arXiv subjects

Sota Asai

Publications and source records attributed to Sota Asai.

14 recordsLinked to original sources

arXiv2026

Mutations of simple-minded collections revisited

Simple-minded collections in the bounded derived category $\mathsf{D}^\mathrm{b}(\operatorname{\mathsf{mod}} A)$ are necessary tools in tilting theory of finite dimensional algebras, as well as silting complexes in the perfect derived category $\mathsf{K}^\mathrm{b}(\operatorname{\mathsf{proj}} A)$. In this paper, we give an explicit mutation formula of simple-minded collections at arbitary direct summands, which is compatible with that of silting complexes. Though this mutation formula may be known to experts, we provide a thorough proof for completeness and future reference. Then we use our formula to revisit $\tau$-tilting theory in terms of mutations of 2-term simple-minded collections, including their relationship with maximal/minimal completions of 2-term presilting complexes and $\tau$-tilting reduction.

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arXiv2026

Left modularity and extremality for (some) infinite lattices

For some important families of complete infinite lattices, we study some generalizations of two fundamental notions which are mostly treated for finite lattices. Specifically, for well-separated $\kappa$-lattices, and also for weakly atomic completely semidistributive lattices, we generalize the notions of left modularity and extremality. These two families of lattices coincide if restricted to finite lattices, but are distinct when infinite lattices are also included. For both families, we prove that extremality and left modularity imply each other. Furthermore, for weakly atomic completely semidistributive lattices, we give several conceptual characterizations of left modular elements, and show that the set of left modular elements form a complete distributive sublattice. Our results, combined with some recent work on finite lattices, imply that the weakly atomic completely semidistributive lattices that are left modular (or extremal) generalize the semidistributive trim lattices; from finite to infinite lattices. We then apply our results to the lattice of torsion classes of finite dimensional algebras, which are known to fall in the intersection of the two families treated in our work. For an algebra $A$, we obtain that the lattice of torsion classes is left modular (equivalently, extremal) if and only if $A$ is brick-directed. This leads to an abundance of concrete examples and non-examples.

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arXiv2026

The interval neighborhoods in the real Grothendieck groups

For a finite dimensional algebra $A$, the TF equivalence on the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ can be regarded as a completion of the $g$-fan. For example, the silting cones $C^\circ(U)$ of 2-term presilting complexes $U$ give the most fundamental family of TF equivalence classes. The next step is studying the TF equivalence classes around each silting cone $C^\circ(U)$. Thus, in this paper, we investigate the closed interval neighborhood $D(U)$ of $C^\circ(U)$. As our main result, we give a $2^{|U|}:1$ correspondence between the TF equivalence classes in $D(U)$ and those in $K_0(\operatorname{\mathsf{proj}} B)_\mathbb{R}$, where $B$ is the algebra appearing in the $\tau$-tilting reduction at $U$. For this purpose, we give an explicit description of defining inequalities and the faces of $D(U)$ as a polyhedral cone, by using 2-term simple-minded collections and $M$-TF equivalences.

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arXiv2026

Bicompact torsion classes and conjectures on brick infinite algebras

A torsion class $\mathcal{T}$ of the module category $\operatorname{\mathsf{mod}} A$ of a finite dimensional algebra $A$ over a field $K$ is said to be compact if there exists a module $M \in \operatorname{\mathsf{mod}} A$ such that $\mathcal{T}$ is the smallest torsion class containing $M$. If a torsion class satisfies this and the dual condition, then we call it a bicompact torsion class. We conjecture that bicompact torsion classes are precisely functorially finite torsion classes, and prove it for hereditary algebras and also for semistable torsion classes. This gives that Demonet Conjecture implies Enomoto Conjecture, both of which are important conjectures on brick infiniteness.

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arXiv2025

Maximal finite semibricks consist only of open bricks

A semibrick is a set of modules satisfying Schur's Lemma, and it is said to be maximal if it is not properly contained in another semibrick. For any finite dimensional algebra $\varLambda$ over an algebracally closed field $K$, we prove that any maximal finite semibrick $\mathcal{S}$ consists only of open bricks $B$, that is, bricks whose orbit closures $\overline{\mathcal{O}_B}$ are irreducible components in the representation schemes.

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arXiv2025

Brick-splitting Torsion Pairs and Left Modularity

We introduce the notion of brick-splitting torsion pairs as a modern analogue and generalization of the classical notion of splitting torsion pairs. A torsion pair is called brick-splitting if any given brick is either torsion or torsion-free with respect to that torsion pair. After giving some properties of these pairs, we fully characterize them in terms of some lattice-theoretical properties, including left modularity. This leads to the notion of brick-directed algebras, which are those for which there does not exist any cycle of non-zero non-isomorphisms between bricks. This class of algebras is a novel generalization of representation-directed algebras. We show that brick-directed algebras have many interesting properties and give several characterizations of them. In particular, we prove that a brick-finite algebra is brick-directed if and only if the lattice of torsion classes is left modular (or equivalently, extremal). We also give a characterization of brick-directed algebras in terms of their wall-and-chamber structure, as well as of a certain Newton polytope associated to them. Moreover, we introduce an explicit construction of an abundance of brick-directed algebras, both of the tame and wild representation types.

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arXiv2024

$M$-TF equivalences on the real Grothendieck groups

For an abelian length category $\mathcal{A}$ with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence on the dual real Grothendieck group $K_0(\mathcal{A})_\mathbb{R}^*=\operatorname{Hom}_\mathbb{R}(K_0(\mathcal{A})_\mathbb{R},\mathbb{R})$, which are defined by semistable subcategories and semistable torsion pairs in $\mathcal{A}$ associated to elements $\theta \in K_0(\mathcal{A})_\mathbb{R}^*$. In this paper, we introduce the $M$-TF equivalence for each object $M \in \mathcal{A}$ as a systematic way to coarsen the TF equivalence. We show that the set $\Sigma(M)$ of closures of $M$-TF equivalence classes is a rational generalized fan in $K_0(\mathcal{A})_\mathbb{R}^*$ which is finite and complete. More precisely, we show that $\Sigma(M)$ is the normal generalized fan of the Newton polytope $\mathrm{N}(M)$ in $K_0(\mathcal{A})_\mathbb{R}$. When $\mathcal{A}$ is the category of finitely generated modules over a finite dimensional algebra $A$, $\Sigma(M)$ can be regarded as a completion of a certain coarsening of the $g$-fan of $A$.

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arXiv2023

Semistable torsion classes and canonical decompositions in Grothendieck groups

We study two classes of torsion classes which generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen-Fei. More strongly, for $E$-tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. We also answer a question by Derksen-Fei negatively by giving examples of algebras which do not satisfy the ray condition. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type $\widetilde{\mathbb{A}}$.

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arXiv2022

Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

In the representation theory of finite-dimensional algebras $A$ over a field, the classification of 2-term (pre)silting complexes is an important problem. One of the useful tool is the g-vector cones associated to the 2-term presilting complexes in the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_{\mathbb{Z}} {\mathbb{R}}$. The aim of this paper is to study the complement $\operatorname{\mathsf{NR}}$ of the union $\operatorname{\mathsf{Cone}}$ of all g-vector cones, which we call the non-rigid region. By the work of Iyama and us, $\operatorname{\mathsf{NR}}$ is determined by 2-term presilting complexes and a certain closed subset $R_0 \subset K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}$, which is called the purely non-rigid region. In this paper, we give an explicit description of $R_0$ for complete special biserial algebras in terms of a finite set of maximal nonzero paths in the Gabriel quiver of $A$. We also prove that $\operatorname{\mathsf{NR}}$ has some kind of fractal property and that $\operatorname{\mathsf{NR}}$ is contained in a union of countably many hyperplanes of codimension one. Thus, any complete special biserial algebra is g-tame, that is, $\operatorname{\mathsf{Cone}}$ is dense in $K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}$.

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arXiv2020

The wall-chamber structures of the real Grothendieck groups

For a finite-dimensional algebra $A$ over a field $K$ with $n$ simple modules, the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_\mathbb{Z} \mathbb{R} \cong \mathbb{R}^n$ gives stability conditions of King. We study the associated wall-chamber structure of $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ by using the Koenig--Yang correspondences in silting theory. First, we introduce an equivalence relation on $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ called TF equivalence by using numerical torsion pairs of Baumann--Kamnitzer--Tingley. Second, we show that the open cone in $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ spanned by the g-vectors of each 2-term silting object gives a TF equivalence class, and this gives a one-to-one correspondence between the basic 2-term silting objects and the TF equivalence classes of full dimension. Finally, we determine the wall-chamber structure of $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ in the case that $A$ is a path algebra of an acyclic quiver.

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arXiv2020

Wide subcategories and lattices of torsion classes

In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal{A}$ from the point of view of lattice theory. Motivated by $τ$-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal{U},\mathcal{T}]$ in the lattice $\operatorname{\mathsf{tors}} \mathcal{A}$ of torsion classes in $\mathcal{A}$ such that $\mathcal{W}:=\mathcal{U}^\perp \cap \mathcal{T}$ is a wide subcategory of $\mathcal{A}$; we call these intervals wide intervals. We prove that a wide interval $[\mathcal{U},\mathcal{T}]$ is isomorphic to the lattice $\operatorname{\mathsf{tors}} \mathcal{W}$ of torsion classes in the abelian category $\mathcal{W}$. We also characterize wide intervals in two ways: First, in purely lattice theoretic terms based on the brick labeling established by Demonet--Iyama--Reading--Reiten--Thomas; and second, in terms of the Ingalls--Thomas correspondences between torsion classes and wide subcategories, which were further developed by Marks--Šťovíček.

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arXiv2018

Bricks over preprojective algebras and join-irreducible elements in Coxeter groups

A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $Δ$, there is a bijection from the Coxeter group $W$ of type $Δ$ to the set of semibricks over the preprojective algebra $Π$ of type $Δ$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $Π$. This paper is devoted to giving an explicit description of these bijections in the case $Δ=\mathbb{A}_n$ or $\mathbb{D}_n$. First, for each join-irreducible element $w \in W$, we describe the corresponding brick $S(w)$ in terms of "Young diagram-like" notation. Next, we determine the canonical join representation $w=\bigvee_{i=1}^m w_i$ of an arbitrary element $w \in W$ based on Reading's work, and prove that $\bigoplus_{i=1}^n S(w_i)$ is the semibrick corresponding to $w$.

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arXiv2018

Semibricks

In representation theory of finite-dimensional algebras, (semi)bricks are a generalization of (semi)simple modules, and they have long been studied. The aim of this paper is to study semibricks from the point of view of $τ$-tilting theory. We construct canonical bijections between the set of support $τ$-tilting modules, the set of semibricks satisfying a certain finiteness condition, and the set of 2-term simple-minded collections. In particular, we unify Koenig-Yang bijections and Ingalls-Thomas bijections generalized by Marks-Šťovíček, which involve several important notions in the derived categories and the module categories. We also investigate connections between our results and two kinds of reduction theorems of $τ$-rigid modules by Jasso and Eisele-Janssens-Raedschelders. Moreover, we study semibricks over Nakayama algebras and tilted algebras in detail.

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arXiv2016

The Grothendieck groups and stable equivalences of mesh algebras

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable categories. As an application, we give an complete classification of the mesh algebras up to stable equivalences.

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