Noetherianity of Diagram Algebras

1. Introduction

In this paper, we prove the local Noetherian property for the linear categories of Brauer and partition algebras as introduced by Patzt in [1]. Additionally, we extend our results to two related categories of diagram algebras: the Rook-Brauer and Rook algebras. Our results are valid without any restrictions on the various parameters of these algebras.

For a commutative Noetherian ring R, the partition algebra $P_n=P_n(R,\delta )$ with parameter $\delta \in R$ is a free R-module over the basis of all partitions of the union of the sets $[-n]=\{-n,\dots ,-1\}$ and $\left[n\right]=\{1,\dots ,n\}$. Each basis element of $P_n$ may be visualized by a diagram $\alpha$ with two vertical columns of n nodes each, with the nodes on the left representing $-n$ through $-1$ and the nodes on the right representing 1 through n, with paths connecting certain nodes. The connected components of $\alpha$ indicate the blocks of the corresponding partition; we identify any diagrams whose components contain the same nodes.

Figure 1. Visualization of the partition $\left\{\right\{-5,5,3\},\{-4,-2,-1,4\},\{-3\},\{1,2\left\}\right\}$.

Figure 1. Visualization of the partition $\left\{\right\{-5,5,3\},\{-4,-2,-1,4\},\{-3\},\{1,2\left\}\right\}$.

For partitions p and q, we define the product $pq$ in the following way. First we conjoin the diagrams p and q by identifying each node k on the right of p with the node $-k$ on the left of q, thus forming a diagram with 3 columns of n nodes each. We then form a new partition s of $[-n]\cup \left[n\right]$ in which two elements belong to the same block if and only if the corresponding nodes in the left or right column of the conjoined diagram are connected. We define $pq={\delta }^r·s$ where r is the number of connected components in the conjoined diagram that only contain nodes in the middle column.

Figure 2. Multiplication in $P_5(R,\delta )$.

Figure 2. Multiplication in $P_5(R,\delta )$.

As an example, below we have a multiplication of two basis diagrams in $P_5(R,\delta )$. Note that there is a factor of $\delta$ in the result to indicate a single connected component (shown in red) that only contains nodes in the middle column.

The Brauer algebra ${\mathrm{Br}}_n={\mathrm{Br}}_n(R,\delta )$ is the free subalgebra of $P_n(R,\delta )$ generated by partitions that are perfect matchings; that is, the blocks in each partition have size 2. Thus, each basis element is represented by a diagram in which each node is connected to exactly one other node.

Two other diagram algebras related to the partition algebra are the Rook-Brauer algebra and the Rook algebra. For a commutative Noetherian ring R, the Rook-Brauer algebra $\mathcal{R}{\mathrm{Br}}_n=\mathcal{R}{\mathrm{Br}}_n(R,\delta ,ϵ)$ with parameters $\delta ,ϵ\in R$ is a free R-module with basis consisting of partitions of $[-n]\cup \left[n\right]$ whose blocks have size $\le 2$. Thus, each basis element is represented by a diagram in which each node is connected to at most one other node. The product of two basis diagrams $p,q\in \mathcal{R}{\mathrm{Br}}_n$ is defined by $pq={\delta }^c{ϵ}^{d}s$ where s is the partition obtained as described above, with c factors of $\delta$ for c loops and d factors of $ϵ$ for d contractible components in the conjoined diagram that only contain nodes in the middle column (see example below).

Figure 3. Multiplication in $\mathcal{R}{\mathrm{Br}}_5(\delta ,ϵ)$.

Figure 3. Multiplication in $\mathcal{R}{\mathrm{Br}}_5(\delta ,ϵ)$.

Here, the final diagram is multiplied by a factor of $\delta$ for the red loop and a factor of $ϵ$ for the isolated node in the middle.

The Rook algebra ${\mathcal{R}}_n={\mathcal{R}}_n(R,ϵ)$ is the free subalgebra of $\mathcal{R}{\mathrm{Br}}_n$ generated by diagrams without any left-to-left or right-to-right edges (i.e. the invertible diagrams of $\mathcal{R}{\mathrm{Br}}_n$). Note that the parameter $\delta$ can be removed as multiplication of diagrams with no left-to-left or right-to-right edges cannot yield any loops in the middle.

2. Main Result

We now introduce the background necessary to discuss our main results. In the following, we fix a commutative Noetherian ring R and parameters $\delta ,ϵ\in R$.

Following Patzt’s [1] convention, given a nonnegative integer $n\in \mathbb{N}$, we will denote by $A_n$ any one of $P_n$, ${\mathrm{Br}}_n$, $\mathcal{R}{\mathrm{Br}}_n$, or ${\mathcal{R}}_n$. For each $A_n$, there is a canonical choice of a “trivial” $A_n$-module R on which all invertible diagrams act trivially and which is annihilated by all other diagrams. We denote by $A_{n-m}$ the subalgebra of $A_n$ generated by diagrams with horizontal connections $\{-k,k\}$ for all $1\le k\le m$.

Denote by $\left\{A_n\right\}$ any one of the sequences $\left\{P_n\right\}$, $\left\{{\mathrm{Br}}_n\right\}$, $\left\{\mathcal{R}{\mathrm{Br}}_n\right\}$, or $\left\{{\mathcal{R}}_n\right\}$. We define a category ${\mathcal{C}}_A$ enriched over the category of R-modules by the following data:

• The objects of ${\mathcal{C}}_A$ are the nonnegative integers $\mathbb{N}$.
• If $m\le n$, we define ${Hom}_{{\mathcal{C}}_A}(m,n):=A_n{\otimes }_{A_{n-m}}R$; otherwise ${Hom}_{{\mathcal{C}}_A}(m,n):=0$ is the trivial R-module. Note that ${End}_{{\mathcal{C}}_A}\left(n\right)\cong A_n$.
• By [1]*Thm. 2.1, there is a surjection

  $${Hom}_{{\mathcal{C}}_A}(m,n)\otimes {Hom}_{{\mathcal{C}}_A}(l,m)\to {Hom}_{{\mathcal{C}}_A}(l,n)(l\le m\le n)$$

  which provides the composition $l\to m\to n$ for $l\le m\le n$; otherwise the composition is 0.

We thereby obtain categories ${\mathcal{C}}_P$, ${\mathcal{C}}_{\mathrm{Br}}$, ${\mathcal{C}}_{\mathcal{R}\mathrm{Br}}$, and ${\mathcal{C}}_{\mathcal{R}}$ enriched over the category of R-modules corresponding to the sequences $\left\{P_n\right\}$, $\left\{{\mathrm{Br}}_n\right\}$, $\left\{\mathcal{R}{\mathrm{Br}}_n\right\}$, and $\left\{{\mathcal{R}}_n\right\}$ respectively.

Definition 1. 

A${\mathcal{C}}_A$-module V is a functor from the category ${\mathcal{C}}_A$ to the category of R-modules. For each $n\in \mathbb{N}$, we denote the R-module $V\left(n\right)$ by $V_n$.

A ${\mathcal{C}}_A$-module map is a natural transformation of ${\mathcal{C}}_A$-modules. Given ${\mathcal{C}}_A$-modules V to W, if $W_n\subseteq V_n$ for each n and the map $W\to V$ induced by the inclusions $W_n\hookrightarrow V_n$ is a ${\mathcal{C}}_A$-module map, then W is a ${\mathcal{C}}_A$-submodule of V. We say that V is finitely generated if there exists a finite set $S\subseteq \bigsqcup _n{V}_n$ such that the smallest ${\mathcal{C}}_A$-submodule of V containing S is V itself.

Definition 2. 

A ${\mathcal{C}}_A$-module V is Noetherian if every ${\mathcal{C}}_A$-submodule of V is finitely generated. We say ${\mathcal{C}}_A$ is locally Noetherian over R if every finitely generated ${\mathcal{C}}_A$-module is Noetherian.

We can now state our main result.

Theorem 1. 

For $A=P$, $\mathrm{Br}$, $\mathcal{R}\mathrm{Br}$, or $\mathcal{R}$, the category ${\mathcal{C}}_A$ is locally Noetherian.

The local Noetherian property for the categories of partition, Brauer, and Temperley-Lieb algebras, with the parameter $\delta$ chosen such that each algebra is semi-simple, was previously established in [2]. Our result extends this by removing restrictions on various parameters, thereby providing a more general statement.

The crux of the main argument is to establish the existence of a functor from the category $\mathrm{FI}$ to the category ${\mathcal{C}}_A$ when $A=P$, $\mathcal{R}\mathrm{Br}$, or $\mathcal{R}$. This implies that any ${\mathcal{C}}_A$-module, when pulled back, also becomes an $\mathrm{FI}$-module. Given that the local Noetherianity property is already known for the category $\mathrm{FI}$[3]*Thm. A, this result consequently extends the local Noetherianity property to the category ${\mathcal{C}}_A$. Unfortunately, this argument does not hold for the Temperley-Lieb algebra, as there is no nontrivial functor from the category $\mathrm{FI}$ to the category of Temperley-Lieb algebras. Therefore, we leave the reader with an open question.

Question: For any parameter δ, is the category of Temperley-Lieb algebras ${\mathcal{C}}_{TL\left(\delta \right)}$ locally Noetherian?

Acknowledgement: We would like to thank our advisor, Dr. Wee Liang Gan for his useful suggestions and helpful discussion.

3. Proof of Main Result

It is known [1]*Prop. 2.3 that ${Hom}_{{\mathcal{C}}_{\mathrm{Br}}}(m,n)$ is a free R-module with a basis which may be represented by diagrams with n nodes on the left, m nodes on the right and an “$(n-m)$-blob” which is connected to exactly $n-m$ nodes, such that every node is connected to exactly one other node or the blob. Similarly, by [1]*Prop. 2.5, ${Hom}_{{\mathcal{C}}_P}(m,n)$ is a free R-module with basis given by the set of all partitions of $[-n]\cup \left[m\right]$ with $n-m$ “marked” blocks.

We now prove an analogous result for categories of Rook and Rook-Brauer algebras.

Proposition 1. 

${Hom}_{{\mathcal{C}}_{\mathcal{R}\mathrm{Br}}}(m,n)$ is a free R-module with a basis consisting of diagrams with n nodes on the left, m nodes on the right and an “$(n-m)$-blob” which is connected to exactly $n-m$ nodes, such that every node is connected to at most one other node or the blob.

Proof. 

Let $J\subset \mathcal{R}{\mathrm{Br}}_n$ be the free R-submodule generated by diagrams which have either an isolated node k with $m+1\le k\le n$ or a right-to-right connection $\{i,j\}$ with $m+1\le i,j\le n$. Note that J is closed under right multiplication by $\mathcal{R}{\mathrm{Br}}_{n-m}$.

We claim that the image of the map $J{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\to \mathcal{R}{\mathrm{Br}}_n{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R$ induced by the inclusion $J\hookrightarrow \mathcal{R}{\mathrm{Br}}_n$ is 0. To prove this, we will show that every basis diagram $\alpha \in J$ may be written $\alpha =\beta \gamma$, where $\beta$ and $\gamma$ are basis diagrams of $\mathcal{R}{\mathrm{Br}}_n$ and $\mathcal{R}{\mathrm{Br}}_{n-m}$ respectively and $\gamma$ is non-invertible. Thus we will have

$$\alpha \otimes 1\mapsto \beta \gamma \otimes 1=\beta \otimes \gamma ·1=\beta \otimes 0=0.$$

Case 1. Assume $\alpha \in J$ has an isolated node k between $m+1$ and n. Since $\alpha$ must have an even number of isolated nodes, we can find another isolated node either $k^{\prime }$ on the left or right of $\alpha$. Let $\beta \in \mathcal{R}{\mathrm{Br}}_n$ be the diagram with the connection $\{k,k^{\prime }\}$ and all other connections are the same as in $\alpha$. Let $\gamma \in \mathcal{R}{\mathrm{Br}}_{n-m}$ be the diagram with a pair of isolated nodes on opposite sides at $\left\{k\right\}$, $\{-k\}$ with all other connections left-to-right $\{-l,l\}$. Then $\gamma$ is non-invertible and $\alpha =\beta \gamma$.

Case 2. Now assume $\alpha$ has no isolated nodes between $m+1$ and n. This implies that $\alpha$ has a right-to-right connection $\{i,j\}$ where $m+1\le i,j\le n$. It follows that $\alpha$ has either a pair of isolated nodes on the left hand side or a left-to-left connection. Denote this pair or connection by $(i^{\prime },j^{\prime })$, let $\beta \in \mathcal{R}{\mathrm{Br}}_n$ be the diagram with connections $\{i,i^{\prime }\}$ and $\{j,j^{\prime }\}$ with all other connections the same as in $\alpha$. If $i^{\prime }$ and $j^{\prime }$ are left-to-left connected in $\alpha$, let $\gamma \in \mathcal{R}{\mathrm{Br}}_{n-m}$ be the diagram with connections $\{-i,-j\}$, $\{i,j\}$, and all other connections left-to-right $\{-l,l\}$. If $i^{\prime }$ and $j^{\prime }$ are isolated nodes in $\alpha$, let $\gamma \in \mathcal{R}{\mathrm{Br}}_{n-m}$ be the diagram with isolated nodes at $\{-i\}$, $\{-j\}$ and the connection $\{i,j\}$ with all other connections left-to-right $\{-l,l\}$. In either case $\gamma$ is non-invertible and $\alpha =\beta \gamma$. This completes the proof of the claim.

The inclusion $J\hookrightarrow \mathcal{R}{\mathrm{Br}}_n$ and projection $\mathcal{R}{\mathrm{Br}}_n\twoheadrightarrow \mathcal{R}{\mathrm{Br}}_n/J$ fit into a short exact sequence

$$0\to J\hookrightarrow \mathcal{R}{\mathrm{Br}}_n\twoheadrightarrow \mathcal{R}{\mathrm{Br}}_n/J\to 0.$$

Applying $-{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R$ yields the exact sequence

$$J{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\to \mathcal{R}{\mathrm{Br}}_n{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\to \mathcal{R}{\mathrm{Br}}_n/J{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\to 0.$$

Since $J{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\to \mathcal{R}{\mathrm{Br}}_n{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}$ is 0, it follows that

$$\mathcal{R}{\mathrm{Br}}_n{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\cong \mathcal{R}{\mathrm{Br}}_n/J{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R.$$

(3.1)

Note that the images in $\mathcal{R}{\mathrm{Br}}_n/J$ of diagrams in $\mathcal{R}{\mathrm{Br}}_n$ such that every node i with $m\le i\le n$ is connected to exactly one other node j with $j\le m$ form a basis of $\mathcal{R}{\mathrm{Br}}_n/J$.

Now let $I\subset \mathcal{R}{\mathrm{Br}}_{n-m}$ be the free R-submodule generated by all noninvertible diagrams in $\mathcal{R}{\mathrm{Br}}_{n-m}$. Then I annihilates R and $\mathcal{R}{\mathrm{Br}}_n/J$. It follows that the action of $\mathcal{R}{\mathrm{Br}}_{n-m}$ on R and $\mathcal{R}{\mathrm{Br}}_{n-m}/J$ factors through $\mathcal{R}{\mathrm{Br}}_{n-m}/I\cong R{S}_{n-m}$, where $S_{n-m}$ is the subgroup of $S_n$ which acts trivially on $\{1,\dots ,m\}$ and acts by permutation on $\{m+1,\dots ,n\}$. Hence

$$\mathcal{R}{\mathrm{Br}}_n/J{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\cong \mathcal{R}{\mathrm{Br}}_n/J{\otimes }_{R{S}_{n-m}}R.$$

(3.2)

Together, (3.1) and (3.2) imply

$$\mathcal{R}{\mathrm{Br}}_n{\otimes }_{\mathcal{R}{\mathrm{Br}}_{n-m}}R\cong \mathcal{R}{\mathrm{Br}}_n/J{\otimes }_{R{S}_{n-m}}R.$$

(3.3)

Since $S_{n-m}$ permutes the basis of $\mathcal{R}{\mathrm{Br}}_n/J$, it follows that $\mathcal{R}{\mathrm{Br}}_n/J{\otimes }_{R{S}_{n-m}}R$ is a free R-module with the desired basis. The result then follows from (3.3). □

The analogous result for ${Hom}_{{\mathcal{C}}_{\mathcal{R}}}(m,n)$ may be obtained by restricting the diagrams to those without left-to-left or right-to-right connections. We adjust the proof slightly to account for this restriction.

Proposition 2. 

${Hom}_{{\mathcal{C}}_{\mathcal{R}}}(m,n)$ is a free R-module with a basis consisting of diagrams with n nodes on the left, m nodes on the right and an “$(n-m)$-blob” which is connected to exactly $n-m$ nodes, such that every node is connected to at most one other node or the blob, and all connections are left-to-right.

Proof. 

Let $J\subset {\mathcal{R}}_n$ be the free R-submodule generated by diagrams which have at least one isolated node k with $m+1\le k\le n$. As in Proposition 1, we claim that the image of $J{\otimes }_{{\mathcal{R}}_{n-m}}R\to {\mathcal{R}}_n{\otimes }_{{\mathcal{R}}_{n-m}}R$ is 0. Note that diagrams in Rook algebras must have the same number of isolated nodes on the left as on the right. Let $\alpha$ be any diagram in J. To each isolated node k between $m+1$ and n, we may associate a unique isolated node $k^{\prime }<0$. Let $\beta \in {\mathcal{R}}_n$ be the diagram where the pairs $(k,k^{\prime })$ are connected and all other nodes are connected as they are in $\alpha$. Let $\gamma \in {\mathcal{R}}_{n-m}$ be the diagram where all the k and $-k$ nodes are isolated and all other nodes are connected horizontally. Then $\gamma$ is noninvertible and $\alpha =\beta \gamma$. The rest of the proof is the same, mutatis mutandis, as the proof of Proposition 1. □

Recall the category $\mathrm{FI}$ whose objects are the nonnegative integers and ${Hom}_{\mathrm{FI}}(m,n)$ consists of injective maps $\left[m\right]\to \left[n\right]$ where $\left[m\right]=\{1,\dots ,m\}$; note that when $m=n$, we have ${Hom}_{\mathrm{FI}}(m,n)=S_n$, the symmetric group on n objects. An $\mathrm{FI}$-module is a functor from the category $\mathrm{FI}$ to the category of R-modules.

For $A=\mathrm{Br}$, $\mathcal{R}\mathrm{Br}$, or $\mathcal{R}$, define a functor $F:\mathrm{FI}\to {\mathcal{C}}_A$ by the following data:

• $F\left(n\right)=n$ for all $n\in {\mathbb{Z}}_{\ge 0}$.
• For all $\alpha \in {Hom}_{\mathrm{FI}}(m,n)$, let $F\left(\alpha \right)\in {Hom}_{{\mathcal{C}}_A}(m,n)$ be the basis diagram with connections $\{-\alpha (i),i\}$ for all $i\in \left[m\right]$, and all $n-m$ remaining nodes on the left are connected to the $(n-m)$-blob.

Figure 4. Example of $F\left(f\right)\in {Hom}_{{\mathcal{C}}_A}(3,5) \mathrm{with} \mathrm{an} \mathrm{injection} f:\left[3\right]\to \left[5\right]$.

Figure 4. Example of $F\left(f\right)\in {Hom}_{{\mathcal{C}}_A}(3,5) \mathrm{with} \mathrm{an} \mathrm{injection} f:\left[3\right]\to \left[5\right]$.

Note that in the above example, the 2 nodes on the left of $F\left(f\right)$ not in the image of f are connected to the $(5-3)$-blob on the right.

When $A=P$, define the functor $G:\mathrm{FI}\to {\mathcal{C}}_P$ by the following data:

• $G\left(\right[n\left]\right)=\left[n\right]$ for all $n\in {\mathbb{Z}}_{\ge 0}$.
• For all $\alpha \in {Hom}_{\mathrm{FI}}(m,n)$, let $F\left(\alpha \right)\in {Hom}_{{\mathcal{C}}_A}(m,n)$ be the basis diagram with connections $\{-\alpha (i),i\}$ for each $i\in \left[m\right]$ and all the remaining $n-m$ nodes on the left constitute the $n-m$ marked blocks.

Figure 5. Example of $G\left(f\right)\in {Hom}_{{\mathcal{C}}_P}(3,5) \mathrm{with} \mathrm{an} \mathrm{injection} f:\left[3\right]\to \left[5\right]$.

Figure 5. Example of $G\left(f\right)\in {Hom}_{{\mathcal{C}}_P}(3,5) \mathrm{with} \mathrm{an} \mathrm{injection} f:\left[3\right]\to \left[5\right]$.

Note that in the above example, $G\left(f\right)$ has $5-3=2$ marked blocks (colored red) in the form of two isolated nodes.

With these functors, any ${\mathcal{C}}_A$-module may be considered as an $\mathrm{FI}$-module by pulling back along either F or G.

For all $m\in \mathbb{N}$, we define the free ${\mathcal{C}}_A$-module$M_A\left(m\right)$ by the following data:

• For all $n\in \mathbb{N}$, $M_A{\left(m\right)}_n$ is the free R-module generated on ${Hom}_{{\mathcal{C}}_A}(m,n)$.
• The action of ${\mathcal{C}}_A$-morphisms is given by postcomposition; that is, for all $\alpha \in {Hom}_{{\mathcal{C}}_A}(n,p)$, the induced map $M_A\left(m\right)\left(\alpha \right):M_A{\left(m\right)}_n\to M_A{\left(m\right)}_p$ is defined by $\beta \mapsto \alpha \beta$ for all $\beta \in {Hom}_{{\mathcal{C}}_A}(m,n)$.

Our objective now is to demonstrate that the free ${\mathcal{C}}_A$-module is finitely generated as an $\mathrm{FI}$-module. We start with the case when $A=P$, for which we establish a quick lemma.

Lemma 1. 

If $j\ge 5m$, then any basis diagram $\alpha \in M_{{\mathcal{C}}_P}{\left(m\right)}_j$ has at least $m+1$ marked blocks consisting of a single node.

Proof. 

Let b be the number of marked blocks in $\alpha$ with a single node. For contradiction, assume $b\le m$. From [1]*Prop. 2.5, $\alpha$ contains $j-m$ marked blocks. So there are another $j-m-b$ marked blocks in $\alpha$, each of which has at least two nodes. Together, these blocks must contain at least $2(j-m-b)$ nodes, out of the $j+m-b$ nodes not already in a block. However, since $j\ge 5m$ and $0\le b\le m$, it follows that $j>3m+b$, or equivalently, $2(j-m-b)>j+m-b$. So there are not enough nodes to fill all the blocks, a contradiction. □

Proposition 3. 

For $m\ge 0$, $M_{{\mathcal{C}}_P}\left(m\right)$ is finitely generated as an $\mathrm{FI}$-module.

Proof. 

The result is immediate for $m=0$; suppose $m\ge 1$. We claim that $M_{{\mathcal{C}}_P}\left(m\right)$ is generated as an $\mathrm{FI}$-module by basis diagrams of $M_{{\mathcal{C}}_P}{\left(m\right)}_k$ where $0\le k\le 5m$.

First, we show that $M_{{\mathcal{C}}_P}{\left(m\right)}_{5m+1}$ is generated by basis diagrams of $M_{{\mathcal{C}}_P}{\left(m\right)}_{5m}$. By Lemma 1, any basis diagram $\alpha \in M_{{\mathcal{C}}_P}{\left(m\right)}_{5m+1}$ has at least $m+1$ marked blocks of size 1. Since $\alpha$ has only m nodes on the right, there must be a marked block of size 1 on the left of $\alpha$.

We can find $\sigma \in S_{5m+1}$ such that $\sigma \alpha$ has the singe-node marked block on the top left. Let $\iota \in {Hom}_{\mathrm{FI}}(5m,5m+1)$ be the inclusion map $\left[5m\right]\hookrightarrow [5m+1]$ and $\overline{\alpha }$ be the diagram obtained from $\sigma \alpha$ by removing the marked block on the top left (see below). Note that $\overline{\alpha }\in M_{{\mathcal{C}}_P}{\left(m\right)}_{5m}$ and $\sigma \alpha =G\left(\iota \right)\overline{\alpha }$, or equivalently $\alpha ={\sigma }^{-1}G\left(\iota \right)\overline{\alpha }$.

Since ${\sigma }^{-1}\in S_{5m+1}$, we have $G\left({\sigma }^{-1}\right)={\sigma }^{-1}$. Thus $\alpha =G\left({\sigma }^{-1}\iota \right)\overline{\alpha }$ with ${\sigma }^{-1}\iota \in {Hom}_{\mathrm{FI}}(5m,5m+1)$. This implies that $M_{{\mathcal{C}}_P}{\left(m\right)}_{5m+1}$ is generated by basis diagrams of $M_{{\mathcal{C}}_P}{\left(m\right)}_{5m}$.

A routine induction now shows that $M_{{\mathcal{C}}_P}\left(m\right)$ is generated as an $\mathrm{FI}$-module by basis diagrams of $M_{{\mathcal{C}}_P}{\left(m\right)}_k$ with $0\le k\le 5m$. Hence, $M_{{\mathcal{C}}_P}\left(m\right)$ is finitely generated as an $\mathrm{FI}$-module. □

Figure 6. $\sigma \alpha \mathrm{to} \overline{\alpha }$.

Figure 6. $\sigma \alpha \mathrm{to} \overline{\alpha }$.

We proceed to prove a similar result for other categories.

Lemma 2. 

For $A=\mathrm{Br}$, $\mathcal{R}$, or $\mathcal{R}\mathrm{Br}$, if $j>2m$, any morphism of $M_{{\mathcal{C}}_A}{\left(m\right)}_j$ has at least a connection from the blob to some node on the left.

Proof. 

By [1]*Prop. 2.3 and Propositions 1 and 2 above, any basis diagram of $M_{{\mathcal{C}}_A}{\left(m\right)}_j$ must have exactly $j-m$ connections to the blob. Since $j-m>m$ and we only have m nodes on the right, there has to be at least one connection from the node on the left to the blob. □

Proposition 4. 

For $A=\mathrm{Br}$, $\mathcal{R}\mathrm{Br}$, or $\mathcal{R}$, the free ${\mathcal{C}}_A$-module $M_{{\mathcal{C}}_A}\left(m\right)$ is finitely generated as an $\mathrm{FI}$-module for all $m\ge 0$.

Proof. 

The case $m=0$ is straightforward; assume $m\ge 1$. We claim that $M_{{\mathcal{C}}_A}{\left(m\right)}_{2m+1}$ is generated as an $\mathrm{FI}$-module by basis diagrams of ${Hom}_{{\mathcal{C}}_A}(m,2m)$. Let $\alpha$ be a basis diagram of ${Hom}_{{\mathcal{C}}_A}(m,2m+1)$. By Lemma 2, there is a connection from the blob to some node on the left of $\alpha$. Furthermore, we can find a $\sigma \in S_{2m+1}$ such that the top left node of $\sigma \alpha$ is connected to the blob. Let $\tilde{\alpha }\in {Hom}_{{\mathcal{C}}_A}(m,2m)$ be the diagram obtained by deleting the top row of $\sigma \alpha$, and let $\iota \in {Hom}_{\mathrm{FI}}(2m,2m+1)$ be the inclusion map. We have $\sigma \alpha =F\left(\iota \right)\tilde{\alpha }$, so that $\alpha ={\sigma }^{-1}F\left(\iota \right)\tilde{\alpha }=F\left({\sigma }^{-1}\iota \right)\tilde{\alpha }$. This establishes the claim.

A standard induction argument then demonstrates that $M_{{\mathcal{C}}_A}\left(m\right)$ is generated as an $\mathrm{FI}$-module by basis diagrams of ${Hom}_{{\mathcal{C}}_A}(m,n)$ where $n\le 2m$. □

Finally, we can prove the main result.

Theorem 2. 

For $A=P$, $\mathrm{Br}$, $\mathcal{R}\mathrm{Br}$, or $\mathcal{R}$, any finitely generated ${\mathcal{C}}_A$-module is Noetherian.

Proof. 

Since any finitely generated ${\mathcal{C}}_A$-module V admits a surjection $⨁M_{{\mathcal{C}}_A}\left(m_i\right)\to V$ for some finite sequence of nonnegative integers $\left\{m_i\right\}$, it suffices to show that $M_{{\mathcal{C}}_A}\left(m\right)$ is a Noetherian ${\mathcal{C}}_A$-module for all $m\ge 0$. Let W be a ${\mathcal{C}}_A$-submodule of $M_{{\mathcal{C}}_A}\left(m\right)$, then W is also an $\mathrm{FI}$-submodule. Propositions 3 and 4, combined with the local Noetherian property for $\mathrm{FI}$[3]*Thm. A, establish that W is finitely generated as an $\mathrm{FI}$-submodule. Consequently, W is also finitely generated as a ${\mathcal{C}}_A$-submodule, thereby demonstrating that $M_{{\mathcal{C}}_A}\left(m\right)$ is a Noetherian ${\mathcal{C}}_A$-module. □