Exceptional collections in surface-like categories

Abstract

Abstract. We provide a categorical framework for recent results of Markus Perling on combinatorics of exceptional collections on numerically rational surfaces. Using it we simplify and generalize some of Perling's results as well as Vial's criterion for existence of a numerical exceptional collection.

Homological mirror symmetry of $\mathbb{F}_1$ via Morse homotopy

“…Following Hille-Perling [12], Elagin-Lunts [5], Kuznesov [15], etc, the triangulated category D b (coh(F 1 )) ≃ Tr(DG(F 1 )) has a series of full strongly exceptional collections 1 We consider the Morse cohomology degree instead of the Morse homology degree.…”

Construction of non-commutative surfaces with exceptional collections of length 4

“…The reason for considering these properties is explained in : it can be shown that the action of the Serre functor on the Grothendieck group for all smooth projective surfaces has the extra property that has rank precisely 2, whilst the unipotency of the Serre functor holds in complete generality [, Lemma 3.1]. Independently, Kuznetsov developed a similar notion in .…”

“…Those properties are described as follows: for a finitely generated free abelian group Λ, a non-degenerate bilinear form −, − : Λ × Λ → Z and an automorphism s ∈ Aut(Λ) we will ask that the extra property that s − id Λ has rank precisely 2, whilst the unipotency of the Serre functor holds in complete generality [14,Lemma 3.1]. Independently, Kuznetsov developed a similar notion in [28].…”

“…In [13] the Serre functor in the case of a surface is studied, and some extra properties are determined: for a finitely generated free abelian group Λ, a nondegenerate bilinear form −, − : Λ × Λ → Z and an automorphism s ∈ Aut(Λ) we will ask that Serre automorphism x, s(y) = y, x for x, y ∈ Λ; unipotency s − id Λ is nilpotent; rank rk(s − id Λ ) = 2; Indeed, it can be shown that the action of the Serre functor on the Grothendieck group for all smooth projective surfaces (besides being unipotent) has the extra property that the rank of s − id Λ is precisely 2. Independently, in [19] a closely related notion is introduced.…”

Integral Chow motives of threefolds with $K$-motives of unit type

“…When X is a surface, this was proved previously by Vial in [41, Theor. 2.7] by a different method based on delicate properties of exceptional collections on surfaces obtained in recent papers by Perling [34] and Kuznetsov [25].…”