In mathematics, Hochschild homology is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Definition of Hochschild homology of algebras

Let k be a ring, A an associative k- algebra, and M an A-bimodule. We will write A^⊗n for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by In mathematics, associativity is a property that a binary operation can have. ... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

with boundary operator d_i defined by

Here a_i is in A for all 1 ≤ i ≤ n and a₀ ∈ M. If we let

then b ° b = 0, so (C_n(A,M), b) is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M. In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

Remark

The maps d_i are face maps making the family of modules C_n(A,M) a simplicial object in the category of k-modules, ie. a functor Δ^o → k-mod, where Δ is the simplicial category and k-mod is the category of k-modules. Here Δ^o is the opposite category of Δ. The degeneracy maps are defined by s_i(a₀ ⊗ ··· ⊗ a_n) = a₀ ⊗ ··· a_i ⊗ 1 ⊗ a_i+1 ⊗ ··· ⊗ a_n. Hochschild homology is the homology of this simplicial module. In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, the simplicial category is the small category with objects the ordered sets for each , and morphisms are monotonic non-decreasing functions. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

Hochschild homology of functors

The simplicial circle S¹ is a simplicial object in the category Fin_* of finite pointed sets, ie. a functor Δ^o → Fin_*. Thus, if F is a functor F: Fin → k-mod, we get a simplicial module by composing F with S¹

The homology of this simplicial module is the Hochschild homology of the functor F. The above definition of Hochschild homology of algebras is the special case where F is the Loday functor.

Loday functor

A skeleton for the category of finite pointed sets is given by the objects In mathematics, a skeleton of a category is a subcategory which, roughly speaking, does not contain any extraneous isomorphisms. ...

n ₊ = {0,1,...,n},

where 0 is the basepoint, and the morphisms are the basepoint preserving set maps. Let A be a k-algebra and M a A-bimodule. The Loday functor L(A,M) is given on objects in Fin_* by In mathematics, a morphism is an abstraction of a structure-preserving mapping between two mathematical structures. ...

.

A morphism

is sent to the morphism f_* given by

where

and b_j = 1 if f^{-1}(j)=∅.

Another description of Hochschild homology of algebras

The Hochschild homology of an algebra A with coefficients in an A-bimodule M is the homology associated to the composition

and this definition agrees with the one above.

References

• Jean-Louis Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0
• A personal note on Hochschild and Cyclic homology
• Hodge decomposition for higher order Hochschild homology

See also

• Cyclic homology