= Second Hochschild cohomology classifies split square-zero extensions
{title2=$HH^2(R,M)$}
Fix an $R$-<bimodule> $M$. A unital ground-ring-linear split extension with square-zero kernel $M$ can be identified with $R\oplus M$, with multiplication $(a,u)(b,v)=(ab,av+ub+f(a,b))$. <Associativity> is exactly $\delta f=0$. Changing a unital linear section by $g:R\to M$ changes $f$ by $\delta g$. Thus equivalence classes of these split extensions are $HH^2(R,M)$. Over a <field> every such vector-space extension splits; over a general ground <ring> a splitting hypothesis is essential for this ordinary cochain classification.
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