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?
Best Answer
Of course, the correct definition of coherence is that in your Question 2. It just so happens that for a sheaf of modules on a scheme it is equivalent to the easier one.
As far as a I know, the notion of coherence is mostly used when one has a sheaf of ring (often non-commutative), different from the structure sheaf. So, for example, the sheaf of D-modules on a smooth variety is not noetherian, but it is a coherent sheaf of rings; this is a very important fact.
There are schemes whose structure sheaf is not coherent, and those are a bit of a mess; for example, the locally finitely presented quasi-coherent sheaves do not form an abelian category. However, in most cases one is not usually bothered by them.
For example, in setting up a moduli problem, it is quite useful to consider non-noetherian base schemes, because the category of locally noetherian schemes has problems (for example, is not closed under fibered products). For example, if $X$ is a projective scheme over a field $k$, and $F$ is a coherent sheaf on $X$, one defined the Quot functor from schemes over $k$ to sets by sending each (possibly non-noetherian scheme) $k$-scheme $T$ into the set of finitely presented quotients of the pullback $F_T$ of $F$ to $T$ that are flat over $T$. Of course, when you actually prove something, one uses that fact that locally on $T$ any finitely presented sheaf comes from one defined a finitely generated $k$-algebra, and works with that, free to use all the results that hold in the noetherian context. Thus, in practice most of the time you don't need to do anything with non-noetherian schemes.