---
tag: W3B4XH5JHZ
kind: definition
name: Exceptional collection
msc: [14F08, 18G80]
defines: G8BQ5THKRZ
uses: [04C5ZBG1WT, T2V7GJJ2EN]
sources:
  - url: https://en.wikipedia.org/wiki/Semiorthogonal_decomposition
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $\mathcal{T}$ be a $k$-linear [[triangulated category|T2V7GJJ2EN]] with shift $X \mapsto X[1]$.
An object $E$ of $\mathcal{T}$ is *exceptional* if $\Hom(E, E) = k$ and $\Hom(E, E[l]) = 0$ for every integer $l \ne 0$.
An *exceptional collection* is a sequence $(E_1, \dots, E_n)$ of objects of $\mathcal{T}$ such that:

1. each $E_i$ is exceptional;
2. $\Hom(E_j, E_i[l]) = 0$ for all $j > i$ and all integers $l$.

The collection is *full* if the smallest triangulated subcategory of $\mathcal{T}$ that contains $E_1, \dots, E_n$ and is closed under isomorphisms is $\mathcal{T}$ itself.
It is *strong* if moreover $\Hom(E_i, E_j[l]) = 0$ for all $i, j$ and all $l \ne 0$.

## Remarks

Example: on the projective space $\mathbb{P}^n$ over $k$, the line bundles $(\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n))$ form a full strong exceptional collection.
It lives in the [[bounded derived category of coherent sheaves|2VP2D51KB8]], and this is a theorem of Beilinson.
Clause 2 holds since $\Hom(\mathcal{O}(j), \mathcal{O}(i)[l]) = H^l(\mathbb{P}^n, \mathcal{O}(i - j))$, and for $-n \le i - j < 0$ all cohomology of $\mathcal{O}(i - j)$ vanishes.
Non-example: on an elliptic curve $C$ over $k$, no line bundle $L$ is exceptional, since $\Hom(L, L[1]) = H^1(C, \mathcal{O}_C) \cong k \ne 0$.
A full exceptional collection gives a semiorthogonal decomposition $\mathcal{T} = \langle \mathcal{A}_1, \dots, \mathcal{A}_n \rangle$, where $\mathcal{A}_i$ is the subcategory generated by $E_i$, equivalent to the bounded derived category of finite-dimensional vector spaces over $k$.