Abstract:

Thick subcategories of a triangulated category form a lattice. These lattices have been actively studied for about 40 years in fields ranging from algebraic topology and algebraic geometry to representation theory.  In many cases, it seems hard to give explicit and complete descriptions of these lattices. Instead of attempting to understand the whole structure, we look at a rather coarse invariant of these lattices and hence of the corresponding triangulated category. Namely, we study the lengths of maximal chains in these lattices. We collect all integers that arise as lengths of maximal chains for a fixed triangulated category T into a set called the "lengths spectrum of T". We will illustrate this concept with examples from algebraic geometry and representation theory and also discuss open questions.