Hochschild Complexes¶
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sage.homology.hochschild_complex.
HochschildComplex
¶ The Hochschild complex.
Let \(A\) be an algebra over a commutative ring \(R\) such that \(A\) a projective \(R\)-module, and \(M\) an \(A\)-bimodule. The Hochschild complex is the chain complex given by
\[C_n(A, M) := M \otimes A^{\otimes n}\]with the boundary operators given as follows. For fixed \(n\), define the face maps
\[\begin{split}f_{n,i}(m \otimes a_1 \otimes \cdots \otimes a_n) = \begin{cases} m a_1 \otimes \cdots \otimes a_n & \text{if } i = 0, \\ a_n m \otimes a_1 \otimes \cdots \otimes a_{n-1} & \text{if } i = n, \\ m \otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n & \text{otherwise.} \end{cases}\end{split}\]We define the boundary operators as
\[d_n = \sum_{i=0}^n (-1)^i f_{n,i}.\]The Hochschild homology of \(A\) is the homology of this complex. Alternatively, the Hochschild homology can be described by \(HH_n(A, M) = \operatorname{Tor}_n^{A^e}(A, M)\), where \(A^e = A \otimes A^o\) (\(A^o\) is the opposite algebra of \(A\)) is the enveloping algebra of \(A\).
Hochschild cohomology is the homology of the dual complex and can be described by \(HH^n(A, M) = \operatorname{Ext}^n_{A^e}(A, M)\).
Another perspective on Hochschild homology is that \(f_{n,i}\) make the family \(C_n(A, M)\) a simplicial object in the category of \(R\)-modules, and the degeneracy maps are
\[s_i(a_0 \otimes \cdots \otimes a_n) = a_0 \otimes \cdots \otimes a_i \otimes 1 \otimes a_{i+1} \otimes \cdots \otimes a_n\]The Hochschild homology can also be constructed as the homology of this simplicial module.
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