= Algebraic cotangent space
{title2=$\mathfrak m_P/\mathfrak m_P^2\cong\Omega^1_{X/k}\otimes k(P)$}
At a <closed point> of a <variety> over an <algebraically closed field>, the cotangent space is $\mathfrak m_P/\mathfrak m_P^2$. The <universal property of Kähler differentials> identifies its <dual vector space> with $k$-derivations of the <local ring> into the residue field, hence with the <Zariski tangent space>. The <Kähler differential sheaf> has the same fibre because a derivation kills constants and products of two elements of the <maximal ideal>.
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