Timeline for Reference request: category of sheaves of O-modules with coherent cohomology
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2023 at 14:29 | answer | added | Tim | timeline score: 1 | |
| Nov 30, 2015 at 3:48 | comment | added | nfdc23 | @AaronBergman: Continuing the theme of my previous comment, it is amusing to note that the result in Huybrechts' book (combine Cor. 3.4 and Prop. 3.5 there) that answers the question posed has as its crucial input a certain Prop. 3.3 whose proof consists of a single sentence: "For a proof, see [44, II, 7.18]." We can all guess what [44] is. :) The parts which Huybrechts does explain more fully (to get from Prop. 3.3 to Cor. 3.4 to Prop. 3.5) are exactly the same arguments as given in [44]. | |
| Nov 30, 2015 at 3:30 | comment | added | nfdc23 | @JasonStarr: The precise reference in SGA6 Exp. II is Cor. 2.2.2.1, but its proof rests on Prop. 3.5 that appears later in Exp. II and is precisely the assertion I had mentioned concerning $D^+$'s. The fun part is that the entire SGA6 proof of Prop. 3.5 (in the noetherian case) is to say: "le cas ou $S$ est noetherian etant bien connu ([H] II 7.19)", where [H] is (of course!) Hartshorne's "Residues and Duality". So T&T really should have referred to the latter (since that is where all of the content is given). | |
| Nov 30, 2015 at 3:02 | comment | added | Jason Starr | In Thomason-Trobaugh, this is Proposition 2.3.1(e), which is attributed there to SGA 6, Expose II. | |
| Nov 30, 2015 at 2:18 | comment | added | Aaron Bergman | A proof that $D^b(\mathrm{Coh(X)}) \simeq D^b_\mathrm{Coh}(\mathrm{QCoh}(X))$ passage to coherent sheaves is in chapter 3 of Huybrechts's book on Fourier-Mukai transforms. Huybrechts also references Gelfand-Manin and Kaschiwara-Schapira for the other results. | |
| Nov 30, 2015 at 1:51 | comment | added | nfdc23 | It is proved in Hartshorne's book "Residues and Duality", in the broader context of the map $D^+({\rm{Qcoh}}(X)) \rightarrow D^{+}_{\rm{qcoh}}(X)$ being an equivalence for any noetherian scheme $X$ (maybe even locally noetherian, but I don't remember offhand), from which the "coherent" version in the bounded case can be easily deduced (as quasi-coherent sheaves on $X$ are exhausted by coherent subsheaves). I always assumed this was widely known to be the go-to reference on such matters; please promote more awareness of the classics! :) | |
| Nov 30, 2015 at 1:02 | comment | added | Jason Starr | Isn't this in Thomason-Trobaugh? | |
| Nov 30, 2015 at 0:57 | history | asked | Dmitry Vaintrob | CC BY-SA 3.0 |