# All terms in a complete exceptional sequence are relatively projective or
relatively injective

Shujian Chen  Department of Mathematics, Brandeis University, Waltham, MA
02454<EMAIL_ADDRESS>and  Kiyoshi Igusa  Department of Mathematics,
Brandeis University, Waltham, MA 02454<EMAIL_ADDRESS>
###### Abstract.

We prove the statement in the title, define the terms and give one
application.

## Introduction

Exceptional sequences have become an active area of research due to several
new uniform proofs for the enumeration formulas for exceptional sequences for
Dynkin quivers. See [12], [3], [5]. The question of which objects in an
exceptional sequence are relatively projective has become important after the
result of [10] where a bijection is given between signed exceptional sequences
(where relatively projective objects are allowed to be “negative”) and ordered
clusters. This has been generalized to the $\tau$-tilting case in [2]. See
also [1] and [4, Theorem 1.19] for a combinatorial version of this.

In [8] it was shown that, for complete exceptional sequences over $A_{n}$ with
straight orientation, every object is either relatively projective or
relatively injective. Theo Douvropoulos explained to us that this statement
should be true more generally and challenged us to prove it. This paper is the
result.

In the finite type case (when $\Lambda=KQ$ for a Dynkin quiver $Q$), this
result follows from [1] and [4] using the correspondence between exceptional
sequences and minimal factorizations of the Coxeter element of the Weyl group
given in [11]. In [1], it is shown that for a covering relation $v<w$ in the
absolute order, either $v^{-1}w$ or $K(w)^{-1}K(v)$ is a one-way reflection,
i.e., agrees with the Bruhat order, where $K(v)=v^{-1}c$, and in [4] it is
shown that one-way reflections in a minimal factorization correspond to
relative projective representations in the complete exceptional sequence.

Throughout this paper, $\Lambda$ will be an hereditary algebra which is finite
dimensional over some field. We will prove the following theorem. The terms
are defined below.

######  Theorem A.

Let $(E_{1},\cdots,E_{k},\cdots,E_{n})$ be a complete exceptional sequence
over $\Lambda$. Then every object $E_{k}$ is either relatively projective or
relatively injective (or both).

By [7, Theorem 4.3], this theorem also holds for complete exceptional
collections which are unordered exceptional sequences since relative
projectiveness and relative injectiveness are independent of the order of the
terms $E_{k}$.

######  Corollary B.

For a random complete exceptional sequence of type $A_{n}$, $B_{n}$, or
$C_{n}$, the probability that the $k$th term $E_{k}$ is both relatively
projective and relatively injective is $\frac{2}{h}$ where $h$ is the Coxeter
number ($h=n+1$ for $A_{n}$ and $h=2n$ in $B_{n},C_{n}$).

###### Proof.

It follows from Theorem A that this probability is equal to the probability
that $E_{k}$ is relatively projective plus the probability that it is
relatively injective minus 1. These probabilities are computed in [6, Theorem
4.2] for type $A_{n}$ and in [9, Corollary 5.5] in types $B_{n},C_{n}$. ∎

##  1\. Exceptional sequences and relative projectivity/injectivity

Recall that a module $E$ is called _exceptional_ is if it is rigid, i.e.,
$\operatorname{Ext}(E,E)=0$ and its endomorphism ring is a division algebra.
In particular $E$ is indecomposable. An _exceptional sequence_ over a
hereditary algebra $\Lambda$ is a sequence of exceptional modules
$(E_{1},\cdots,E_{n})$ so that, for all $i<j$, $E_{i}\in E_{j}^{\perp}$ by
which we mean

$\operatorname{Hom}(E_{j},E_{i})=0=\operatorname{Ext}(E_{j},E_{i}).$

The sequence is called _complete_ if it is maximal, i.e., of length $n$ where

######  Lemma 1.1.

Let $A\to B\to C$ be an almost split sequence over a hereditary algebra so
that $C$ is exceptional. Then $B\in C^{\perp}$, i.e.,

$\operatorname{Hom}(C,B)=0=\operatorname{Ext}(C,B).$

###### Proof.

This follows from the 6-term exact sequence

$\operatorname{Hom}(C,A)\to\operatorname{Hom}(C,B)\to\operatorname{Hom}(C,C)\to\operatorname{Ext}(C,A)\to\operatorname{Ext}(C,B)\to\operatorname{Ext}(C,C)$

Since $C$ is rigid and $A=\tau C$, $\operatorname{Hom}(C,\tau
C)=0=\operatorname{Ext}(C,C)$. Also,
$\operatorname{Hom}(C,C)\to\operatorname{Ext}(C,A)$ is an isomorphism since it
is a nonzero map between one-dimensional vector spaces over the division
algebra $\operatorname{End}(C)$. Therefore, the remaining two terms are zero:
$\operatorname{Hom}(C,B)=0=\operatorname{Ext}(C,B)$. ∎

For any complete exceptional sequence $(E_{1},\cdots,E_{k},\cdots,E_{n})$ for
a hereditary algebra $\Lambda$, we use the notation

${\mathcal{A}}_{k}=(E_{k+1}\oplus\cdots\oplus E_{n})^{\perp}$

to denote the wide subcategory of all $\Lambda$-modules $X$ so that

$\operatorname{Hom}(E_{j},X)=0=\operatorname{Ext}(E_{j},X)$

for all $j>k$. A _wide subcategory_ of $mod\text{-}\Lambda$ is an exactly
embedded abelian subcategory which is closed under extensions [11]. An object
$E_{k}$ in the exceptional sequence is called _relatively projective_ if
$E_{k}$ is a projective object of the wide subcategory ${\mathcal{A}}_{k}$.

######  Lemma 1.2.

$E_{k}$ is relatively projective if and only if there is no epimorphism
$X\twoheadrightarrow E_{k}$ where $X$ is an object of ${\mathcal{A}}_{k-1}$.

###### Proof.

($\Rightarrow$) If $E_{k}$ is projective in ${\mathcal{A}}_{k}$, then any such
epimorphism would split since ${\mathcal{A}}_{k-1}\subset{\mathcal{A}}_{k}$.
But $E_{k}\notin{\mathcal{A}}_{k-1}$, so no such $X$ exists.

($\Leftarrow$) If $E_{k}$ is not projective in ${\mathcal{A}}_{k}$ then there
is an almost split sequence

$A\to X\to E_{k}$

in ${\mathcal{A}}_{k}$ where $A\neq 0$. Lemma 1.1 implies that $X\in
E_{k}^{\perp}$. Thus, $X\in{\mathcal{A}}_{k}\cap
E_{k}^{\perp}={\mathcal{A}}_{k-1}$. ∎

An object $E_{k}$ in the complete exceptional sequence $(E_{1},\cdots,E_{n})$
is _relatively injective_ if it is an injective object of the left
perpendicular category

${\mathcal{B}}_{k}=\,^{\perp}(E_{1}\oplus\cdots\oplus E_{k-1})$

It is well-known that ${\mathcal{B}}_{k}$ is dual to ${\mathcal{A}}_{k-1}$ in
the sense that ${\mathcal{A}}_{k-1}={\mathcal{B}}_{k}^{\perp}$ and
${\mathcal{B}}_{k}=\,^{\perp}{\mathcal{A}}_{k-1}$. Thus ${\mathcal{B}}_{k}$ is
the wide subcategory of $mod\text{-}\Lambda$ generated by
$E_{k},\cdots,E_{n}$.

######  Lemma 1.3.

$E_{k}$ is relatively injective if and only if there is no monomorphism
$E_{k}\rightarrowtail Y$ where $Y\in{\mathcal{B}}_{k+1}$.

###### Proof.

Similar to the proof of Lemma 1.2. ∎

##  2\. Proof of Theorem A

Suppose that $E_{k}$ is neither relatively projective nor relatively
injective. Then, by Lemmas 1.2 and 1.3, we have short exact sequences:

$A\to X\to E_{k}$ $E_{k}\to Y\to B$

where $X\in{\mathcal{A}}_{k-1}={\mathcal{B}}_{k}^{\perp}$ and
$Y\in{\mathcal{B}}_{k+1}\subset{\mathcal{B}}_{k}$. This implies in particular
that $\operatorname{Ext}(Y,X)=0$. However, the following diagram will show
that this is not true.

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{E_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$

Since $\operatorname{Ext}$ is right exact, we have an epimorphism

$\operatorname{Ext}(B,X)\twoheadrightarrow\operatorname{Ext}(B,E_{k})$

So, there is a short exact sequence $X\to Z\to B$ so that $E_{k}\to Y\to B$ is
the pushout of this sequence along $X\twoheadrightarrow E_{k}$. This gives a
short exact sequence:

$X\to E_{k}\oplus Z\to Y.$

But this sequence does not split since $E_{k}$ is not a direct summand of
$X\oplus Y$ since $X\in E_{k}^{\perp}$ and $Y\in\,^{\perp}E_{k}$. So,
$\operatorname{Ext}(Y,X)\neq 0$. This contradiction proves the theorem.

## Acknowledgments

The authors thank Theo Douvropoulos for many discussion about this problem.
The second author thanks the Simons Foundation for its support: Grant #686616.

## References

  * [1] Biane, Philippe, and Matthieu Josuat-Vergès. _Noncrossing partitions, Bruhat order and the cluster complex._ Annales de l’Institut Fourier. Vol. 69. No. 5. 2019.
  * [2] Buan, Aslak Bakke, and Bethany Rose Marsh. _$\tau$ -exceptional sequences._ Journal of Algebra 585 (2021): 36–68.
  * [3] Chapuy, Guillaume, and Theo Douvropoulos. _Counting chains in the noncrossing partition lattice via the W-Laplacian._ Journal of Algebra 602 (2022): 381–404.
  * [4] Chen, Shujian and Kiyoshi Igusa, _Generalized Goulden-Yong duals and signed minimal factorizations._ arXiv:2311.05732.
  * [5] Douvropoulos, Theo. _On enumerating factorizations in reflection groups._ Algebr. Comb. 6.2 (2023), pp. 359–385. issn: 2589–5486.
  * [6] Igusa, Kiyoshi. _Probability distribution for exceptional sequences of type $A_{n}$._ arXiv:2112.04996.
  * [7] Igusa, Kiyoshi and Ray Maresca. _On Clusters and Exceptional Sets in Types $\mathbb{A}$ and $\tilde{\mathbb{A}}$._ arXiv:2209.12326.
  * [8] Igusa, Kiyoshi, and Emre Sen. _Exceptional sequences and rooted labeled forests._ arXiv:2108.11351.
  * [9] Igusa, Kiyoshi, and Emre Sen. _Rooted labeled trees and exceptional sequences of type $B_{n}/C_{n}$._ arXiv:2310.01700.
  * [10] Igusa, Kiyoshi and Gordana Todorov. Signed exceptional sequences and the cluster morphism category. arXiv: 1706.2222.
  * [11] Ingalls, Colin, and Hugh Thomas. _Noncrossing partitions and representations of quivers._ Compositio Mathematica 145.6 (2009): 1533–1562.
  * [12] Michel, Jean. _Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element._ Proc of Amer Math Soc 144.3 (2016): 937–941.