We will introduce the (new) notion of approximability in triangulated categories and show its power.

We will introduce the notion by example: the derived category D(R-Mod) of all complexes of R-modules is an approximable triangulated for any ring R. And this is because any module has a projective resolution. Thus approximability can be thought of as a fanciful version of projective resolutions – Auslander was a master at using
such things.

The main nontrivial example (to date) is that the derived category of quasicoherent sheaves on a separated,
quasi-compact scheme is an approximable triangulated category. As relatively easy corollaries one can:

(1) Prove an old conjecture of Bondal and Van den Bergh, about a strong generation in D^{perf}(X).

(2) Generalize an old theorem of Rouquier about a strong generation in D^b_{coh}(X). Rouquier proved
the result only in equal characteristic, we can extend to a mixed characteristic.

(3) Generalize a representability theorem of Bondal and Van den Bergh, from schemes proper over fields to
schemes proper over any noetherian rings.

After stating these results and explaining what they mean, we will (time permitting) also mention structural theorems. It turns out that approximable triangulated categories have a fair bit of intrinsic, internal structure that comes for free.

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