Timeline for What are the merits of the different finiteness conditions on quasi-coherent sheaves?
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| Feb 15, 2014 at 23:32 | comment | added | Ingo Blechschmidt | A small correction: You write "For instance, probably 1 and 2 can be expressed in terms of some allowable syntax in topos theory, but 3 can't.". This is not true: In fact, a sheaf $F$ of $\mathcal{O}_X$-modules fulfils (3) if and only if from the point of view of the internal language of the sheaf topos $\mathrm{Sh}(X)$, $F$ is a coherent module (as usually defined in an ordinary course on commutative algebra with no sheaves in sight). | |
| May 12, 2010 at 15:23 | comment | added | BCnrd | 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. | |
| May 10, 2010 at 4:50 | comment | added | JBorger | OK, I'm not completely convinced, but at least it seems pretty reasonable. So things are only as bad as I thought. | |
| May 9, 2010 at 16:58 | comment | added | BCnrd | 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. | |
| May 9, 2010 at 13:49 | comment | added | JBorger | 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! | |
| May 9, 2010 at 12:52 | comment | added | BCnrd | 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. :) | |
| May 9, 2010 at 10:08 | answer | added | Sasha | timeline score: 1 | |
| May 9, 2010 at 10:01 | answer | added | Angelo | timeline score: 9 | |
| May 9, 2010 at 9:31 | history | asked | JBorger | CC BY-SA 2.5 |