The interval neighborhoods in the real Grothendieck groups

Abstract: 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 $τ$-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.

• The paper’s main contribution is establishing a precise 2^(|U|):1 correspondence between interval-neighborhood TF-classes and reduced TF-classes.
• The methodology employs explicit polyhedral constructions via semibrick labelings and silting mutation theory to detail the structure of facets and faces.
• The implications include enhanced tools for stability analysis and moduli computations by connecting categorical data with combinatorial and geometric techniques.

Interval Neighborhoods and TF-Equivalence Structures in Real Grothendieck Groups

Introduction and Motivation

The study centers on the geometry and combinatorics associated with silting theory, $\tau$-tilting theory, and Grothendieck groups for finite-dimensional algebras. The principal object of investigation is the space $K_0(\proj A)_{\mathbb{R}}$, the real Grothendieck group of the category of finitely generated projective $A$-modules for a finite-dimensional algebra $A$. The paper addresses the refinement and local structure of the TF (torsion-free) equivalence relations arising in this real vector space, within which silting theory and $g$-vector fans play a central role.

The $g$-fan, which encodes mutations of 2-term silting complexes as rational polyhedral cones, is typically incomplete except for brick-finite algebras. To overcome this, TF-equivalence provides a completion, partitioning $K_0(\proj A)_{\mathbb{R}}$ into finer polyhedral pieces parametrized in terms of induced torsion pairs. The analysis focuses on "interval neighborhoods" $D(U)$ of silting cones corresponding to 2-term presilting complexes $U$, aiming to describe the structure of TF-equivalence classes and faces in these neighborhoods.

Main Contributions

Definition and Structure of Interval Neighborhoods

For $U$ a basic 2-term presilting complex, the interval neighborhood $K_0(\proj A)_{\mathbb{R}}$0 is defined as the set of points in $K_0(\proj A)_{\mathbb{R}}$1 where the associated semistable torsion pairs satisfy $K_0(\proj A)_{\mathbb{R}}$2 and $K_0(\proj A)_{\mathbb{R}}$3. The open version, $K_0(\proj A)_{\mathbb{R}}$4, further requires $K_0(\proj A)_{\mathbb{R}}$5 and $K_0(\proj A)_{\mathbb{R}}$6. These sets are rational polyhedral cones whose combinatorics reflect local wall-and-chamber behavior near the silting cone $K_0(\proj A)_{\mathbb{R}}$7. The paper shows that $K_0(\proj A)_{\mathbb{R}}$8 is generated by inequalities associated with the semibricks derived from $K_0(\proj A)_{\mathbb{R}}$9, encoding categorical torsion and torsion-free closures.

Bijections between TF-Equivalence Classes

A central achievement is the establishment of a bijective correspondence between the set of TF-equivalence classes contained in $A$0 (the local analytic structure near $A$1) and a product parametrization by subsets of direct summands of $A$2 and TF-classes in the reduced algebra $A$3 (arising from $A$4-tilting reduction at $A$5). Specifically, there is a $A$6 correspondence:

$A$7

where $A$8 parametrizes subsets of indecomposable summands, $A$9 is a TF-equivalence class for $A$0, and $A$1 is an explicit, piecewise-linear section induced from the Grothendieck group surjection. This result provides a clear and highly structured account of how the local TF-equivalence structure in the interval neighborhood decomposes into data from the $A$2-tilting reduction.

Complete Face and Facet Description

The faces and facets of the cone $A$3 are rigorously characterized in terms of semibrick labelings, submodule and factor module structure, and morphisms induced by silting mutation theory. In a key result, every facet of $A$4 corresponds uniquely to a pair $A$5, distinguishing walls arising from either factor modules of $A$6 or submodules of $A$7. Each such facet is normal to a brick module $A$8, which is explicitly computable from $A$9, and the structure of $g$0 is completely determined by a system of inequalities $g$1 over these bricks. The construction aligns these facet labelings with the brick-labeling of the exchange quiver for 2-term silting complexes, achieving a categorical and combinatorial compatibility.

Generalized Fans, M-TF Equivalence, and Connections to Reduction

An important further development is the construction of finite, complete generalized fans in $g$2 corresponding to faces of $g$3, parametrized by subsets $g$4 of summands. These fans are realized as the coarsenings associated to $g$5-TF-equivalence, where $g$6 is a direct sum of the modules $g$7 constructed from the 2-term simple-minded collections linked to the completions of $g$8. The explicitness of this connection allows for a comprehensive combinatorial account of the face structure of $g$9 as lifts from the reduction algebra $g$0.

Numerical and Structural Highlights

• $g$1 parametrization: The paper establishes a precise $g$2 correspondence between interval-neighborhood TF-classes and reduced TF-classes, with complete explicit formulae for the maps involved.
• Dimension formulae: For each face in $g$3 corresponding to $g$4, the dimension equals $g$5, decomposing the stratification into contributions from direct summand choices and the reduced algebra's geometry.
• Strong explicit descriptions: In practice, all faces, facets, and their normal vectors are described via semibricks and simple-minded collections, and the section $g$6 is realized constructively.

Theoretical and Practical Implications

These results have several immediate consequences for silting/tilting theory and the study of wall-and-chamber structures:

• Categorical Clarity: The intricate combinatorial-lattice structures (fan decompositions, intervals of torsion classes, and bricks) controlling module categories are made manifest in the geometry of $g$7 through explicit, functorially compatible correspondences.
• Reduction Compatibility: The entire local fan structure near a silting cone is governed by the combinatorics of the $g$8-tilting reduced algebra, enabling inductive and recursive approaches to the structure of TF-equivalence classes and their stratifications.
• Applications to Stability and Moduli: The clarified polyhedral geometry provides concrete tools for the study of moduli spaces of (semi)stable modules, stability scattering diagrams, and the relationship between silting theory and spaces of Bridgeland stability conditions.
• Computational and Algorithmic Utility: With the explicit recipes for faces and facets, practical algorithms for computing the wall-chamber decompositions for given algebras and presilting data become feasible.

Further Directions and Speculative Outlook

The methods and results suggest multiple extensions:

• Analysis of interval neighborhoods beyond the context of 2-term presilting complexes, possibly to the broader context of derived or extriangulated categories.
• Interplay between TF-equivalence class decompositions and cluster-theoretic models, particularly for wild or infinite type algebras.
• Deeper investigation into the algebraic and geometric meaning of the generalized fans produced, and their connection with mirror symmetry, scattering diagrams, and the modular representation theory of finite-dimensional algebras.

Conclusion

This work delivers a comprehensive and rigorous analysis of the local and global polyhedral geometry of real Grothendieck groups under TF-equivalence, with particular emphasis on the interval neighborhoods of silting cones. Through explicit and constructive correspondences, it bridges categorical, combinatorial, and geometric perspectives, yielding new tools for both the foundational understanding and the practical calculation of stability spaces and wall-chamber structures in the representation theory of finite-dimensional algebras.