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Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split short exact sequence $0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0$.

Question 1: Is there an easy example of $A$ such that $\sup \{ \xi(M,N) | M, N $ indecomposable $\}$ is infinite?

 

Question 2: Is there an easy example of $A$ such that $\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$ is infinite?

(here $\tau$ denotes the Auslander-Reiten translate. One might also ask for an algebra where the number of middle term indecomposable summands of Auslander-Reiten sequences can be arbitrarily large)

Question 3: Do we have $\sup \{ \xi(M,N) | M, N $ indecomposable $\}=\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$?

Bonusquestion: Can $\xi(M,N)$ be calculated with the GAP-package QPA?

Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split short exact sequence $0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0$.

Question 1: Is there an easy example of $A$ such that $\sup \{ \xi(M,N) | M, N $ indecomposable $\}$ is infinite?

Question 2: Is there an easy example of $A$ such that $\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$ is infinite?

(here $\tau$ denotes the Auslander-Reiten translate. One might also ask for an algebra where the number of middle term indecomposable summands of Auslander-Reiten sequences can be arbitrarily large)

Question 3: Do we have $\sup \{ \xi(M,N) | M, N $ indecomposable $\}=\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$?

Bonusquestion: Can $\xi(M,N)$ be calculated with the GAP-package QPA?

Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split short exact sequence $0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0$.

Question 1: Is there an easy example of $A$ such that $\sup \{ \xi(M,N) | M, N $ indecomposable $\}$ is infinite?

Question 2: Is there an easy example of $A$ such that $\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$ is infinite?

(here $\tau$ denotes the Auslander-Reiten translate. One might also ask for an algebra where the number of middle term indecomposable summands of Auslander-Reiten sequences can be arbitrarily large)

Question 3: Do we have $\sup \{ \xi(M,N) | M, N $ indecomposable $\}=\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$?

Question 4: Is it true that $\sup \{ \xi(M,N) | M, N $ indecomposable $\} \leq 2$ for Nakayama algebras?

Bonusquestion: Can $\xi(M,N)$ be calculated with the GAP-package QPA?

Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split short exact sequence $0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0$.

Question 1: Is there an easy example of $A$ such that $\sup \{ \xi(M,N) | M, N $ indecomposable $\}$ is infinite?

Question 2: Is there an easy example of $A$ such that $\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$ is infinite?

(here $\tau$ denotes the Auslander-Reiten translate. One might also ask for an algebra where the number of middle term indecomposable summands of Auslander-Reiten sequences can be arbitrarily large)

Question 3: Do we have $\sup \{ \xi(M,N) | M, N $ indecomposable $\}=\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$?

Bonusquestion: Can $\xi(M,N)$ be calculated with the GAP-package QPA?

Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split short exact sequence $0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0$.

Question 1: Is there an easy example of $A$ such that $\sup \{ \xi(M,N) | M, N $ indecomposable $\}$ is infinite?

Question 2: Is there an easy example of $A$ such that $\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$ is infinite?

(here $\tau$ denotes the Auslander-Reiten translate. One might also ask for an algebra where the number of middle term indecomposable summands of Auslander-Reiten sequences can be arbitrarily large)

Question 3: Do we have $\sup \{ \xi(M,N) | M, N $ indecomposable $\}=\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$?

Question 4: Is it true that $\sup \{ \xi(M,N) | M, N $ indecomposable $\} \leq 2$ for Nakayama algebras?

Let $A$ be a finite dimensional algebra and $M$ and $N$ indecomposable $A$-modules. Denote by $\xi(M,N)$ the maximal number of indecomposable summands of a modules $X$ such that there is a non-split short exact sequence $0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0$.

Question 1: Is there an easy example of $A$ such that $\sup \{ \xi(M,N) | M, N $ indecomposable $\}$ is infinite?

Question 2: Is there an easy example of $A$ such that $\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$ is infinite?

(here $\tau$ denotes the Auslander-Reiten translate. One might also ask for an algebra where the number of middle term indecomposable summands of Auslander-Reiten sequences can be arbitrarily large)

Question 3: Do we have $\sup \{ \xi(M,N) | M, N $ indecomposable $\}=\sup \{ \xi(M,\tau(M)) | M $ indecomposable $\}$?

Question 4: Is it true that $\sup \{ \xi(M,N) | M, N $ indecomposable $\} \leq 2$ for Nakayama algebras?

Bonusquestion: Can $\xi(M,N)$ be calculated with the GAP-package QPA?