Normalized Hochschild cochain complex

For a unital associative algebra, normalized positive-degree cochains vanish whenever an argument is $1$. They form a subcomplex of the Hochschild cochain complex computing the same Hochschild cohomology. The unit-insertion contracting homotopy on degenerate bar terms proves the normalization equivalence. A normalized degree-two Hochschild cocycle defines a unital square-zero extension of an algebra.