Sparseness of t-structures and negative Calabi-Yau dimension in triangulated categories generated by a spherical object

TL;DR

This paper investigates the structure of triangulated categories generated by spherical objects, revealing how their t-structures and Calabi-Yau properties vary with the parameter w.

Contribution

It characterizes the existence of t-structures and co-t-structures in categories generated by w-spherical objects and explores their potential negative Calabi-Yau dimension.

Findings

For w ≤ 0, no non-trivial t-structures exist.

For w ≥ 1, t-structures are present while co-t-structures are absent.

For w ≤ -1, the category may have negative Calabi-Yau dimension.

Abstract

Let k be an algebraically closed field and let T be the k-linear algebraic triangulated category generated by a w-spherical object for an integer w. For certain values of w this category is classical. For instance, if w = 0 then it is the compact derived category of the dual numbers over k. As main results of the paper we show that for w \leq 0, the category T has no non-trivial t-structures, but does have one family of non-trivial co-t-structures, whereas for w \geq 1 the opposite statement holds. Moreover, without any claim to originality, we observe that for w \leq -1, the category T is a candidate to have negative Calabi-Yau dimension since \Sigma^w is the unique power of the suspension functor which is a Serre functor.