Timeline for What are the merits of the different finiteness conditions on quasi-coherent sheaves?
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| May 9, 2010 at 11:45 | comment | added | Angelo | (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". | |
| May 9, 2010 at 11:31 | comment | added | Angelo | 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). | |
| May 9, 2010 at 10:46 | comment | added | JBorger | 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? | |
| May 9, 2010 at 10:34 | comment | added | JBorger | 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) | |
| May 9, 2010 at 10:01 | history | answered | Angelo | CC BY-SA 2.5 |