It's my understanding that there's no disagreement about the right way to define a quasi-coherence for a sheaf $F$ of $O_X$-algebras (over a scheme, locally ringed space, or even locally ringed topos). It means that, after passing to some cover, it's isomorphic to a cokernel of a map of free $O_X$-modules.

But now there are several different finiteness conditions you can put on these.

- $F$ is of finite type if, after passing to some cover, it is isomorphic to a quotient of a finite free module.
- $F$ is of finite presentation if, after passing to some cover, it is isomorphic to a cokernel of a map of finite free modules.
- $F$ is coherent if it is of finite type and for every open $U$, every integer $n>0$, and every map $O_X^n\to F$, the kernel is of finite type.

Then 3 ==> 2 ==> 1. They are all equivalent if $X$ is Spec of a noetherian ring.

The first two conditions seem very natural and are of the standard kind in sheaf theory: there exists a cover over which some property exists. But the definition of coherence is very different, and purely on formal grounds, we might expect that the class of coherent sheaves would be not so well behaved. (For instance, probably 1 and 2 can be expressed in terms of some allowable syntax in topos theory, but 3 can't.) Sure enough, the early sections of EGA are a mess when they talk about coherent sheaves, with noetherian hypotheses all over. For instance, if $X=\mathrm{Spec}(R)$, then $O_X$ is proved to be coherent only when $R$ is noetherian, as far as I can tell, whereas it's obviously finitely presented. Also, I think a quasi-coherent sheaf on an affine scheme is finitely presented if and only if the corresponding module is. And finite presentation is stable under pull back, but coherence isn't (e.g. $X\to\mathrm{Spec}(\mathbf{Z})$, where $O_X$ isn't coherent).

So coherence seems like a bad condition in the absence of some other hypotheses which make it collapse into one of the good ones. My feeling is that such foundational things should be very formal and tight, and if they're not, it's probably because we're using approximations of the right concepts.

Question #1: What is coherence actually good for? Suppose we tried to replace it with finite presentation everywhere. Would anything go wrong? (Is there difference between algebraic and analytic geometry here?)

Question #2: If finite presentation has its problems (which it does, I think, but I can't remember them now), are there any known variants that are better behaved? For instance, what about this condition: For an integer $n\geq 0$, let's say that $F$ is of $n$-finite type if, after restricting to some cover, there exists an exact sequence of $O_X$-modules $$M_n\to M_{n-1} \to \cdots \to M_0 \to F \to 0,$$ where each $M_i$ is $O_X^{r_i}$ for some integer $r_i\geq 0$. So finite type = 0-finite type, and finite presentation = 1-finite type. Then say $F$ is of $\infty$-finite type if there exists an $n$ such that $F$ is of $n$-finite type. Is there any chance that being of $\infty$-finite type is well behaved?

yoursis the right one. Things are worse than I thought! – JBorger May 9 '10 at 13:49