There is some sort of symmetry in the definition of flat module and pure short exact sequence which can be made precise as follows.
Let $R$ be a ring (with unit), $\mathcal{R}$ be the class of all right $R$-modules, and $\mathcal{S}$ be the class of all short exact sequences of left $R$-modules.
For $\mathcal{A}\subseteq\mathcal{R}, \mathcal{B}\subseteq\mathcal{S} $, define a binary relation as $$ \mathcal{A}\perp \mathcal{B}:\Leftrightarrow ~\forall M\in\mathcal{A},\forall\mathcal{E}\in\mathcal{B},M\otimes_R \mathcal{E}~~\text{is still exact}.$$ which induces an (antitone) Galois connection as
$$\mathcal{A}^\perp=\{\mathcal{E}\in\mathcal{B}|\mathcal{A}\perp\{\mathcal{E}\}\}$$ $$^\perp\mathcal{B}=\{M\in\mathcal{A}|\{M\} \perp\mathcal{B}\}$$
Observe that $$^\perp\mathcal{S}=\{\text{flat modules}\}$$ $$\mathcal{R}^\perp=\{\text{pure short exact sequences}\}$$
My question is what are the closure operators corresponding to this Galois connection? i.e. how to describe $$^\perp(\mathcal{A}^\perp), (^\perp\mathcal{B})^\perp$$ explicitly? I think it’s common to focus on characterizing the closure operator once a Galois connection is found(one famous example is the Hilbert’s Nullstellensatz).
For the former, a rough guess is that $^\perp(\mathcal{A}^\perp)$= (modules in) the smallest full subcategory containing $\mathcal{A}$ which is closed under taking coproducts, filtered colimits, direct summands, and contains a generator.
Explanation:
the $\supseteq$ part is easy. Firstly coproducts and filtered colimits of modules in $\mathcal{A}$ should appear in the LHS since $^\perp(\mathcal{A}^\perp)$ is closed under these colimits due to the exactness of taking them. Secondly, all flat modules should appear in the LHS since
$$^\perp(\varnothing^\perp)=\{\text{flat modules}\}.$$
So we should add these to the RHS “categorically”. Flat modules can be seen as filtered colimit of free modules of finite rank by the Lazard’s theorem, while the regular module $R$ can be seen as a direct summand of a finite coproduct of copies of a generator. Note that the LHS is closed under taking direct summand(by a cute use of the snake lemma) and always contains the generator $R$, so the RHS is contained in the LHS.
My question lies in showing the $\subseteq$ part. I did some research on the internet but found no relavant results on this problem, although this seems like a natural question. If it’s right, how to prove it(the proof would be constructive I guess); if not, how to reformulate the guess?
Of course there is a dual statement of the guess lurking in the s.e.s. side, but I’m not sure what’s the counterpart of $R$ in the category of s.e.s..
Further explanation:
The problem is natural also in the sense that it’s slightly reminiscent of the Quillen’s small object argument, in which there is a similar Galois connection between two classes of morphisms $\mathcal{A},\mathcal{B}$ in a category, and a direct result is $^\perp(\mathcal{A}^\perp)$ = the smallest class of morphisms containing $\mathcal{A}$ which contains all isomorphisms and is closed under push-outs, retracts and transfinite composition (as well as coproducts, since it can be constructed from the former three). They are similar in the way that the “closure operator” are both described by smallest subsomething closed under certain colimits(but not all, of course). I hope mentioning this could shed some light on the original problem, since the hard part in the proof of the small object argument is also the forward inclusion, as we need to show that morphisms in $^\perp(\mathcal{A}^\perp)$ can actually be reached by a finite process of applying colimit constructions in the RHS to morphisms in $\mathcal{A}$. I guess that in the original problem the proof is supposed to be something like that. Also, the supposed answer may add some restrictions to $\mathcal{A}$, as in the small object argument.
I posted this question on Math StackExcange and have been waiting for an answer for a month so I think it’s reasonable to repost it here. I think this question is hard, so it’s just fine and well appreciated if someone could point out whether there is a similar technique solving a similar problem. I guess in the theory of cotorsion pairs there maybe some similar questions of concern.
