Ideal filtrations and stabilization in alternating HRS tilting

Abstract

We prove that the realization functors obtained by alternating forward and backward HRS tilts of any torsion pair over a finite-dimensional algebra are equivalences from some step onward. The number of steps is bounded by the lengths of the indecomposable projective modules. The even hearts define an increasing filtration of ideals contained in a square-zero ideal. For a torsion pair induced by a two-term silting complex, we identify the successive layers with Buan-Zhou kernel ideals and compute them by reduction modulo the successive ideals. We also describe the largest quotient over which the complex becomes tilting and determine the exact stabilization index from the two parity heights. We construct algebras for which the HRS height is arbitrarily large, and compute their ideal filtrations.