A certain type of perfection…

Yesterday, on the last day of 2012, I finished writing some material on the existence of approximations by perfect complexes on algebraic spaces which I committed to the project on the first day of 2013, i.e., today. This covers the results necessary to proceed with the strategy described here to apply Artin’s axioms to the existence of Quot spaces. As mentioned there, the approximation result (in the case of schemes) can be found in the paper “Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor” by Lipman and Neeman. Thanks to Jack Hall for pointing out the reference.

The new (as in “it wasn’t in the Stacks project before”) material is mainly in the chapters Derived Categories of Schemes and Derived Categories of Spaces. It should be easy to add a lot more basic material in these chapters, but for now I’ve basically only added the material leading up to

  1. Proposition Tag 08DB which says that D(QCoh(X)) = D_{QCoh}(X) for quasi-compact schemes with affine diagonal,
  2. Theorem Tag 08ES which says we have approximation by perfect complexes for quasi-compact and quasi-separated schemes,
  3. Proposition Tag 08H1 which says that D(QCoh(X)) = D_{QCoh}(X) for quasi-compact algebraic spaces with affine diagonal, and
  4. Theorem Tag 08HP which says that we have approximation by perfect complexes for quasi-compact and quasi-separated algebraic spaces.


It turns out that there is an induction principle for quasi-compact and quasi-separated algebraic spaces which I missed formulating clearly and using in writing the material in the chapter on cohomology of sheaves on algebraic spaces. For example, in order to prove one has finite cohomological dimension for quasi-coherent sheaves on quasi-compact and quasi-separated algebraic spaces, we introduced (with help from Bhargav Bhatt) the alternating Cech complex and used that. However, it may makes sense to revise and replace this by an application of the induction principle. On the other hand, I think that carefully applying the alternating Cech complex, you can sometimes prove more (i.e., you can get explicit bounds which I do not see how to get otherwise and which may even be interesting in the case of schemes). I hope to return to this sometime in the future.

3 thoughts on “A certain type of perfection…

  1. If I am not mistaken, you actually prove 4: approximation by perfect complexes for any qc+qs algebraic space, not necessarily with affine diagonal. Right?

    I also wanted to point out that my paper “Étale dévissage, descent and pushouts of stacks” is devoted to induction principles like the one you mention. In particular, Thm D is this induction principle, except for the étale topology instead of the Nisnevich topology (I also include finite étale maps) so that I can treat Deligne-Mumford stacks as well. I’m also guessing, but I’m to lazy to check, that your induction for algebraic spaces is essentially identical to Lurie’s “scallop decompositions”.

    Of course, I don’t claim that this induction procedure is new at all: it’s implicit in Raynaud-Gruson’s paper and more explicit in papers on motives (Morel-Voevodsky etc) where they are called upper or elementary distinguished squares just as you call them (I use the term “étale neighborhoods”).

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