Abstracts and Available Videos
Abstract: 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 category 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, quasicompact scheme is an approximable
triangulated category. As relatively easy corollaries one
can:
(1) Prove an old conjecture of Bondal and
Van den Bergh, about strong generation in D^{perf}(X).
(2) Generalize an old theorem of of
Rouquier about strong generation in D^b_{coh}(X). Rouquier
proved the result only in equal characteristic, we can extend to
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.
