# Do I know what "coherent sheaf" means if I know what it means on locally Noetherian schemes?

I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely presented modules agree. For quasicoherent sheaves over a locally Noetherian scheme $X$, coherent and locally finitely presented sheaves agree. In general, coherent sheaves are locally finitely presented, and hence they pull back along morphisms $f : X \to Y$ where $X$ is locally Noetherian.

Is this already enough information to tell me what coherent sheaves must be?

More precisely, if $Y$ is a scheme, let $N_Y$ be the category whose objects are pairs $(X, f)$ of a locally Noetherian scheme and a morphism $f : X \to Y$ and whose morphisms are commuting triangles. The category of coherent sheaves $\text{Coh}(N_Y)$ on $N_Y$ is the category whose objects are assignments, to each $(X, f) \in N_Y$, of a coherent sheaf $F_X \in \text{Coh}(X)$ and assignments, to each morphism $g : (X_1, f_1) \to (X_2, f_2)$ in $N_Y$, of an isomorphism

$$F_{X_1} \cong g^{\ast} F_{X_2}$$

satisfying the obvious compatibility condition, with the obvious notion of morphism. Since coherent sheaves are locally finitely presented, they pull back to locally finitely presented sheaves in a way satisfying the obvious compatibility conditions, so there is a natural functor

$$\text{Coh}(Y) \to \text{Coh}(N_Y)$$

given by taking pullbacks along all morphisms $f : X \to Y$ where $X$ is locally Noetherian. The more precise version of my question is:

Is this functor an equivalence?

I think there is a slicker way to ask this using descent and another slicker way to ask this using Kan extensions, but I'll refrain from both to be on the safe side. If the above is true, I'd also be interested in knowing to what extent I can restrict "locally Noetherian" to a smaller subcategory. Does it suffice to use Noetherian schemes? Affine Noetherian schemes? Over an affine Noetherian base $\text{Spec } S$, does it suffice to use $\text{Spec } R$ where $R$ is finitely generated over $S$?

• I want to point out that the notion of coherence originated in complex analysis rather than algebraic geometry. Specifically, Oka says that the sheaf of holomorphic functions on a complex manifold is coherent, although it would not be "approximated" by (locally noetherian) schemes in any reasonable sense. This doesn't answer you asked, but it does suggest that maybe it isn't the right question. May 8, 2015 at 18:21
• Let $Y$ be any scheme such that $\mathcal{O}_Y$ is not coherent. Now consider the object $\mathcal{O}_{N_Y}$ of $\text{Coh}(N_Y)$ given by assigning to each $(X, f)$ in $N_Y$ the sheaf $\mathcal{O}_X$. Surely this is not in the essential image of your functor... May 8, 2015 at 20:38
• Second, in general I think young mathematicians have an incentive to think about things differently from our illustrious predecessors. After all, they already spent decades thinking about things deeply from their point of view and wringing insights out of it. If I want to have different insights from them, I should think about things differently from them and see if anything comes of it. If it doesn't, no worries, I can still learn the classical stuff. If it does, though, that would be great! May 9, 2015 at 5:22
• @QiaochuYuan: My point is that useful new concepts and insights do not arise out of a void. For schemes and Artin stacks, coherence is generally the wrong "finiteness" notion away from the locally noetherian case. The systematic limit techniques in EGA IV are used all the time to handle that issue for work with schemes and Artin stacks that aren't locally noetherian. Into what phenomenon or example for derived stacks away from a "locally noetherian" setting do you seek an insight that has no shadow in our prior experience with schemes and Artin stacks beyond the locally noetherian case? May 9, 2015 at 11:41
• @QiaochuYuan: Now that you have identified a source of motivation, I have taken a look at it. I think you are reading more meaning into Gaitsgory's generality of definition than he intended; he doesn't seem to prove any hard theorem away from the locally noetherian setting, and there is no evidence that he gave serious thought to what is a "right" definition for such extra generality. So rather than develop a general definition for a purpose which does not yet exist, it seems more suitable to email Gaitsgory and ask for his thoughts. My prediction is that he'll say he sees no motivation. May 10, 2015 at 3:35

Take $A$ to be Brian Conrad's universal counter example, i.e., the infinite countable product of copies of $\mathbf{F}_2$. Then $A$ is absolutely flat and every finitely presented module is finite locally free. On the other hand, every element of $A$ is an idempotent. Hence if $B$ is a Noetherian ring whose spectrum is connected, then any ring map $A \to B$ factors through a quotient $\kappa = A/\mathfrak m$ for some maximal ideal $\mathfrak m \subset A$. (Please observe that $\kappa \cong \mathbf{F}_2$.) Thus if we choose for every maximal ideal $\mathfrak m$ an integer $n_\mathfrak m \geq 0$, then we can assign to $A \to B$ as above the module $B^{\oplus n_\mathfrak m}$. I leave it up to you to see that this produces an object of $\text{Coh}(N_Y)$ for $Y = \text{Spec}(A)$. But of course, this object does not come from a coherent $A$-module if we take the function $\mathfrak m \mapsto n_\mathfrak m$ to take an infinite number of distinct values.