In Commutative Algebra: with a View Toward Algebraic Geometry Eisenbud describes the Kähler-Differential as a functor that assigns $\Omega_{S/R}$ to an $R$-Algebra $S$ and to a commutative diagramm $$ S \xrightarrow{\hspace{1cm}} S' \\ \uparrow \hspace{1.5cm} \uparrow \\ R \xrightarrow{\hspace{1cm}} R' $$ of rings the corresponding morphism $\Omega_{S/R} \to \Omega_{S'/R'}$ which makes the diagramm $$ \Omega_{S/R} \xrightarrow{\hspace{.5cm}} \Omega_{S'/R'} \\ \uparrow \hspace{1.5cm} \uparrow \\ S \xrightarrow{\hspace{1cm}} S' $$ commute. Then he writes

It is right-exact in the same sense that the zeroth relative homology functor is a right-exact functor of pairs of spaces in topology.

Question: What is that supposed to mean? Does he maybe mean the relative cotangent sequence? For ringmorphisms $R \to S \to T$ we get an exact sequence of $T$-modules $$ T \otimes_S \Omega_{S/R} \to \Omega_{T/R} \to \Omega_{T/S} \to 0.$$

As I understand this, the two left morphisms in this exact sequence come from the construction above. However I don't understand how this statement can be understood as right-exactness, because we would have to start with an exact sequence but we don't. Also I couldn't find anything to understand his statement about the zeroth relative homology functor.


I think he is referring to the relative cotangent sequence, indeed. I doubt that his statement about "right exactness" should be taken too literally. What does it even mean to say that "the zeroth relative homology functor is a right-exact functor of pairs of spaces"? It means that if you have pair of spaces $(X,A)$, hence a sequence of maps of pairs $(A,\emptyset)\rightarrow(X,\emptyset)\rightarrow(X,A)$, you obtain an exact sequence $H_0(A)\rightarrow H_0(X)\rightarrow H_0(X,A)\rightarrow 0$. (But of course $(A,\emptyset)\rightarrow(X,\emptyset)\rightarrow(X,A)$ is not really an exact sequence in the sense of classical homological algebra, what does it even mean to say this for a sequence of maps?). I believe that he is referring to the evident formal similarity with this sequence; this interpretation is reasonable, as you can see from what he says on page 389 (just after the relative cotangent sequence), that the Andre-Quillen homology functors play the role of the higher homology functors.


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