On the representation dimension and finitistic dimension of special multiserial algebras

TL;DR

This paper proves that all special multiserial algebras have a representation dimension at most three, confirming the finitistic dimension conjecture for this class by constructing radical embeddings into finite type algebras.

Contribution

It establishes a bound on the representation dimension of monomial and self-injective special multiserial algebras and confirms the finitistic dimension conjecture for all special multiserial algebras.

Findings

Representation dimension ≤ 3 for monomial and self-injective special multiserial algebras

Construction of radical embeddings into finite representation type algebras

Finitistic dimension conjecture holds for all special multiserial algebras

Abstract

For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less or equal to three. This implies that the finitistic dimension conjecture holds for all special multiserial algebras.

Equations

$$\operatorname{repdim}\nolimits(A)={\rm inf}\{\operatorname{gldim}\nolimits(\operatorname{End}\nolimits_{A}(M)^{op})\mid M\mbox{ is a generator-cogenerator of $A$ }\}.$$

$${\rm findim}(A)={\rm sup}\{{\rm pd}(M)\mbox{ for all $M$ such that }{\rm pd}(M)<\infty\}.$$

$$Q^{Sp}=(Q_0^{Sp},Q_1^{Sp},s^{Sp},e^{Sp})$$

$$Q_0^{Sp}=\{v_1,v_2\}\cup Q_0\setminus \{v\}$$

$$Q_1^{Sp}=Q_1.$$

$$s^{Sp}(a)=\{\begin{array}{ll}{v}_i& \text{mbox}ifa\in S_i,i=1,2,\\ s(a)& \text{mbox}otherwise.\end{array}$$

$$e^{Sp}(a)=\{\begin{array}{ll}{v}_i& \text{mbox}ifa\in E_i,i=1,2,\\ e(a)& \text{mbox}otherwise.\end{array}$$

$${\rho }^{Sp}=\rho \setminus (\{ab|a\in E_i\text{mbox}andb\in S_j,\text{mbox}fori\ne j\}$$

$$\begin{array}[]{lll}S_{1}&=&\{b_{1}\},\\ S_{2}&=&\{b_{2},\ldots,b_{n}\},\\ E_{2}&=&\{a\in E(v)|ab_{1}\in I\},\\ E_{1}&=&E(v)\setminus E_{2}.\end{array}$$

Full text

Part of this work took place during a visit of the author to the University of São Paulo: the author would like to thank Eduardo Marcos for his hospitality. This work was supported through the EPSRC fellowship grant EP/P016294/1.

Introduction

Many of the important open conjectures in representation theory of Artin algebras are of a homological nature, such as the finitistic dimension conjecture, Nunke’s condition and Nakayma’s conjectures. Amongst these conjectures there is a logical hierarchy, in that if the finitistic dimension conjecture holds then Nunke’s condition holds which in turn implies the Nakayama conjectures; for an overview, see, for example [8, 11, 14].

The finitistic dimension conjecture states that for any Artin algebra $A$, the supremum of the projective dimensions of the finitely generated right $A$-modules of finite projective dimension is finite. This conjecture was originally posed as a question by Rosenberg and Zelinsky and then published by Bass in 1960 [1].

Although the finitistic dimension conjecture is open in general, there has been much related work in recent years reducing the problem to simpler classes of algebras [12, 13]. There are many classes of algebras where the conjecture has been shown to hold [2, 9]. For classes of algebras of mostly wild representation type, the two most prominent examples where the finitistic dimension conjecture is known to hold are the monomial algebras [3, 10] and the radical cubed zero algebras [7].

In this paper, we will show that the finitistic dimension conjecture holds for special multiserial algebras, a large class of mostly wild algebras, containing many other important and well-studied classes of algebras such as, for example, special biserial algebras, symmetric radical cubed zero algebras and almost gentle algebras [4, 5].

It is well known that most finite dimensional algebras are of wild representation type implying that their representation theory is at least as complicated as the representation theory of the free associative algebra in two variables. Special multiserial algebras form a class of mostly wild finite dimensional algebras. It was shown in [4] that the radical of their indecomposable modules is a sum of uniserial modules whose pairwise intersection is either a simple module or zero. This is an indication that uniserial modules play an important role in the study of their representation theory. In this paper, we show that for a monomial special multiserial algebra $A$ of infinite representation type, the direct sum of all uniserial submodules of $A$ gives rise to an Auslander generator of $A$.

In order to show this, we construct radical embeddings from monomial special multiserial algebras to a direct product of representation finite string algebras whose quivers are linearly oriented Dynkin diagrams of type $\mathbb{A}$ and cyclically oriented Dynkin diagrams of type $\mathbb{\tilde{A}}$. Therefore by [2] we obtain that the representation dimension of a monomial special multiserial algebra is less or equal to three.

We further show that for any special multiserial algebra $A$, a relation is either monomial or is a linear combination of elements in the socle of $A$. We then apply the results in [2] in combination with our results on monomial special multiserial algebras, to show that the representation dimension of self-injective special multiserial algebras is less or equal to three.

To summarise, in this paper we show the following:

Theorem 1**.**

Let $A$ be a monomial special multiserial algebra. Then there exists a radical embedding $f:A\to B$ where $B$ is an algebra of finite representation type.

Corollary 2**.**

Let $A$ be a monomial special multiserial algebra. Then $repdim(A)\leq 3$.

In [6] Brauer configuration algebras are defined as generalisations of Brauer graph algebras. Brauer configuration algebras are symmetric algebras, so in particular they are self-injective and it follows from the next result that their representation dimension is less or equal to 3.

Corollary 3**.**

Let $A$ be a self-injective special multiserial algebras. Then $repdim(A)\leq 3$. In particular, the representation dimension of a Brauer configuration algebra is less or equal to $3$.

Corollary 4**.**

Let $A$ be a special multiserial algebra. Then the finitistic dimension of $A$ is finite.

Acknowledgements: I would like to thank Ed Green for the many wonderful hours we have spent talking mathematics. In particular, I would like to thank him, Karin Erdmann and Jan Geuenich for the discussions involving the results of this paper. I also would like to thank Julian Külshammer for alerting me to a mistake in an earlier version of the paper.

1. Background

Let $K$ be an algebraically closed field. A quiver $Q=(Q_{0},Q_{1},s,e)$ consists of a finite set of vertices $Q_{0}$, a finite set of arrows $Q_{1}$ and maps $s,e:Q_{1}\to Q_{0}$ where, for $a\in Q_{1}$, $s(a)$ denotes the vertex at which $a$ starts and $e(a)$ denotes the vertex at which $a$ ends. For $a,b\in Q_{1}$, such that $e(a)=s(b)$, we write $ab$ for the element in $KQ$ given by the concatenation of $a$ and $b$. For $v\in Q_{0}$, denote by $e_{v}$ the associated idempotent. We call an element $x\in KQ$ uniform if there exists $v,w\in Q_{0}$ such that $e_{v}xe_{w}=x$. All modules considered are finitely generated right modules and for $A$ a finite dimensional $K$-algebra, we denote by $\mod A$ the category of finitely generated right $A$-modules. Furthermore, set $D(A)=\operatorname{Hom}\nolimits_{K}(A,K)$ and denote by $J_{A}$ the Jacobson radical of $A$. We call a finite dimensional $K$-algebra basic, if $A=KQ/I$ for $I$ an admissible ideal in $KQ$.

From now on, whenever we write $A=KQ/I$, we assume that $I$ is admissible.

Recall that $\operatorname{gldim}\nolimits(A)={\rm sup}\{{\rm pd}(M)\mid M\in\mod A\}$ and that $M$ is a generator-cogenerator of $A$ if $A\oplus D(A)\in{\rm add}\;M$ where ${\rm add}\;M$ is the subcategory of $\mod A$ generated by direct sums of direct summands of $M$. Then

[TABLE]

Moreover, $M$ is an Auslander generator of $A$ if $\operatorname{gldim}\nolimits(End_{A}(M)^{op})=\operatorname{repdim}\nolimits(A)$. The finitistic dimension of $A$ is given by

[TABLE]

Let $A=KQ/I$, we say that condition (S) holds for $A$ if the following holds:

(S) For all $a\in Q_{1}$ there exists at most one arrow $b\in Q_{1}$ such that $ab\notin I$ and there exists at most one arrow $c\in Q_{1}$ such that $ca\notin Q_{1}$.

Definition 5**.**

A finite dimensional algebra $A$ is special multiserial if it is Morita equivalent to an algebra $KQ/I$ such that (S) holds. **

We recall the following results and definitions from [2]. For $v\in Q_{0}$, set $S(v)$ to be the subset of $Q_{1}$ consisting of arrows starting at $v$ and set $E(v)$ to be the set of arrows of $Q_{1}$ ending at $v$. Note that if there is a loop $a$ at $v$ then $a\in E(v)\cap S(v)\neq\emptyset$.

Suppose $S(v)=S_{1}\sqcup S_{2}$ and $E(v)=E_{1}\sqcup E_{2}$ are disjoint unions. The collection $Sp=(S_{1},S_{2},E_{1},E_{2})$ is a splitting datum at $v$ (for $I$) if

1. (1)

$ab\in I$, for all $a\in E_{i}$ and $b\in S_{j}$ with $i\neq j$, 2. (2)

$I=\langle\rho\rangle$ where $\rho$ is a set of relations of the form $\sum\lambda apb$ such that none of the $a$ are in $E_{1}$ or none of the $a$ are in $E_{2}$ and such that none of the $b$ are in $S_{1}$ or none of the $b$ are in $S_{2}$.

An algebra $KQ/I$ is called monomial if $I$ is monomial, that is if $I$ is generated by paths. Remark that condition (2) always holds if $I$ is monomial.

Let $Sp=(S_{1},S_{2},E_{1},E_{2})$ be a splitting datum at $v$. Then we define a new quiver

[TABLE]

by setting

[TABLE]

and

[TABLE]

The map $s^{Sp}:Q_{1}^{Sp}\to Q_{0}^{Sp}$ is given by

[TABLE]

The map $e^{Sp}:Q_{1}^{Sp}\to Q_{0}^{Sp}$ is given by

[TABLE]

We define $A^{Sp}=KQ^{Sp}/I^{Sp}$ for $I^{Sp}=\langle\rho^{Sp}\rangle$ where

[TABLE]

A radical embedding $f:A\to B$ is an algebra monomorphism such that $f(J_{A})=J_{B}$. It is shown in [2] that a splitting datum gives rise to a radical embedding.

Proposition 6**.**

[2]** Let $A=KQ/I$ with $I$ admissible. Let $Sp=(E_{1},E_{2},S_{1},S_{2})$ be a splitting datum at some vertex $v$ of $Q$. Then there exists a radical embedding $f:A\to A^{Sp}$.

Also recall the following results from [2].

Theorem 7**.**

[2]** Let $A$ and $B$ be basic algebras.

1. (1)

If $f:A\to B$ is a radical embedding with $B$ a representation finite algebra then $repdim(A)\leq 3$. 2. (2)

Let $P$ be an indecomposable projective-injective $A$-module and set $A/soc(P)$. Then $repdim(A)\leq 3$ if $repdim(A/soc(P))\leq 3$.

2. Some results on special multiserial algebras

In the following proposition we show that the relations in a special multiserial algebra are of a particular form.

Proposition 8**.**

Let $A=KQ/I$ be a special multiserial algebra with $I=\langle\rho\rangle$ satisfying condition (S). Let $r=\sum\lambda_{p}p\in\rho$ be uniform with $\lambda_{p}\in K$ such that almost all $\lambda_{p}=0$ and where each $p$ is a path in $Q$. Then either $r$ is a path or for all $\lambda_{p}\neq 0$, $p$ is in the socle of $A$ as a right and left $A$-module.

Proof.

We will start by showing that the result holds when considering the socle of $A$ as a right $A$-module. Suppose there exists a unique $\alpha_{p}\neq 0$, then $r=p$.

Suppose that $r=\lambda_{p}p-\lambda_{q}q$ with $p,q\notin I$. Then without loss of generality we can assume that $\lambda_{p}=1$.

Now suppose that $p\notin\operatorname{soc}\nolimits A$ and that $q\in\operatorname{soc}\nolimits A$. Then there exists $a\in Q_{1}$ such that $pa\notin I$ but since $q\in\operatorname{soc}\nolimits A$, $qa\in I$ and this a contradiction. Suppose now that $p,q\notin\operatorname{soc}\nolimits A$. Then there exist $a,b\in Q_{1}$ such that $pa,qb\notin I$. Since $p-\lambda_{q}q\in I$, by condition (S) we have $a=b$. Therefore if $p=p^{\prime}c$ and $q=q^{\prime}d$ for $c,d\in Q_{1}$ then $ca,da\notin I$. This implies by condition (S) that $c=d$ and hence $(p^{\prime}-\lambda_{q}q^{\prime})c\in I$. Moreover, $p^{\prime}c,q^{\prime}c\notin I$. Now let $p^{\prime}=p^{\prime\prime}c^{\prime}$ and $q^{\prime}=q^{\prime\prime}d^{\prime}$ which implies that $c^{\prime}c,d^{\prime}c\notin I$ and $c^{\prime}=d^{\prime}$. Continuing in this way, we see that $p=q$.

Suppose now that $r=\sum\lambda_{p}p$ and suppose that $\lambda_{q}\neq 0$ and $q\notin\operatorname{soc}\nolimits A$. Then there exists $a\in Q_{1}$ such that $qa\notin I$ and therefore there exists $q^{\prime}$ with $\lambda_{q^{\prime}}\neq 0$ and $q^{\prime}a\notin I$. Since $A$ is special, this implies $q=q^{\prime}$. Inductively it then follows that $pa\notin I$ for any $p$ such that $\lambda_{p}\neq 0$ and using that $A$ is special it follows that $r=q$.

Note that we have only used specialness on the right side. Using specialness on the left side, we obtain the result for the socle of $A$ as a left $A$-module. ∎

The following follows directly from condition (S).

Lemma 9**.**

Let $A=KQ/I$ be monomial special multiserial and let $Sp=(S_{1},S_{2},E_{1},E_{2})$ be a splitting datum at some vertex $v$ in $Q$.

1. (1)

Suppose that $S_{1}=\{b\}$, for $b\in Q_{1}$. Then $E_{1}$ consists of the unique arrow $a$ such that $ab\notin I$ if such an arrow $a$ exists, otherwise $E_{1}$ is empty. 2. (2)

Suppose that $E_{1}=\{c\}$, for $c\in Q_{1}$ then $S_{1}=\{d\}$ where $d$ is the unique arrow such that $cd\notin I$ if such an arrow $d$ exits and $S_{1}$ is empty otherwise.

Moreover, for $Sp$ as in (1) or (2) above, $A^{Sp}$ is monomial special multiserial.

3. Proof of Theorem 1

We show that for any monomial special multiserial algebra $A=KQ/I$ there is a radical embedding of $A$ into a disjoint union of representation finite string algebras whose underlying quiver is either a linearly oriented quiver of type ${\mathbb{A}}$ and or a cyclically oriented quiver of type $\widetilde{\mathbb{A}}$.

Proof of Theorem 1 and Corollary 2: Let $A=KQ/I$ be a monomial special multiserial algebra such that $I$ is generated by paths. Define $c(A)=|\{v\in Q_{0}|S(v)>1\}|+|\{v\in Q_{0}|E(v)>1\}|$.

If $c(A)=0$ then $Q$ is a disjoint union of quivers where each quiver is either a linearly oriented quiver of type ${\mathbb{A}}$ or a cyclically oriented quiver of type $\widetilde{\mathbb{A}}$. So $A$ is a product of representation finite string algebras, and it therefore is of finite representation type.

Suppose that $c(A)\geq 1$. Let $v\in Q_{0}$ such that $|S(v)|>1$ or $|E(v)|>1$. Suppose that $S(v)=\{b_{1},\ldots,b_{n}\}$ with $n\in\mathbb{N},n>1$. Set

[TABLE]

Note that $E_{1}$ consists of the unique arrow $a\in Q_{1}$ such that $ab_{1}\notin I$ if such an arrow exists. That $Sp=(S_{1},S_{2},E_{1},E_{2})$ is a splitting datum at $v$ follows directly (S) and from the fact that $I$ is monomial.

By Lemma 9, $A^{Sp}$ is again a monomial special multiserial algebra and $c(A^{Sp})\leq c(A)-1$.

We treat the case $|E(v)|>1$ in a similar way.

Repeating this a finite number of times and setting $A=A_{1}$ and $A_{2}=A^{Sp}$, we obtain by Proposition 6 a sequence of radical embeddings $A_{1}\to A_{2}\to\cdots A_{k}=B$ such that $B$ is a string algebra with $c(B)=0$ and $B$ is therefore representation finite. Then it follows from Theorem 7 (1) that $repdim(A)\leq 3$. $\Box$

Let $f:A\to B$ be the radical embedding constructed in the proof of Theorem 1 above. By the proof of Theorem 1.1 in [2] an Auslander generator of $A$ is given by $A\oplus D(A)\oplus N$ where $N$ is the direct sum of isomorphism class representatives of the indecomposable $B$-modules considered as $A$-modules. Therefore in the case of a monomial special multiserial algebra $N$ is given by the direct sum of all uniserial submodules of $A$. $\Box$

Proof of Corollary 3: Since $A$ is self-injective, every projective is injective. Applying Proposition 8 we obtain that iteratively factoring out the socles of the projective injective indecomposable modules gives rise to a monomial special multiserial algebra. Thus the result follows from Theorem 1 and by the successive application of Theorem 7 (2). $\Box$

Proof of Corollary 4: It follows from Proposition 8 that $A/\operatorname{soc}\nolimits(A)$ is a monomial special multiserial algebra and the result follows from [13, 4.3] and Theorem 1. $\Box$