is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

Question

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent sheaves be constructed categorically from the category of vector bundles? I am thinking of Coh being some kind of 'abelian envelope' of Vect.

(in the affine case this question can be rephrased as: can the category of finitely generated modules be defined via the category of projective modules of finite rank)?)

Answer 1

Here are a few comments that might be useful. I don't think there is a chance that this can work unless the scheme in question has the resolution property (meaning every coherent sheaf is a quotient of a locally free sheaf of finite rank). Otherwise the category of locally free sheaves does not even form a generator of the category of all quasi-coherent sheaves, so it clearly contains more information.

Secondly, Quiaochu Yuan's construction for the affine case does not work globally for most schemes. What he does is indeed freely adding cokernels (to get to coherent sheaves) or freely adding all colimits (to get to quasi-coherent sheaves). The free cocompletion under all colimits of an additive category is given by taking the category of additive presheaves on it. (The free cocompletion under cokerenels is simply the closure of the representables under cokernels.) So, if we do that to the category of vector bundles on a scheme, we obtain a category of presheaves. However, any category of presheaves has a projective generator, while the category of quasi-coherent sheaves rarely does.

Finally, something positive that can be said: If you do assume that your scheme satisfies the resolution property (and I'll assume it is quasi-compact, not sure if that's necessary), then the full subcategory of vector bundles is a dense subcategory of the category of quasi-coherent sheaves. This is actually a quite amazing result: in a Grothendieck abelian category, any strong generator is dense, see

Brian Day and Ross Street, Categories in which all strong generators are dense, J. Pure Appl. Algebra 43 (1986), no. 3, 235–242. MR 868984

Thus we know that the category of quasi-coherent sheaves is a reflective subcategory of the free cocompletion of the category of vector bundles. Any reflective subcategory is the localization of the surrounding category at the morphisms that the reflector inverts (that is, it can be obtained by formally inverting a class of morphisms). Since we're dealing with a locally finitely presentable category, this can be further reduced to inverting a generating set of these morphisms. In some sense this says that the category of quasi-coherent sheaves can be obtained by first freely adding colimits, and then imposing some relations (formally turn a certain set of morphisms into isomorphisms).

It seems however rather difficult to get an explicit such set of morphisms in general.

Edit: I noticed that you're also interested in algebraic spaces and algebraic stacks. The above argument about the category of quasi-coherent sheaves also works at that level of generality as long as the resolution property holds. Specifically, if you have a quasi-compact stack $X$ on the fpqc-site of affine schemes which has the resolution property (in algebraic topology these are sometimes called Adams stacks, since they are precisely the stacks associated to Adams Hopf algebroids), then the category of quasi-coherent sheaves on $X$ is given by a localization of the free cocompletion of the category of dualizable quasi-coherent sheaves on $X$ at a set of morphisms.

Note that it is not clear from this argument wether or not this set of morphisms is entirely determined by the subcategory of dualizable quasi-coherent sheaves. If that is not the case, there could be Adams stacks in the above sense with equivalent categories of dualizable sheaves but inequivalent categories of quasi-coherent sheaves.

Answer 2

Here's a negative answer: there can be no self-dual way to pass from the category of finitely generated projective modules to the category of all finitely generated modules (over, say, a Noetherian ring). To see this, note that the category of finitely generated projective modules is self-dual via the functor $Hom(-,R)$, but the inclusion of projective modules into all modules is obviously not usually self-dual (for instance, since projective modules are not the same as injective modules).

Thus you cannot say that coherent sheaves are the "abelian envelope" of vector bundles; any construction of coherent sheaves from vector bundles must in some way care about which direction maps are going (e.g. ya-tayr's suggestion, which adjoins formal cokernels of maps of vector bundles but not formal kernels).

Answer 3

(obviously, in the affine case this question translates into: can the category of (finitely generated) modules be defined via the category of projective modules (of finite rank)?)

Yes (you're assuming Noetherian here, right?). We will need to combine two observations. Let $A$ be an $\text{Ab}$-enriched category and let $\text{Mod}(A)$ be the category of additive functors $A^{op} \to \text{Ab}$ (generalizing either right modules or presheaves according to taste). The Cauchy completion $\hat{A}$ of $A$ is the category obtained from $A$ by first formally adjoining biproducts and then splitting all idempotents. When $A$ has one object with endomorphism ring $R$ then $\hat{A}$ is the category of finitely generated projective right $R$-modules.

Observation 1: The natural restriction map $\text{Mod}(\hat{A}) \to \text{Mod}(A)$ is an equivalence.

(This is the easy direction of Morita theory for $\text{Ab}$-enriched categories.)

In particular, the category of right modules over (finitely generated projective right $R$-modules) is $\text{Mod}(R)$.

To prove this it suffices to check that a right module $F : A^{op} \to \text{Ab}$ uniquely extends both to formal biproducts and to split idempotents, or in other words that biproducts and split idempotents are both absolute colimits for $\text{Ab}$-enriched functors. Existence follows from the fact that $\text{Ab}$ has all biproducts and all idempotents split in it. For uniqueness the essential point is that both being a biproduct and being a split idempotent are defined by equations among morphisms and that equations among morphisms are always preserved; see this blog post for more details in the case of biproducts, and the case of split idempotents is even easier (these are already absolute colimits for ordinary functors).

So starting from finitely generated projective modules / vector bundles, we've recovered all modules / quasicoherent sheaves. But we wanted to recover just the finitely generated modules. There are several options from here; for example, there is a canonical inclusion $\hat{A} \to \text{Mod}(A)$ and we can take the smallest abelian subcategory containing its image. Perhaps the most categorical answer is the following.

Observation 2: Let $R$ be a ring. A module $M \in \text{Mod}(R)$ is finitely generated iff $\text{Hom}(M, -)$ preserves filtered colimits where all of the maps in the diagram are monomorphisms.

(In particular, being finitely generated is a Morita invariant property: it does not depend on the choice of ring $R$.) For a proof of a closely related fact see this math.SE answer.

---

Edit: Actually there is an easier construction. If $A = \hat{R}$ is the category of finitely generated projective right $R$-modules then we can just consider "finitely generated (right) $A$-modules," namely those which are a quotient of a finite direct sum of representables. This reproduces the usual notion of finite generation.

Answer 4

Under the Noetherian hypothesis, the category of coherent sheaves is the smallest abelian subcategory (say, in the category of $\mathcal{O}_X$-modules), containing all line bundles. I don't have a reference in mind, but any standard algebraic geometry text should work (Harstsorne, Liu, Vakil's notes, Stacks project etc.)

UPD. As for the question of getting rid of the ambient category, this is a tuff (and, probably, not very natural) one. You still have to keep track of some structure. In particular, vector bundles form an exact category and one may take its abelian hull (this is some kind of an adjoint functor, though, you have to deal with 2-categories). This has appeared in some work of Keller. Probably, you still get the right answer, but I can't say anything more precise.

Answer 5

Mimicking the theory of projective resolutions, try this:

Start with the category whose objects are pairs $(V_1,V_0,d:V_1 \to V_0)$ where the $V_i$ are vector bundles, and whose morphisms are pairs $(f_i:V_i \to W_i)_{i \in \{0,1\}}$ intertwining with $d$.

Now divide each Hom group by the subgroup of maps where $f_0:V_0 \to W_0$ lifts to $H:V_0 \to W_1$. Taking cokernels gives a functor to coherent sheaves, and I think its an equivalence.

PS: correcting something. I forgot the possibility that, on some non-projective schemes, a coherent sheaf might not admit any surjection from a vector bundle. (Or perhaps, might not admit any nontrivial vector bundles at all.)