Canonical tilting relative generators

Abstract

Given a relatively projective birational morphism $f\colon X\to Y$ of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over $Y$) generators $T_{X,f}$ and $S_{X,f}$ in $\mathcal{D}^b(X)$. We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that $\mathcal{D}^b(X)$ has such a filtration $\mathcal{L}$ where the lattice is the set of all birational decompositions $f \colon X \xrightarrow{g} Z \xrightarrow{h} Y$ 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 $\mathcal{L}$. We realise all such $Z$ as the fine moduli spaces of simple quotients of $\mathcal{O}_X$ 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 $\mathcal{D}^b(X)$.