5
$\begingroup$

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.


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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ This is very helpful, thanks! In the $\operatorname{GP}(R)$ case that I was thinking about, the admissibility of the relevant epi and mono is even clearer, because the kernel/cokernel are obviously in GP again (but I am very happy that this is a more general fact!). $\endgroup$ – Matthew Pressland Jun 21 '16 at 15:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.