Iterated HRS-tilting and derived equivalences for commutative notherian rings

Let R be a commutative noetherian ring. It is well known that the assignment of
support allows one to characterise many subcategories of Mod(R) and D(R) in
terms of certain subsets of Spec(R): hereditary torsion classes in Mod(R)
(see eg [4]), localising (and smashing) subcategories of D(R) ([3]) and, more
generally, compactly generated t-structures of D(R) ([1]).

In this talk we will see that also in the heart of any (nondegenerate) compactly
generated t-structure of D(R), hereditary torsion pairs are classified by
support. This fact can be used to show that if two compactly generated
t-structures have finite distance from each other, then they are linked by a
chain of HRS-tilting procedures, with respect to hereditary torsion pairs of
finite type at each step.

We will also address the question of whether these HRS-tilts provide derived
equivalences, using a criterion from [2]. We will see that if R is a commutative
noetherian ring, every hereditary torsion pair in Mod(R) provides derived
equivalence; and that if an intermediate (nondegenerate) compactly generated
t-structure satisfies some conditions (i.e. it is restrictable, [1]) then the
iterated HRS-tilting procedure found above provides a derived equivalence at
each step.

This talk is based on an ongoing joint work with J. Vitória.

[1] Alonso Tarrio, Jeremias Lopez, Saorin. ``Compactly generated t-structures on the
    derived category of a noetherian ring'', J. Algebra 324, 313-346
[2] Chen, Han, Zhou. ``Derived equivalences via HRS-tilts'', Adv. Math. 345 (2019)
[3] Neeman. ``The chromatic tower for D(R)'', Topology 31 (1992), no. 3, 519-532
[4] Stenström. ``Rings of quotients'', Springer-Verlag 1975