# Reference request: category of sheaves of O-modules with coherent cohomology

Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of $\mathcal{O}_X$-modules whose cohomology is coherent. The only reference I could find are these notes, which are written in some exotic language I can't parse. Does anyone know a more canonical reference?

Maybe a more general question is "what is a general reference to cite when using equivalences between classical (Grothendieck-era) derived categories and their more modern analogues?"

(Lurie's Higher Algebra will sometimes mention such equivalences without proving them).

• Isn't this in Thomason-Trobaugh? – Jason Starr Nov 30 '15 at 1:02
• 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! :) – nfdc23 Nov 30 '15 at 1:51
• In Thomason-Trobaugh, this is Proposition 2.3.1(e), which is attributed there to SGA 6, Expose II. – Jason Starr Nov 30 '15 at 3:02
• @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). – nfdc23 Nov 30 '15 at 3:30
• @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]. – nfdc23 Nov 30 '15 at 3:48