Abstract
The second Hochschild cohomology group of rings (that is algebras over k = Z) classifies the extensions of a ring by a bimodule provided that the ex-tensions are split as abelian groups. In order to classify non-split extensions, Mac Lane introduced in the fifties the so-called Mac Lane (co)homology theory, that we denote by HML and which is closely related to the cohomology of the Eilenberg-Mac Lane spaces. Hochschild (co)homology and Mac Lane (co)homology coincide when the ring contains the rational numbers, but they differ in general.
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Loday, JL., Pirashvili, T. (1998). Mac Lane (co)homology. In: Cyclic Homology. Grundlehren der mathematischen Wissenschaften, vol 301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11389-9_13
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