Hochschild homology - Alchetron, The Free Social Encyclopedia

In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Cartan & Eilenberg (1956).

Definition of Hochschild homology of algebras

Let k be a ring, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product A^e=A⊗A^o of A with its opposite algebra. Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as A^e-modules. Cartan & Eilenberg (1956) defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by

H H n ( A , M ) = Tor n A e ( A , M ) H H n ( A , M ) = Ext A e n ( A , M )

Hochschild complex

Let k be a ring, A an associative k-algebra that is a projective k-module, 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

C n ( A , M ) := M ⊗ A ⊗ n

with boundary operator d_i defined by

d 0 ( m ⊗ a 1 ⊗ ⋯ ⊗ a n ) = m a 1 ⊗ a 2 ⋯ ⊗ a n d i ( m ⊗ a 1 ⊗ ⋯ ⊗ a n ) = m ⊗ a 1 ⊗ ⋯ ⊗ a i a i + 1 ⊗ ⋯ ⊗ a n d n ( m ⊗ a 1 ⊗ ⋯ ⊗ a n ) = a n m ⊗ a 1 ⊗ ⋯ ⊗ a n − 1

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

b = ∑ i = 0 n ( − 1 ) i d i ,

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.

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, i.e. a functor Δ^o → k-mod, where Δ is the simplex 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.

Hochschild homology of functors

The simplicial circle S¹ is a simplicial object in the category Fin_* of finite pointed sets, i.e. 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

Δ o ⟶ S 1 Fin ∗ ⟶ F k - mod .

The homology of this simplicial module is the Hochschild homology of the functor F. The above definition of Hochschild homology of commutative 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

n + = { 0 , 1 , … , n } ,

where 0 is the basepoint, and the morphisms are the basepoint preserving set maps. Let A be a commutative k-algebra and M be a symmetric A-bimodule. The Loday functor L(A,M) is given on objects in Fin_* by

n + ↦ M ⊗ A ⊗ n .

A morphism

f : m + → n +

is sent to the morphism f_* given by

f ∗ ( a 0 ⊗ ⋯ ⊗ a n ) = ( b 0 ⊗ ⋯ ⊗ b m )

where

b j = ∏ f ( i ) = j a i , j = 0 , … , n ,

and b_j = 1 if f⁻¹(j) = ∅.

Another description of Hochschild homology of algebras

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

Δ o ⟶ S 1 Fin ∗ ⟶ L ( A , M ) k - mod ,

and this definition agrees with the one above.

References

Hochschild homology Wikipedia

(Text) CC BY-SA

Contents