Structures de Poisson sur une intersection complète à singularités isoléesPoisson structures over a complete intersection with isolated singularities

We study Poisson structures over singular varieties. For this purpose, we consider the Koszul complex associated to the equations of a complete intersection. This complex forms a differential graded algebra which is equivalent to the algebra of the variety. We show that a Poisson structure is equivalent to a sequence of multiderivations over the Koszul complex. If the variety has isolated singularities, then we can construct a sequence of multiderivations of reduced form. To cite this article: B. Fresse, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 5–10.     

Homologies d'algèbres Artin–Schelter régulières cubiques

Hochschild homology of cubic Artin–Schelter regular algebras of type A with generic coefficients is computed. We follow the method used by Van den Bergh (K-Theory 8 (1994) 213–230) in the quadratic case, by considering these algebras as deformations of a polynomial algebra, with remarkable Poisson brackets. A new quasi-isomorphism is introduced. De Rham cohomology, cyclic and periodic cyclic homologies, and Hochschild cohomology are also computed. To cite this article: N. Marconnet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The primitive cohomology lattice of a complete intersection

We describe the primitive cohomology lattice of a smooth even-dimensional complete intersection in projective space. To cite this article: A. Beauville, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Hodge type of the exotic cohomology of complete intersections

If $\text{X?}\text{P}{}^{\mathrm{n}}$ is a smooth complete intersection, its cohomology modulo the one of $\text{P}{}^{\mathrm{n}}$ is supported in middle dimension. If the complete intersection is singular, it might also carry exotic cohomology beyond the middle dimension. We show that for this exotic cohomology, one can improve the known bound for the Hodge type of its de Rham cohomology. To cite this article: H. Esnault, D. Wan, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

ON THE STRUCTURE OF COHOMOLOGY RINGS OF p-NILPOTENT LIE ALGEBRAS

In this paper the authors investigate the structure of the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose E ₂-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen–Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as a ring with the E ₂-term. We present criteria for the collapsing of this spectral sequence and provide some examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.     

The Quantum Cohomology Ring of Flag Varieties

We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263-277). We also give a geometric proof of the quantum Monk formula.

     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. To cite this article: F.M. Bleher, T. Chinburg, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. A Riemannian interpretation of the Reeb class will give some geometric criteria which enables one to tell whether the modular class vanishes or not. It also enables one to construct examples of unimodular Poisson manifolds and others which are not unimodular. Finally, we prove that the first leafwise cohomology space is an invariant of Morita equivalence. To cite this article: A. Abouqateb, M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Topological André-Quillen Cohomology and E∞ André-Quillen Cohomology

We show that the André-Quillen cohomology of an E_∞ simplicial algebra with arbitrary coefficients and the topological André-Quillen cohomology of an E_∞ ring spectrum with Eilenberg-Mac Lane coefficients may be calculated as the André-Quillen cohomology of an associated E_∞ differential graded algebra.     

Eigenvalues of Frobenius acting on the ℓ-adic cohomology of complete intersections of low degree

We show that the eigenvalues of Frobenius acting on ?-adic cohomology of a complete intersection of low degree defined over the finite field $\text{F}{}_{\mathrm{q}}$ modulo the cohomology of the projective space are divisible as algebraic integers by q^κ, where the natural number κ is predicted by the theorem of Ax and Katz (Amer J. Math. 93 (1971) 485–499) on the congruence for the number of rational points. To cite this article: H. Esnault, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the cohomology of the Weyl algebra,the quantum plane,and the q-Weyl algebra

Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler–Poincaré characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.     

Hochschild Cohomology of <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We compute the Hochschild cohomology of any block of q-Schur algebras. We focus on the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of q-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all q-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derive equivalence.     

Hopf-cyclic homology and cohomology with coefficients

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter–Drinfeld compatibility condition leading to the concept of a stable anti-Yetter–Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Hochschild cohomology of ? n -Galois coverings of an algebra

We consider the ? _n -Galois covering ?? _n of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872?C1893]. We calculate the dimensions of all Hochschild cohomology groups of ?? _n and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg??s conjecture.     

A universal deformation ring which is not a complete intersection ring

Bleher and Chinburg recently used modular representation theory to produce an example of a linear representation of a finite group whose universal deformation ring is not a complete intersection ring. We prove this by using only elementary cohomological obstruction calculus. To cite this article: J. Byszewski, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The Lie algebra structure of the first Hochschild cohomology group for monomial algebrasLa structure d'algèbre de Lie du premier groupe de la cohomologie de Hochschild pour des algèbres monomiales

We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra Λ, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of Λ in characteristic zero. Criteria for the solvability, the (semi-) simplicity, the commutativity and the nilpotency are given. To cite this article: C. Strametz, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 733–738.     

L'anneau de cohomologie d'une variété de Seifert

All manifolds M considered in this Note are orientable Seifert 3-manifolds with base surface S² and infinite fundamental group π₁ (M). Our goal is to compute the cohomology ring H^* (M; Z/2Z). The ring structure will enable us to determine whether M admits a degree 1 map into RP³ or not. We describe the equivariant chain complex for the universal cover M of M, and give a diagonal approximation. The cohomology ring H^* (M; Z/2Z) is computed.