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For a given field $k$, let $A$ be an associative, unital $k$-algebra. Let $M$ be an $A$-bimodule. Define the Hochschild cochain complex of $A$ with coefficients in $M$ as

$$C^n(A;M):=\operatorname{Hom}_k(A^{\otimes n},M)$$ with the usual differential. One sometimes considers normalized $n$-cochains, namely those $f\in C^n(A;M)$ with $f(a_1\otimes \ldots \otimes a_n)=0$ if $a_i=1_A$ for some $i=1,\ldots, n$.

Why are these cochains called normalized?

Margaret
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  • It is just a name. Such cocycles are useful in many contexts — for example, when one classifies split extensions using $HH^2(A)$ the fact that a 2-cocycle is normalized means that the unit element of the extension is the same as the unit element of the original algebra. The same thing happens when one uses group cohomology to construct central simple algebras. The name normalized does not mean anything, though, just like normal spaces and normal subgroups and normal linear maps. – Mariano Suárez-Álvarez Nov 21 '25 at 23:36

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