DQCoh(X) = D(E)

Over the last week I added the final arguments to show the result of the title of this post when X is a quasi-compact and quasi-separated scheme or algebraic space X.

Let X be a quasi-compact and quasi-separated algebraic space. There exists a differential graded algebra E with only a finite number of nonzero cohomologies, such that DQCoh(X) = D(E).

For schemes this result is due to Bondal and van den Bergh and can be found in their wonderful paper Generators and representability of functors in commutative and noncommutative geometry. The proof for algebraic spaces is exactly the same (we claim no originality, as always). You can find more precise statements and proofs in the Stacks project here:

  1. the case of schemes is in Section Tag 09M2
  2. the case of algebraic spaces is in Section Tag 09M9

The statement involves the derived category of differential graded modules over a differential graded ring. Moreover, the proof of the result as given in the paper by Bondal and van den Bergh invokes a general result a la Gabriel-Popescu of Keller which can be found in Keller’s paper Deriving DG categories, in Section 4.3 to be precise. A very general (perhaps the most general available) version of Keller’s result is in a more recent paper by Marco Porta, entitled “The Popescu-Gabriel theorem for triangulated categories”.

So, this got me a bit worried as I am not an expert in differential graded categories, Frobenius categories, etc. But it turns out, as Michel van den Bergh hinted at in an email, that one needs only the most basic material on differential graded algebras, differential graded modules, and a tiny bit about differential graded categories. This material can be found in the new chapter on differential graded algebra. The key construction needed for the proof of the theorem of the title can be found in Section Tag 09LU.


PS: In my graduate student seminar I will lecture on this and related material during the semester and I will go over the new material a second time, adding a few more details and corrections. But as always, I’d be most happy if you can find mathematical errors (this will earn you a Stacks project mug) or have suggestions for improvements of exposition. Thanks!

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