Showing the category of perfect complexes on a scheme is essentially small - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:58:00Z http://mathoverflow.net/feeds/question/39279 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39279/showing-the-category-of-perfect-complexes-on-a-scheme-is-essentially-small Showing the category of perfect complexes on a scheme is essentially small Dylan Wilson 2010-09-19T06:11:18Z 2011-02-14T17:57:58Z <p>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...</p> <p>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. </p> <p>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?</p> <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/39279/showing-the-category-of-perfect-complexes-on-a-scheme-is-essentially-small/55431#55431 Answer by Leo Alonso for Showing the category of perfect complexes on a scheme is essentially small Leo Alonso 2011-02-14T17:57:58Z 2011-02-14T17:57:58Z <p>I have just noticed this question. There is (reasonably) self-contained proof in section 3 of <a href="http://arxiv.org/abs/0905.2063v1" rel="nofollow">arXiv:0905.2063v1</a>, 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".</p> <p>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. </p> <p>The globalization is done by patching and I think is described with some detail in the paper.</p>