---
tag: G8BQ5THKRZ
kind: concept
name: Exceptional collection
msc: [14F08, 18G80]
sources:
  - url: https://en.wikipedia.org/wiki/Semiorthogonal_decomposition
---

## Summary

An exceptional collection in a linear [[triangulated category|T2V7GJJ2EN]] is a finite ordered list of exceptional objects.
Each of them has only scalar endomorphisms and no self-extensions.
There are no morphisms, in any degree, from later objects to earlier ones.
A full exceptional collection splits the category into a semiorthogonal decomposition whose pieces are copies of the derived category of vector spaces.
The basic example is Beilinson's collection $\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n)$ on projective space.

## Explanations

### L1 human

An exceptional object behaves like a single point: its only endomorphisms are scalars and it has no deformations or extensions with itself.
An exceptional collection is a list of such objects in which maps only go forward along the list.
When the collection generates everything, the whole category is built from these simple pieces in a triangular way, much like a matrix that is upper triangular.
For projective space, the line bundles $\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n)$ form such a list, which makes its derived category very concrete.