Lemma 10.140.1. Let k be an algebraically closed field. Let S be a finite type k-algebra. Let \mathfrak m \subset S be a maximal ideal. Then
\dim _{\kappa (\mathfrak m)} \Omega _{S/k} \otimes _ S \kappa (\mathfrak m) = \dim _{\kappa (\mathfrak m)} \mathfrak m/\mathfrak m^2.
Proof. Consider the exact sequence
\mathfrak m/\mathfrak m^2 \to \Omega _{S/k} \otimes _ S \kappa (\mathfrak m) \to \Omega _{\kappa (\mathfrak m)/k} \to 0
of Lemma 10.131.9. We would like to show that the first map is an isomorphism. Since k is algebraically closed the composition k \to \kappa (\mathfrak m) is an isomorphism by Theorem 10.34.1. So the surjection S \to \kappa (\mathfrak m) splits as a map of k-algebras, and Lemma 10.131.10 shows that the sequence above is exact on the left. Since \Omega _{\kappa (\mathfrak m)/k} = 0, we win. \square
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