= Conormal module of the diagonal
{title2=$\Omega^1_{A/k}\cong I/I^2$}
For a commutative $k$-algebra $A$, let $I$ be the kernel of multiplication $\mu:A\otimes_k A\to A$. The formula $a\mapsto[1\otimes a-a\otimes1]$ is a <derivation of an algebra> into the <quotient module> $I/I^2$, with $A$ acting through the first factor. It induces an isomorphism from the <module of Kähler differentials>. An inverse is induced by $q(x\otimes y)=x\,dy$: the identity $q(rs)=\mu(r)q(s)+\mu(s)q(r)$ shows $q(I^2)=0$. For $z=\sum_i x_i\otimes y_i\in I$, the equality $\sum_i x_i y_i=0$ gives $[z]=\sum_i x_i[1\otimes y_i-y_i\otimes1]$, proving that the two maps are inverse. For an <affine variety>, $I$ is the defining ideal of its <diagonal morphism>.
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