# Hypotheses of the Conormal Exact Sequence

On Wikipedia, in the description of the conormal exact sequence, it is described as arising from a closed immersion, which corresponds in the affine case to a surjection of algebras. However, in Eisenbud (CA with a View Towards AG), we have proposition 16.3 (page 389) as follows:

If $\pi:S\to T$ is an epimorphism of $R$-algebras, with kernel $I$, then there is an exact sequence of $T$-modules $$I/I^2 \overset d\to T\otimes_R\Omega_{S/R} \overset {D\pi}\to\Omega_{T/R}\to0$$ where the right-hand map is given by $D\pi:c\otimes db\mapsto cdb$ and the left-hand map takes the class of $f$ to $1\otimes df$.

An epimorphism of algebras is certainly not in general a surjection, and yet Eisenbud later writes $S/I= T$, making it quite clear that this map is intended to be a surjection. I've checked Eisenbud's published list of errata, and this is not there. Is this a typo or am I missing something?

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This abuse of language is quite common. But it should be clear from the context that no epimorphisms of commutative rings are considered, they are only relevant in a small part of algebraic geometry (namely monomorphisms of schemes). Anyway, this leads us to the following question: Let $S \to T$ be an epimorphism of commutative $R$-algebras. Is $T \otimes_R \Omega_{S/R} \to \Omega_{T/R}$ an epimorphism of $T$-modules? Remark that for localizations it is even an isomorphism. –  Martin Brandenburg Feb 4 '13 at 11:54
This choice of language probably stems from the fact that the notions of mono- and epimorphisms are often introduced in the context of $R$-modules as a shorthand for injective/surjective linear map. Thus one way to interpret the sentence you mention is "Let $\pi: S \rightarrow T$ be a homomorphism of $R$-algebras that is an epimorphism as a map of $R$-modules".