Splitting torsion classes in higher homological algebra

Abstract

Let $d \geqslant 1$ be an integer, $\mathscr{M}$ a suitable $d$-abelian category. There is a notion of higher torsion classes in $\mathscr{M}$, also known as $d$-torsion classes. We generalise a classic theorem of Hoshino by providing criteria for a $d$-torsion class $\mathscr{U}$ to be splitting, notably that $\mathscr{U}$ is stable under $τ_d^-$, the inverse $d$-Auslander--Reiten translation. We apply the criteria to show that under additional assumptions, which are satisfied in Dynkin type $A$, the elements of the spine of the lattice of $d$-torsion classes are precisely the splitting $d$-torsion classes. The spine consists of the elements which belong to a chain of maximal length.