= Ground-ring qualification in Hochschild cohomology
{title2=$HH_k^q(R,M)=\operatorname{Ext}_{R\otimes_kR^{\mathrm{op}}}^q(R,M)\text{ if }R\text{ is }k\text{-projective}$}
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
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