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## Canonical tilting relative generators

Advances in Mathematics. 2018. Vol. 323. P. 226-278.
Bondal A. I., Bodzenta-Skibinska A.

Given a relatively projective birational morphism f : X → Y
of smooth algebraic spaces with dimension of fibers bounded
by 1, we construct tilting relative (over Y) generators TX,f
and S_X,f in D^b(X). We develop a piece of general theory of
strict admissible lattice filtrations in triangulated categories
and show that D^b(X) has such a filtration L where the lattice
is the set of all birational decompositions f : X −→^g Z
−→^h Y with smooth Z. The t-structures related to T_X,f and S_X,f are
proved to be glued via filtrations left and right dual to L.
We realise all such Z as the fine moduli spaces of simple
quotients of O_X in the heart of the t-structure for which S_X,g
is a relative projective generator over Y . This implements the
program of interpreting relevant smooth contractions of X in
terms of a suitable system of t-structures on D^b(X).