When is the category of Gorenstein projective $R$-modules Frobenius?

Question

Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a complete projective resolution, i.e. that are expressible as the image of the map $P_{-1}\to P_0$ in some sequence

$$\cdots\to P_{-2}\to P_{-1}\to P_0\to P_1\to P_2\to\cdots$$

of projective $R$-modules, such that this sequence remains exact under $\operatorname{Hom}_R(-,P)$ for any projective $P$. (For extra emphasis: I do not make any finite-generation assumptions on $M$, or on the $P_i$.)

My questions are then the following:

• Is $\operatorname{GP}(R)$ a Frobenius category?

Depending on the answer, there are natural follow-up questions:

• If yes, is there a good reference for this, ideally with a complete argument?

• If not, can we get this property back by assuming a little more about $R$? As a target, I would like to deal with the case that $R$ is a complete preprojective algebra of an arbitrary finite quiver (which is not typically Noetherian).

Henrik Holm has a very nice paper called 'Gorenstein homological dimensions', which proves lots of things about $\operatorname{GP}(R)$ for a general ring $R$, but does not directly address this question. He does show that this category is resolving, which helps a bit (in particular, it is closed under extensions and so inherits an exact structure from $\operatorname{Mod}{R}$). Assuming I haven't made any mistakes, I think the facts that $\operatorname{GP}(R)$ contains all projective $R$-modules, which are also injective in $\operatorname{GP}(R)$, and that every Gorenstein projective admits both an epimorphism from and a monomorphism to such a module are all pretty much clear from the definition. So the only thing that might go wrong (I think!) is that there could be more projectives/injectives that don't agree with each other.

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There is some extra context which might give a flavour of the kind of sources I would most appreciate (although of course any answer is appreciated!). I am aware that many authors consider the case that $R$ is Iwanaga–Gorenstein, meaning that $R$ is Noetherian and of finite injective dimension as a module on each side, and then consider the Frobenius category

$$\operatorname{GP}'(R):=\{X\in\operatorname{mod}{R}:\operatorname{Ext}_R^i(M,R)=0\ \forall\ i>0\}.$$

I am interested in certain (probably) non-Noetherian rings $R$, but that still have finite injective dimension as a module on each side, and in certain (possibly not finitely-generated) $R$-modules $M$ such that $\operatorname{Ext}^i_R(M,P)=0$ for any projective $P$ and any $i>0$. My feeling is that there should be some reasonable Frobenius category of 'Gorenstein projective-like' modules associated to $R$ and containing $M$, by focussing on the homological conditions and forgetting about any finiteness (even something like $\operatorname{GP}'(R)$ as defined above, but with $\operatorname{mod}{R}$ replaced by $\operatorname{Mod}{R}$, and $R$ replaced by an arbitrary projective in the condition – one of the things that Holm proves is that this category is then very close to $\operatorname{GP}(R)$ as defined at the top of the question, but might also include some modules with no finite resolution by Gorenstein projectives). However, I am not very familiar with what can go wrong when you drop finiteness conditions, and am concerned that I may lose the Frobenius property somewhere.

If it helps, I may in the end want my category to be Krull–Schmidt (in the strong sense that indecomposables are characterised by having local endomorphism rings) which means I will have to require that $R$ is semi-perfect. This gives a bit more control over $\operatorname{Proj}{R}$, as it means that there are finitely many indecomposable projectives such that every projective is a (possibly infinite) direct sum of these.

Answer 1

It's always a Frobenius category and its projective-injective objects are the projective modules. Your analysis is essentially right. In fact, this holds very generally. I worked out the following when I read a bit in Enochs-Yenda, but I suppose it's well-known:

Given an exact category $(\mathcal{A}, \mathcal{E})$, let $\mathcal{P}$ be the full subcategory of its projective objects. You can induce a new exact structure $\mathcal{E}_{\mathcal{P}}$ on $\mathcal{A}$ which consists of the $\mathcal{E}$-exact sequences $0 \to A' \to A \to A'' \to 0$ such that

$$0 \to \hom(A'', P) \to \hom(A, P) \to \hom(A', P) \to 0$$

is exact for all $P \in \mathcal{P}$.

Note that the objects in $\mathcal{P}$ are projective and injective with respect to the exact structure $\mathcal{E}_{\mathcal{P}}$ (there may be more $\mathcal{E_P}$-projective and injectives, of course).

Define $\mathcal{G_P}$ to be the category of Gorenstein projective objects with respect to $\mathcal{E}_{\mathcal{P}}$: these are the objects $G$ appearing as the image of a differential of a bi-infinite $\mathcal{E}_{\mathcal{P}}$-acyclic complex with components from $\mathcal{P}$.

We have $\mathcal{P} \subseteq \mathcal{G_P}$ because we can just take $\cdots \to 0 \to P \xrightarrow{\rm id} P \to 0 \to \cdots$.

The key fact is:

An $\mathcal{E}$-admissible epic $p\colon A \to G$ with $G \in \mathcal{G_P}$ is necessarily $\mathcal{E_P}$-admissible.

Since $G \in \mathcal{G_P}$ there is an $\mathcal{E_P}$-admissible epic $q\colon P \to G$ with $P \in \mathcal{P}$. Since $P$ is projective, there is $f \colon P \to A$ such that $q = pf$. Since $p$ has a kernel, Quillen's "obscure axiom" on exact categories implies that $p$ is an $\mathcal{E_P}$-admissible epic. You can also play around with long exact sequences and $\operatorname{Ext}^i(G,P)$ to see this.

It follows from this and the horseshoe lemma that $\mathcal{G_P}$ is closed under extensions in $(\mathcal{A}, \mathcal{E})$. Hence $\mathcal{G_P}$ is also closed under extensions in $(\mathcal{A}, \mathcal{E_P})$.

Proposition. The two exact structures on $\mathcal{G_P}$ induced by $\mathcal{E}$ and by $\mathcal{E_P}$ coincide. With respect to this exact structure $\mathcal{G_P}$ is a Frobenius category whose projective-injective objects are precisely the objects from $\mathcal{P}$.

That the two induced exact structures coincide on $\mathcal{G_P}$ follows again from the key observation.

Since the objects in $\mathcal{P}$ are projective and injective in $(\mathcal{A}, \mathcal{E_P})$, they are also projective and injective in $\mathcal{G_P}$.

If $G$ is either projective or injective in $\mathcal{G_P}$, it is a retract of an object of $\mathcal{P}$, hence it belongs to $\mathcal{P}$. To see this, suppose $G$ is projective in $\mathcal{G_P}$. Then there is an $\mathcal{E_P}$-admissible epic $P_{-1} \to G$, which splits since $G$ is projective. So $G$ is a retract of $P_{-1}$. It follows that $G$ is both projective and injective in $\mathcal{G_P}$, and also projective in $(\mathcal{A},\mathcal{E})$. Similarly, if $G$ is injective, it is a retract of $P_0$, hence projective and injective in $\mathcal{G_P}$ and also projective in $(\mathcal{A}, \mathcal{E})$.

That there are enough projectives and injectives in $\mathcal{G_P}$ holds by definition.