# Showing the category of perfect complexes on a scheme is essentially small

So I've tried to set up a few meetings with professors to talk about this, but I think one of them forgot about it after a big conference, and the other is still on vacation... in the meantime...

At one point in Thomason's paper on the classification of thick subcategories of the derived category of perfect complexes over a (EDIT: quasi separated, quasi compact!) scheme, he needs to use the fact that this category is essentially small (that is: it has a SET of isomorphism classes). The reference for this is yet another paper by Thomason (and ghost buddy Trobaugh) that is HUGE (and very important) called "Higher algebraic K-theory of schemes and derived categories." The trouble is that the proof of essential smallness is tucked away in some appendix, paraphrased, and uses results that are scattered throughout this paper in a not-so-clear manner.

So my question is: does anyone know of a proof, or an explanation of the proof, of the fact that the derived category of perfect complexes on a (quasi-separated, quasi-compact!) scheme is essentially small that doesn't require me to learn algebraic K-theory and read all of Thomason's big paper?

EDIT: A "strictly perfect complex" is a bounded complex of locally free $O_x$-modules of finite type. A "perfect complex" on $X$ is a complex $E$ of sheaves of $O_X$-modules such that there is an open cover of $X$ where $E$ restricted to each neighborhood in the cover is quasi-isomorphic to a strict perfect complex.

Some background: be gentle, I know very little algebraic geometry... I learned everything backwards, so triangulated category talk is fine, but algebro-geometric talk should be dumbed down, and perhaps illustrated with some examples.

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I am neither very familiar with algebraic geometry nor with category theory. I know what a scheme is. What is a perfect complex over a scheme? If the complexes are indexed by integers and the objects that can occur in the complexes have a set of representatives of their isomorphism classes, then each complex is isomorphic to a complex with objects coming from a fixed set. Countable sequences with values in a fixed set form a set. Countable sequences together with morphisms between the elements of the sequence that are actually maps are again a set. –  Stefan Geschke Sep 19 '10 at 7:05
So, what this boils down to is this: If the objects that occur in the complexes under consideration come from a fixed set, the size of the complexes is bounded by some cardinal and the morphisms that occur in the complexes also come from a set of things (such as maps between objects in a given set of objects), then the whole category is small. If the above only holds up to isomorphism, then the category is essentially small. –  Stefan Geschke Sep 19 '10 at 7:08
At the risk of looking like a fool(I don't know much AG or any set theory), presumably this is a lot easier with some kind of Noetherian and quasi-separated hypo.? Can't we then just take a Cech cover and then we know every locally free can be represented up to isomorphism by an iso. class of proj. modules + descent data and then a complex is a bd. complex of these mod.s, which can be described the same way. Does this not work or you're really serious about the level of generality your asking for(any scheme)? Anyways, on the plus side I doubt the argument needs tough things from that paper. –  Daniel Pomerleano Sep 19 '10 at 22:27
just as a small additional note any fg projective module on an an affine scheme can be embedded as a submodule of a free one so those iso classes form a set... Maybe the above idea works without any of those hypotheses actually... –  Daniel Pomerleano Sep 19 '10 at 22:53
@Stefan- added a definition for you. @Daniel- We assume quasi-compact quasi-separarted, which I added... and your argument looks like it could work. Mind putting a slightly more detailed version as an answer? Like- can you give a reference for the correspondence between locally free complexes and projective modules with descent data? –  Dylan Wilson Sep 20 '10 at 0:27

## 1 Answer

I have just noticed this question. There is (reasonably) self-contained proof in section 3 of arXiv:0905.2063v1, though it is phrased in the language of formal schemes. I think it works replacing every time that "noetherian formal scheme" occurs by "quasi-compact and quasi-separated scheme".

The affine case is easy enough, every perfect complex is quasi-isomorphic to a strict perfect complex which in turn is a bounded complex of finitely generated projective modules. Choosing a set of representatives of isomorphism classes of projective modules (as direct summands of free modules, for instance) you are done.

The globalization is done by patching and I think is described with some detail in the paper.

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