The Stacks project

Definition 89.3.6. Let $R \to S$ be a local homomorphism of local rings. The relative cotangent space1 of $R$ over $S$ is the $S/\mathfrak m_ S$-vector space $\mathfrak m_ S/(\mathfrak m_ R S + \mathfrak m_ S^2)$.

[1] Caution: We will see later that in our general setting the tangent space of an object $A \in \mathcal{C}_\Lambda $ over $\Lambda $ should not be defined simply as the $k$-linear dual of the relative cotangent space. In fact, the correct definition of the relative cotangent space is $\Omega _{S/R} \otimes _ S S/\mathfrak m_ S$.

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