# Graded version of Baer's Criterion

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a graded version for this? I mean:

Let $R$ be a graded ring. Let $M$ be a graded $R$-module.

If $Ext_R^1(R/I,E)=0$ for all homogeneous ideals $I$ of $R$ then $Ext_R^1(M,E)=0$ for every graded $R$-module $M$?

• Can you not just copy the usual proof inserting the word "graded" everywhere? Feb 20 '15 at 8:25
• @Eric: No - cf. my answer. But in some way also Yes - using the right interpretation of "graded Ext" and "graded R"... Feb 20 '15 at 12:12

To get the correct graded version you have to observe all possible shifts. More precisely:

Let $$G$$ be an abelian group, let $$R$$ be a $$G$$-graded ring, and let $$M$$ be a $$G$$-graded $$R$$-module. Then, the following statements are equivalent:

(i) $$M$$ is injective;

(ii) For every $$g\in G$$, every monomorphism $$v\colon N\rightarrow R(g)$$ and every morphism $$w\colon N\rightarrow M$$ there is a morphism $$u\colon R(g)\rightarrow M$$ such that $$u\circ v=w$$.

(Note that in the above, "morphism" means morphism in the category of $$G$$-graded $$R$$-modules.)

In order to proof this you have to observe that the category of $$G$$-graded $$R$$-modules is an AB5-category and that $$\bigoplus_{g\in G}R(g)$$ is a generator of this category, and then you can apply Lemma 1 in Section 1.10 of Grothendieck's Tohoku paper (Link).

• I added a link to the Tohoku paper (where you can also find a definition of AB5). Feb 20 '15 at 14:20
• I do not know of any translation. But do not fear - it is not french but maths! Feb 20 '15 at 14:55
• Michael Barr has an English translation available here: math.mcgill.ca/barr/papers/gk.pdf Feb 21 '15 at 9:08