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Pseudo-torsion classes

Kiyoshi Igusa, Ray Maresca

Abstract

For a finite dimensional algebra $Λ$, we consider a torsion class $G$ in $mod$-$Λ$, which is not necessarily finitely generated. We construct a wall-and-chamber structure for $G$ where the chambers are the connected components of the complement of the union of walls. We also consider ``infinitesimal chambers". To each chamber we associate a ``pseudo-torsion class'' and a ``pseudo-torsionfree class'' and show that they are all distinct. We consider ``green paths'' in the stability space and associate to them Harder-Narasimhan stratifications of $G$. This paper is part of a series of papers whose goal is to study the ``ghosts'' which are remnants of the indecomposable $Λ$-modules which do not lie in $G$. In the special case when our torsion class is all of $mod$-$Λ$, we are in the classical well-known setting. All of our results apply to this classical setting. The ``pseudo-torsion classes'' are torsion classes. We point out that we do not take the closure of the set of walls. So, we get more chambers and our results are new even in this classical setting.

Pseudo-torsion classes

Abstract

For a finite dimensional algebra , we consider a torsion class in -, which is not necessarily finitely generated. We construct a wall-and-chamber structure for where the chambers are the connected components of the complement of the union of walls. We also consider ``infinitesimal chambers". To each chamber we associate a ``pseudo-torsion class'' and a ``pseudo-torsionfree class'' and show that they are all distinct. We consider ``green paths'' in the stability space and associate to them Harder-Narasimhan stratifications of . This paper is part of a series of papers whose goal is to study the ``ghosts'' which are remnants of the indecomposable -modules which do not lie in . In the special case when our torsion class is all of -, we are in the classical well-known setting. All of our results apply to this classical setting. The ``pseudo-torsion classes'' are torsion classes. We point out that we do not take the closure of the set of walls. So, we get more chambers and our results are new even in this classical setting.

Paper Structure

This paper contains 21 sections, 52 theorems, 46 equations, 3 figures.

Table of Contents

  1. Basic definitions
  2. Strict morphisms and strict extensions
  3. Pseudo-torsion pairs
  4. Pseudo-wide subcategories
  5. Relative pseudo-torsion classes
  6. Wall and chamber structure
  7. Walls thick and thin
  8. Pseudo-torsion classes given by stability conditions
  9. Perpendicular classes
  10. Pseudo-chambers
  11. Examples
  12. $B_2$ example
  13. A torsion class in $A_3$
  14. Ghosts in $A_3$
  15. Walls have two sides
  16. ...and 6 more sections

Key Result

Lemma 1.2

If $A$ is a strict subobject of $B$ and $B'$ is another subobject of $B$. Then $A\cap B'$ is a strict subobject of $B'$.

Figures (3)

  • Figure 1: The wall-and-chamber structure for $mod\text{-}\Lambda$ is shown on the left. On the right we have that of ${\mathcal{G}}={\mathcal{G}}_3$. The pseudo-torsion classes ${\mathcal{P}}_0,{\mathcal{P}}_1,{\mathcal{P}}_2,{\mathcal{P}}_3$ are equal to the torsion classes ${\mathcal{G}}_0,{\mathcal{G}}_1,{\mathcal{G}}_2,{\mathcal{G}}_3={\mathcal{G}}$ on the left. These are given by ${\mathcal{G}}_0=\{0\}$, ${\mathcal{G}}_1=add(I_1)$, ${\mathcal{G}}_2=add(I_1,I_2)$ and ${\mathcal{G}}_3={\mathcal{G}}=add(I_1,I_2,P_1)$. Also, ${\mathcal{G}}_5=add(P_2)$. The pseudo-torsion classes ${\mathcal{P}}_4,{\mathcal{P}}_5$ are discussed below.
  • Figure 2: The full picture for $mod\text{-}\Lambda$ is on the left. The torsion classes ${\mathcal{G}}_i$ contained in ${\mathcal{G}}={\mathcal{G}}_6$ for $i\le 6$ are equal to the pseudo-torsion classes ${\mathcal{P}}_i$ for $i\le 6$ on the right. There are three more pseudo-torsion classes ${\mathcal{P}}_7,{\mathcal{P}}_8,{\mathcal{P}}_9$ which are all characterized as the left perpendicular of the green walls of their chamber and, also, they are generated by the red walls. For example, ${\mathcal{P}}_7=S_1^\perp$ is generated by $I_2$ and $S_3$. Vertices $\color{blue}a$ and $\color{red}b$ are "ghosts" explained in Example \ref{['eg: A3 ghosts']}.
  • Figure 3: This is the figure for Example \ref{['eg: A3 ghosts']}. The two back spots are the $g$-vectors of $S_1$ at bottom and $S[1]$ at top. In the quadrilateral we have vertices with balance $g$-vector notation ${\dfrac{\color{blue}S_3}{\color{red}X}}$ and ${\dfrac{\color{blue}Y}{\color{red}0}}$ explained in the text. For lack of space, we have shortened the pseudo-wall notation in this figure from $D_{\mathcal{G}}(X)$ to $D(X)$

Theorems & Definitions (121)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Theorem 1.3
  • proof
  • Definition 1.4
  • Lemma 1.5: Main Lemma
  • proof
  • Lemma 1.6
  • Proposition 1.7
  • ...and 111 more