Spherical subcategories in representation theory

Abstract

We introduce a new invariant for triangulated categories: the poset of spherical subcategories ordered by inclusion. This yields several numerical invariants, like the cardinality and the height of the poset. We explicitly describe spherical subcategories and their poset structure for derived categories of certain finite-dimensional algebras.

Discrete derived categories I: homomorphisms, autoequivalences and t-structures

“…To any reasonable k-linear triangulated category, [23] associates a poset derived from indecomposable spherelike objects. In [23, §6], these posets are computed for discrete derived algebras.…”

Formality of -objects

“…P 1 [k]-objects are well-known as spherical objects; see [45]. Without the Calabi-Yau property they are called spherelike objects and studied in [21,22] by Kalck, Ploog and the first author. P n [2]-objects are known as P n -objects and studied in [25] by Huybrechts and Thomas.…”

Spherical subcategories in algebraic geometry

“…All triangulated categories occurring in this article and its representation‐theoretic counterpart are of this type: bounded derived categories of abelian categories with enough injectives have an enhancement by [, §3]. In fact, by and enhancements also exist for the derived category of coherent sheaves on a quasi‐projective scheme, even though there are no injectives and these enhancements are, moreover, unique.…”

“…This example shows that can contain infinitely many pairwise incomparable spherical subcategories. See [, §2] for a further study of this and related questions.…”

“…Our theory applies to many triangulated categories. For examples from the representation theory of finite‐dimensional algebras, we refer to the forthcoming article . In this article, applications to algebraic geometry are given in the final Section .…”