= Conormal injectivity for a generically smooth Cartier divisor
{title2=$0\to\mathcal I_W/\mathcal I_W^2\to\Omega_V^1|_W\to\Omega_W^1\to0$}
Let $W$ be an integral <effective Cartier divisor> in an integral <variety> $V$ over a <perfect field>, and suppose $W$ is not contained in the singular locus of $V$. On a dense open subset both <varieties> are smooth. The <Zariski tangent space> of $W$ there has codimension one in that of $V$, so the differential of the local defining equation is nonzero. Thus the <conormal sheaf> map is injective generically. Its <kernel> is a subsheaf of a <line bundle> on the integral <variety> $W$ and is therefore a <torsion-free sheaf>; generic vanishing implies zero everywhere. Right exactness is the <Conormal exact sequence for Kähler differentials>.
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