Hochschild Complex Research Articles

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

Cohomology of Quantum General Linear Groups

This paper deals with the cohomology of infinitesimal quantum general linear groups. We prove that Hodd((Gq)1,K)=0 and Hev((Gq)1,K)≅K[N]. Our approach is to consider theq-analogue of a unipotent group scheme and construct a filtration of the Hochschild complex forK[(Uq)1]. Meanwhile a certainq-deformation of a polynomial coalgebra arises naturally, which is also believed to be interesting.

Microlocalisation de l'homologie de Hochschild

We construct a homotopy operator on the. Hochschild complex of the algebra of smooth functions, which concentrates the chains of the complex, living on the various powers of the base manifold, onto jets of functions all along the principal diagonals of the powers. The construction can be adapted to compute the Hochschild homology of the algebras of continuous functions having partial derivatives of finite order in L p. In particular, our method allows one to lift the compactness hypothesis in A. Connes' theorem concerning the Hochschild homology of the algebra of smooth functions on smooth manifolds.

Dihedral homology of commutative algebras

Let A be an associative k-algebra with involution, where k is a commutative ring of characteristic not equal to two. Then the dihedral groups act on the Hochschild complex and, following Loday, a new homological theory, called dihedral homology, can be defined generalizing the notion of cyclic homology defined by Connes. Here we give a model to compute dihedral homology of a commutative algebra over a characteristic zero field. As, for an involutive algebra, we have a decomposition of Hochschild homology into a direct sum of two k-modules: Z 2- equivariant and skew Z 2- equivariant Hochschild homologies, we give smoothness criteria in terms of vanishing of some Z 2- equivariant Hochschild homology groups.

Filtrations, mixed complexes, and cyclic homology in mixed characteristic

In [5], we computed the cyclic homology of finite extensions of Dedekind domains. The crucial result was a formula for the cyclic homology, relative to a ground ring R, of A = R[x]/(P (x)). This paper presents two extensions of the methods and results of this paper. Recall that Hochschild homology, though generally defined in terms of a bar complex, is almost never, in practice, computed from it. A smaller complex, often a direct summand (Y·, bY ) of the reduced Hochschild complex (X·, b), is usually used. For example, it is well known (cf., e.g., [2], [11], [12]) that the Hochschild homology of A = R[x]/(P (x)) is the homology of the periodic complex

Higher operations on the Hochschild complex

Higher operations on the Hochschild complex

Cyclic homology of Dedekind domains

The purpose of this paper is to calculate the cyclic homology of rings of integers of global fields. We accomplish this by explicitly computing the homology of the simple complex associated to Tsygan’s double complex. To accomplish this, we first compute the cyclic homology of cyclic algebras, i.e., algebras of the form A = R[t]/(P (t)), where P is a monic polynomial with coefficients in R. More precisely, we give a complex (2.9) of finite free A-modules and R-linear maps, whose homology gives the cyclic homology HCr(A/R) when A is an integral domain. This complex enjoys a great advantage over the usual complex, (2.1), for cyclic homology, in that the rank of the modules grows linearly rather than exponentially in r. As a result, one can compute the homology groups explicitly in many interesting cases. If R is a Dedekind domain whose field of fractions K is a global field (i.e. a number field or the fraction field of a curve over a finite field), and S is the integral closure of R in a finite separable extension L/K, the cyclic homology of S/R can be obtained from the cyclic homology of certain cyclic algebras, by a kind of adelization procedure (3.2). To establish the relation between our cyclic complex and the “standard” cyclic complex of Tsygan, Loday, and Quillen [12], we describe a subcomplex of the latter, the small Tsygan complex, which is quasi-isomorphic to the full complex and which admits a quasiisomorphism to our cyclic complex. The construction of the latter quasi-isomorphism is the key technical feature of our paper. It is motivated by our attempts to solve the extension problem arising from the spectral sequence which computes the homology of the Tsygan complex. The differentials of this spectral sequence are implicitly worked out in the proof of Th. 2.10. The solution of the extension problem follows from the technical results Prop. 2.6 and Prop. 2.8. The first section discusses the Hochschild theory for cyclic algebras and Dedekind domains. For cyclic algebras, Hochschild homology can be computed from a periodic complex (1.6). This complex seems to be part of the folklore; but it seems to have been discovered independently, in varying degrees of generality, by a number of people, including Zack [18]; Cortinas, Guccione, and Villamayor [4]; Burghelea and ViguePoirrier [3]; Goodwillie (unpublished); Masuda and Natsume [14]; Wolffhardt [17]; and perhaps others) By adelizing, we obtain (Prop. 1.9) the periodicity of Hochschild homology for Dedekind domains. By constructing explicit quasi-isomorphisms between the Zack complex and the standard complex for Hochschild homology, we can study the maps HH∗(S/R) → HH∗(T/R) → HH∗(T/S), where R ⊂ S ⊂ T is a triple of Dedekind domains (Props. 1.11, 1.13), as well as the ring structure of HH∗(S/R) (1.16). We conclude with a construction of the small subcomplex of the standard Hochschild complex, which motivates the small subcomplex of the Tsygan complex.

Splittings of Hochschild's Complex for Commutative Algebras

Splittings of Hochschild's Complex for Commutative Algebras