Secondary Hochschild homology and differentials. (English) Zbl 1534.13010
Let \(\Bbbk\) be a field of characteristic 0, and consider a triple \((A,B,\varepsilon)\) where \(B\) is a commutative \(\Bbbk\)-algebra and \(A\) an associative \(\Bbbk\)-algebra with a \(B\)-algebra structure induced by a morphism of \(\Bbbk\)-algebras \(\varepsilon:B\rightarrow A\) with \(\varepsilon(B)\) contained in the center of \(A\). To such a triple one can associate a homology theory known as secondary Hochschild homology, which generalizes the classic Hochschild homology since when \(B=\Bbbk\) it reduces to the Hochschild homology of \(A\). It is known that the first Hochschild homology of a \(\Bbbk\)-algebra \(A\) is isomorphic to the first module of Kähler differentials \(\Omega^1_{A|\Bbbk}\). The authors generalize this results by first defining the module of secondary Kähler differentials \(\Omega^1_{\mathcal{T}|\Bbbk}\), where \(\mathcal{T}=(A,B,\varepsilon)\), and then proving that this module is isomorphic to \(HH_1(A,B,\varepsilon)\), the first secondary Hochschild homology module, provided \(A\) is also commutative. This constitutes the main result of the paper.
Reviewer: Luigi Ferraro (Edinburg)
MSC:
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 13N05 | Modules of differentials |
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