (T) ∖ σ_{e} (T)$ be the non-essential spectrum of T. We show that, for each connected component M of the manifold $R e g (σ_{F} (T))$ of all smooth points of $σ_{F} (T)$ , there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups $H^{p} ({(z - T)}^{k}, E)$ grow at least like the sequence ${(k^{d})}_{k \geq 1}$ with d = dim M.

Local-global principle for annihilation of general local cohomology

J. Asadollahi, K. Khashyarmanesh, Sh. Salarian (2001)

Colloquium Mathematicae

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Let A be a Noetherian ring, let M be a finitely generated A-module and let Φ be a system of ideals of A. We prove that, for any ideal in Φ, if, for every prime ideal of A, there exists an integer k(), depending on , such that $^{k ()}$ kills the general local cohomology module $H_{Φ_{}}^{j} (M_{})$ for every integer j less than a fixed integer n, where $Φ_{} : =_{} : \in Φ$ , then there exists an integer k such that $^{k} H_{Φ}^{j} (M) = 0$ for every j < n.

On the minimaxness and coatomicness of local cohomology modules

Marzieh Hatamkhani, Hajar Roshan-Shekalgourabi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$ -module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$ -minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$ -module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${(H_{I}^{i} (M))}_{𝔪}$ is a minimax $R_{𝔪}$ -module for all $𝔪 \in Max (R)$ and for all $i < n$ , then the set ${Ass}_{R} (H_{I}^{n} (M))$ is finite. Also, if $H_{I}^{i} (M)$ is...

Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type $𝐀_{1}$

Bo Hou, Yanhong Guo (2015)

Czechoslovak Mathematical Journal

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The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let $Λ_{t}$ be the Yoneda algebra of a reconstruction algebra of type $𝐀_{1}$ over a field $. I n t h i s p a p e r, a m i n i m a l p r o j e c t i v e b i m o d u l e r e s o l u t i o n o f$ t $i s c o n s t r u c t e d, a n d t h e$ -dimensions of all Hochschild homology and cohomology groups of $Λ_{t}$ are calculated explicitly.