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 $S$?
Best Answer
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.