Jasso Reduction in Finite-Dimensional Algebra

• Jasso reduction is a technique in representation theory that simplifies the study of support τ-tilting modules, torsion classes, semibricks, and wide subcategories via idempotent quotients.
• It employs an exact functor to create an equivalence between subcategories and reduced algebras, establishing canonical bijections among key combinatorial structures.
• The method supports inductive classification by recursively decomposing module categories through the sequential removal of τ-rigid summands.

Jasso reduction, also known as Jasso’s $\tau$-rigid reduction theorem, is a structural method in the representation theory of finite-dimensional algebras for producing new module categories via idempotent quotients, facilitating the analysis of support $\tau$-tilting modules, functorially finite torsion classes, semibricks, and wide subcategories. This reduction is central in understanding the interplay between these combinatorial and categorical structures, providing canonical bijections and inductive classification techniques (Asai, 2016).

1. Background and Fundamental Notions

Let $A$ be a basic finite-dimensional $K$-algebra over a field $K$. Denote by $\operatorname{mod} A$ the category of finite-dimensional right $A$-modules. Key objects underlying the Jasso reduction include:

• Auslander–Reiten translation ($\tau$): $\tau = D\mathrm{Ext}^1_A(-, A)$, where $D = \mathrm{Hom}_K(-, K)$.
• $\tau$-rigid modules: $U \in \operatorname{mod} A$ is $\tau$-rigid if $\operatorname{Hom}_A(U, \tau U) = 0$.
• Support $\tau$-tilting modules: $U$ is support $\tau$-tilting if, additionally, there is some projective $P$ with $|U| + |P| = |A|$ and $\operatorname{Hom}_A(P, U) = 0$.
• Torsion classes: Full subcategories $\mathcal{T} \subseteq \operatorname{mod} A$ closed under extensions and factor modules; functorially finite torsion classes are those that are both covariantly and contravariantly finite.

These definitions are standard and serve as foundational terminology for stating Jasso’s theorem (Asai, 2016).

2. Statement of Jasso’s $\tau$-Rigid Reduction Theorem

Let $U$ be a $\tau$-rigid $A$-module and $T$ its unique Bongartz completion, i.e., the support $\tau$-tilting module with $U \in \operatorname{add} T$ and $$\operatorname{Fac} T = {}^\perp(\tau U) = U^\perp \cap \operatorname{Fac} U$$. Define $\Gamma = \operatorname{End}_A(T)$ and let $[\!U\!] \subseteq \Gamma$ be the two-sided ideal generated by all endomorphisms factoring through $\operatorname{add} U$; set $C := \Gamma/[\!U\!]$ (the idempotent reduction).

Consider the index sets:

• $\s\tau\text{-}\mathrm{tilt}_U\,A = \{ M \in \s\tau\text{-}\mathrm{tilt}\,A \mid U \in \operatorname{add} M \}$
• $\f\text{-}\mathrm{tors}_U\,A = \{ \mathcal{T} \in \f\text{-}\mathrm{tors}\,A \mid \operatorname{Fac} U \subseteq \mathcal{T} \subseteq U^\perp \cap \operatorname{Fac} T \}$

Proposition 1.20 (Asai, 2016):

There are bijections

$\f\text{-}\mathrm{tors}_U\,A \longleftrightarrow \f\text{-}\mathrm{tors}\,C, \qquad \s\tau\text{-}\mathrm{tilt}_U\,A \longleftrightarrow \s\tau\text{-}\mathrm{tilt}\,C,$

given by

$$\mathcal{T} \mapsto \mathcal{T} \cap U^\perp, \qquad M \mapsto \mathfrak{f} M$$

where $\mathfrak{f} M$ is the image of $M$ under the canonical functor $$U^\perp \cap \operatorname{Fac} T \to \operatorname{mod} C$$. These bijections commute with support–tilting mutation and induce a commutative diagram relating the sets of modules and torsion classes.

3. Construction and Functorial Properties

The constructive heart of Jasso reduction is the exact functor

$$F: U^\perp \cap \operatorname{Fac} T \longrightarrow \operatorname{mod} C, \qquad F(X) = \operatorname{Hom}_A(T, X) / [\!U\!] (\operatorname{Hom}_A(T, X))$$

which yields an equivalence of exact categories:

$$U^\perp \cap \operatorname{Fac} T \overset{\sim}{\longrightarrow} \operatorname{mod} C.$$

This equivalence aligns:

• Torsion classes in the interval $$\operatorname{Fac} U \subseteq \mathcal T \subseteq U^\perp \cap \operatorname{Fac} T$$ with functorially finite torsion classes in $\operatorname{mod} C$.
• Support $\tau$-tilting modules containing $U$ with support $\tau$-tilting $C$-modules.
• Support–tilting mutation in $\s\tau\text{-}\mathrm{tilt}_U A$ with that in $\s\tau\text{-}\mathrm{tilt} C$.

A critical technical point is that $F$ is exact and respects torsion pair structures, and that formation of $C$ via $\Gamma/[\!U\!]$ does not introduce or annihilate projective summands beyond those corresponding to $U$.

4. Role in the Theory of Semibricks and Wide Subcategories

Jasso’s reduction theorem, as analyzed in Asai’s study of semibricks, provides the categorical framework for relating support $\tau$-tilting modules, functorially finite torsion classes, left-finite semibricks, and left-finite wide subcategories. In particular:

• Semibricks in $U^\perp \cap \operatorname{Fac} M$ correspond to bricks in the reduced algebra $C = \operatorname{End}_A(M)/[\!U\!]$.
• The formula $$\operatorname{ind}(M/\operatorname{rad}_B M) \setminus \operatorname{ind}(U/\operatorname{rad}_B U) \leftrightarrow \operatorname{ind}(\mathfrak{f} M / \operatorname{rad}_C (\mathfrak{f} M))$$ exemplifies this correspondence.

The reduction procedure allows an explicit realization of any left-finite wide subcategory $\mathcal{W} = \operatorname{Filt}(S)$ (for a semibrick $S$) as $\operatorname{mod} B/(e)$ for an idempotent $e$ in $B = \operatorname{End}_A(M)$—effectively constructing module categories by deleting bricks associated with $U$ (Asai, 2016).

5. Inductive Constructions and Applications

Through repeated application of Jasso reduction—removing partial $\tau$-rigid summands $U$ from larger modules—one recursively reduces to algebras of smaller dimension. This drives an inductive method for:

• Classifying all left-finite semibricks and associated wide subcategories of $A$.
• Realizing any semibrick $S \subset \operatorname{brick} A$ by starting with the full semibrick of simples, performing a sequence of idempotent reductions to delete unwanted bricks, and descending to semisimple or zero algebras.

Example: For $A = K(1 \to 2 \to 3)$ and $U = (2)$, Bongartz completion yields $T = (3 \oplus 2 \oplus 1)$, with $\Gamma = \operatorname{End}_A(T) \cong K(1 \to 2)$, and $C = K(1\to 2)/(\text{vertex }2) \cong K(1)$. The interval of torsion classes between $\operatorname{Fac}(2)$ and $\operatorname{Fac}(3\oplus 2\oplus 1)$ is in bijection with $\f\text{-}\mathrm{tors}\,C$, which here has only two elements (Asai, 2016).

A direct corollary is that all left-finite semibricks are obtainable via a finite sequence of Jasso reductions.

6. Broader Significance

Jasso reduction endows the study of support $\tau$-tilting modules, torsion classes, semibricks, and wide subcategories with a powerful mechanism for both structural simplification and inductive classification. It provides canonical bijections between restricted support $\tau$-tilting objects and their images in quotient endomorphism algebras and offers a practical and theoretical toolkit for analyzing representation-finite situations and decomposing module categories into elementary, combinatorial pieces (Asai, 2016).