# What are the merits of the different finiteness conditions on quasi-coherent sheaves?

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.

1. $F$ is of finite type if, after passing to some cover, it is isomorphic to a quotient of a finite free module.
2. $F$ is of finite presentation if, after passing to some cover, it is isomorphic to a cokernel of a map of finite free modules.
3. $F$ 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?

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Jim, coherence is crucial for analytic spaces (clarifies Oka's deep results) but your def. of qcoh is poor in analytic geometry (see Rem. 2.1.5ff in "Relative ampleness in rigid geometry"). "Noetherian" in comm alg makes sense for modules & is useful when ring is noetherian over itself; likewise coh. is significant when $O_X$ is coh. over itself: it's a version of "noetherian" for ringed spaces. Non-noeth. formal schemes in Raynaud's approach to rigid geom. have coh. structure sheaf. Read Serre's FAC paper for good formal properties of coh! I disagree with your "right" way to define qcoh. :) –  BCnrd May 9 '10 at 12:52
Thanks. I had a quick look at that paper. So not only are you saying that my definition isn't the right one, you also say that you're not sure yours is the right one. Things are worse than I thought! –  JBorger May 9 '10 at 13:49
Jim, I don't think qcoh is an important notion away from "scheme-like" objects, so I don't view it as bad that there may be no "good" (or "right") definition of qcoh in total generality (including analytic spaces, topoi, etc.). In that paper I gave a definition useful for my purposes (which your proposed definition is not), and ultimately I think that's always the test of a definition: is it useful for some purpose? As I said in that paper, lack of further application kills motivation to search for a "better" def'n of qcoh in the analytic setting. So doesn't seem "worse" than you thought. –  BCnrd May 9 '10 at 16:58
OK, I'm not completely convinced, but at least it seems pretty reasonable. So things are only as bad as I thought. –  JBorger May 10 '10 at 4:50
Jim, I should clarify that upon further reflection, our different-looking definitions of quasi-coherence (in the analytic setting) do coincide. Your definition implies my definition by a little argument with coherence, and by 2.1.8(3) in my paper (mentioned above), over a sufficiently refined admissible affinoid covering a qcoh sheaf in sense of my definition arises from a module and so by choosing a gigantic presentation of the module it recovers your definition. For the purposes of my paper, the def'n I used was more convenient than the one you suggest above; but ultimately they agree. –  BCnrd May 12 '10 at 15:23

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.

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Thanks. But I'm not quite sure what you're actually saying. First, in your second sentence, you must have forgotten an assumption on the scheme. Second, are you saying that it's well known that the definition in my question #2 is the right one and that the existing definition of coherence in the literature should be scrapped? If so, is this discussed anywhere, prehaps with some actualy theorems proved? Is it generally true that noetherianness hypotheses can be dropped if you work with this definition? (continued) –  JBorger May 9 '10 at 10:34
Finally, I think I agree with what you're getting at your 3rd and 4th paragraphs. Definitions should work without any hypotheses and should allow you to reduce things to finite cases, where they coincide with some perhaps more naive definitions. (That was the whole reason for the question--what are the right definitions in general?) So is the point of paragraphs 3 and 4 that finite presentation has some good properties, but not as many as you might hope, so we want to keep it but we still need something else? If so, that's good--it confirms what I expected. And that is the def in my Q #2? –  JBorger May 9 '10 at 10:46
Dear James, sorry for being unclear. I was trying to make a couple of points. The first is that the useful notion, in the contexts in which I work, is that of finitely presented sheaf, not that of coherent sheaf. On the other hand, essentially all you use is that locally they come from noetherian schemes, and thus all the real work is done with noetherian schemes. The second is that the coherence condition is important in other contexts, for example when working with D-modules. In my second sentence I was just trying to give an example of when it is important (continued below). –  Angelo May 9 '10 at 11:31
(continued from above) About my first sentence, sorry, I thought I meant something, but I am not sure anymore. In any case, I have seen your notion of $\infty$-finite type somewhere (I can't recall where, maybe Borel's book on D-modules), and it seems to me that it should always give an abelian category. Also, sorry again, "the sheaf of D-modules" in my second sentence should be "the sheaf D of differential operators". –  Angelo May 9 '10 at 11:45
OK. That makes sense. Thanks! –  JBorger May 10 '10 at 4:52

Coherent sheaves form an abelian category --- this is why they are used.

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