# Questions in the paper “The category of good modules over a quasi-hereditary algebra has almost split sequences”

I am reading the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences", the link is here:https://pub.uni-bielefeld.de/publication/1780235.

In the paper, $A$ is an artin algebra, and $A$-mod the category of (finitely generated) $A$-modules. Let $\Theta=\{\Theta(1), \dots, \Theta(n) \}$ be a finite set of $A$-modules with $Ext^1_A(\Theta(j), \Theta(i))=0$ for $j \geq i$. We denote $\mathcal{F}(\Theta)$ the full subcategory of $A$-mod of direct summands of modules having a filtration with factors in $\Theta$. I have some questions about $\mathcal{F}(\Theta)$:

1. It says that"$M$ belongs to $\mathcal{F}(\Theta)$ if and only if $M$ has submodules $0=M_0 \subseteq M_1 \subseteq \cdots \subseteq M_t=M$ such that $M_s/M_{s-1}$ is isomorphic to a module in $\Theta$"(By the definition, if $M \in \mathcal{F}(\Theta)$, there is an $A$-module N such that $M \oplus N$ has submodules $0=M'_0 \subseteq M'_1 \subseteq \cdots \subseteq M'_t=M \oplus N$ and $M'_s/M'_{s-1}$ is isomorphic to a module in $\Theta$, then how to get $M$ also has submodules $0=M_0 \subseteq M_1 \subseteq \cdots \subseteq M_t=M$ such that $M_s/M_{s-1}$ is isomorphic to a module in $\Theta$?)
2. It also defines $\mathcal{X}(\Theta)$ the full subcategory of $A$-mod of all modules which are direct summands of modules in $\mathcal{F}(\Theta)$. (By the definition, I think $\mathcal{F}(\Theta)$ has already been closed under direct summands, what is the difference between $\mathcal{X}(\Theta)$ and $\mathcal{F}(\Theta)$?)
3. It gives an example to show $\mathcal{X}(\Theta)$ and $\mathcal{F}(\Theta)$ are not same: $A=\begin {pmatrix} k & k & k \\ 0 &k &0 \\ 0 &0& k\end {pmatrix}$,where k is a field, $\Theta(2)$ is the simple projective $A$-module, $\Theta(1)$ is its injective hull: here the indecomposable modules of length 2 belong to $\mathcal{X}(\Theta)$ but not to $\mathcal{F}(\Theta)$. (I don't know how to get the conclusion)

Thank you for any help.

## 1 Answer

This is probably just a mistake in the preprint. If you look at the corresponding paragraph in the published version

Claus Michael Ringel, MR 1128706 The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), no. 2, 209--223.),

the category ${\mathscr F}(\Theta)$ is defined differently. Quote:

We denote by ${\mathscr F}(\Theta)$ the full subcategory of $A$-mod of modules having a filtration with factors in $\Theta$.