Ground-ring qualification in Hochschild cohomology ($H{H}_k^q(R,M)={\mathrm{Ext}}_{R{\otimes }_k{R}^{\mathrm{op}}}^q(R,M)\text{ if }R\text{ is }k\text{-projective}$)

The ordinary Hochschild cochain complex is defined using a chosen commutative ground ring $k$. The bar complex is exact via insertion of $1$, but its terms are projective over the enveloping algebra when $R$ is projective over $k$, in particular when $k$ is a field. Without that hypothesis the ordinary cochain groups compute relative Hochschild cohomology and cannot automatically be identified with absolute Ext. Likewise, ordinary second cohomology classifies ground-ring-linearly split square-zero extensions, not arbitrary nonsplit ring extensions.

// Destination: module-theory.bigb