Аннотация:
I will consider a weakly localising Serre subcategory $B$ in an abelian
category $A$, i.e. a Serre subcategory such that the quotient $A/B$ admits
a torsion pair with the torsion-free part equivalent to the category $E$
of $B$-closed objects. I will give sufficient conditions for $B$ to be
weakly localising in terms of torsion-tilting chains in $A$. I will
also argue that $T$-consistent pairs of t-structures of amplitude 2 are
equivalent to (strongly) torsion-tilting chains.
Given a scheme $X$ of dimension $n$, the derived category $\mathrm{D}(X)$ admits a
$T$-consistent pair of t-structures of amplitude $n$ which yields a pair
of amplitude 2. As a result, the category $\mathrm{Coh}_2(X)$ of sheaves
supported in codimension 2 is weakly localising. I will prove that,
under additional assumptions on $X$, the additive category $E_2(X)$ of
locally $\mathrm{Coh}_2(X)$-closed objects allows us to reconstruct $X$ up to an
isomorphism outside of codimension 2.
For a normal surface $X$ I will construct its final model $X'$ from the
additive category $E_2(X)$. I will argue that $X$ admits an open embedding
into $X'$ with complement of codimension two and I will give conditions
under which $X$ is isomorphic to $X'$.
This is based on a joint work with A. Bondal.
