Algebraic cotangent space (${\mathfrak{m}}_P/{\mathfrak{m}}_P^2\cong {\Omega }_{X/k}^1\otimes k(P)$)

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.