Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.

Then, given an exact sequence $\eta: 0\to X \to Z_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $\eta$ by $f$ gives an exact sequence $f\eta: 0\to X \to Z'_{\eta} \to M \to 0$ in $Ext^1(M,Y)$.

Similarly, given an exact sequence $\eta: 0\to M \to W_{\zeta} \to Y \to 0$ in $Ext^1(Y,M)$, the pullback of $\zeta$ by $f$ gives an exact sequence $\zeta f: 0\to M \to W'_{\zeta} \to X \to 0$ in $Ext^1(X,M)$.

It is well known that these assignments actually define functors $Ext^1(M,-); Ext^1(-,M): \text{Mod } R \to \text{Mod } R$, where for $f \in Hom(X,Y)$;

$$Ext^1(M,f): Ext^1(M,X) \xrightarrow{\eta \mapsto f\eta} Ext^1(M,Y)$$

$$Ext^1(f,M): Ext^1(Y,M) \xrightarrow{\zeta \mapsto \zeta f} Ext^1(X,M). $$

My question is: Are the functors $Ext^1(M,-)$ and $Ext^1(-,M)$ $R$-linear?

I think this should be well known, but I am looking for an explicit reference.

Please help.

Update: I have accepted Mark Grant's answer which gives a reference for additivity of the functor. If someone has an explicit reference for the fact that multiplication by ring elements is preserved too, please do post an answer; I'd really appreciate it and upvote your answer too!

  • $\begingroup$ (Note : noetherian and finitely generated are not important here, but commutative is. I don't know a reference, that's why I'm not answering, but it should be in any reference discussing the equivalence between this construction and the functoriality of Ext as a derived functor) $\endgroup$ Apr 26, 2021 at 8:39
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    $\begingroup$ @Maxime Ramzi: Indeed, I think you are very much right that any Noetherian hypothesis is irrelevant. The Commutativity is needed otherwise $Hom$ is not a module in general. Even commutativity could be disregarded I think if I only asked about "additivity" instead of "linearity" ... $\endgroup$
    – sdey
    Apr 26, 2021 at 8:45

1 Answer 1


An explicit reference for the additivity is

Mac Lane, Saunders, Homology, Classics in Mathematics. Berlin: Springer-Verlag. x, 422 p. (1995). ZBL0818.18001

in particular Chapter III, Theorem 2.1. The book seems to be treating rings $R$ which need not be commutative, hence there is no mention of the $R$-module structure since it need not exist!


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