Bicompact torsion classes and conjectures on brick infinite algebras

Abstract: 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.

• The paper establishes that a torsion class is bicompact if and only if it is functorially finite by leveraging silting cones and the geometry of the Grothendieck group.
• It proves multiple equivalences among torsion classes, showing that rigidity, compactness, and homological conditions are deeply interconnected.
• The study connects non-rigid lattice points to the generation of infinite semibricks, thereby addressing conjectures on brick infinite algebras.

Bicompact Torsion Classes and Conjectures on Brick Infinite Algebras

Introduction and Motivation

The classification of torsion classes in module categories of finite-dimensional algebras has significant implications for the structure theory of representations and the combinatorial theory of lattices. This work investigates the relationship between bicompact torsion classes and functorially finite torsion classes, providing a framework that addresses conjectures regarding the structure of brick infinite algebras. Central to this development are the precise relationships between compactness notions for torsion classes, the geometry of the Grothendieck group, and the lattice-theoretic properties of module categories.

Bicompact Torsion Classes and Functorial Finiteness

A torsion class $T \subset \mathsf{mod}\,A$ is defined as compact if there exists $M$ with $T$ the smallest torsion class containing $M$; cocompactness is dually defined. A torsion class is bicompact if it is both compact and cocompact. Functorially finite torsion classes, which correspond to those of the form $\mathsf{Fac} M$, are always bicompact by classical results due to Smålø.

The main conjecture addressed posits an equivalence: a torsion class is bicompact if and only if it is functorially finite. This assertion encapsulates a crucial structural question for the lattice of torsion classes, connecting internal generation properties to homological finiteness conditions.

Main Structural Results

The paper establishes several equivalences for torsion classes associated with stability conditions. For $\theta \in K_0(\mathsf{proj}\,A)_\mathbb{R}$, torsion classes of the form $T_\theta$ (modules with all nonzero factor modules $N$ satisfying $\theta(N)>0$) and $_\theta$ (all factor modules $M$0 with $M$1) are studied in detail. The main theorem demonstrates the following nine-way equivalence for such torsion classes:

• $M$2 is rigid (lies in a $M$3 cone for some 2-term silting complex $M$4);
• $M$5 (resp. $M$6) is bicompact, compact, functorially finite, or cocompact.

The proof leverages the geometry of silting cones (chambers of the $M$7-fan) and their relationship with torsion pairs, the wall-and-chamber structure on $M$8, and combinatorial properties of module categories. A key argument is the identification of open sets in the Grothendieck group corresponding to compact torsion classes, which ensures that compactness implies rigidity, and thus functorial finiteness.

For hereditary algebras, the conjecture is fully established: all bicompact torsion classes are functorially finite. The proof exploits the irreducibility properties of representation varieties over hereditary algebras and openness conditions for bricks and torsion classes in their generic strata.

Brick Infiniteness and Conjectural Implications

The study of bricks (indecomposable modules with division endomorphism algebra) and semibricks (collections of pairwise Hom-orthogonal bricks) is entrenched in the examination of representation-theoretic finiteness. The paper focuses on two conjectures:

• Enomoto’s Conjecture: Every brick infinite algebra possesses an infinite semibrick.
• Demonet’s Conjecture: For brick infinite $M$9, there exists a non-rigid lattice point $T$0.

The manuscript demonstrates that Demonet’s conjecture implies Enomoto’s. Given a non-rigid lattice point, one constructs an infinite semibrick generating the associated torsion class by functoriality and the structure theorems established for bicompact torsion classes. This relies on the fact that for lattice points, the associated torsion classes of non-rigid $T$1 cannot be compact, else they would be functorially finite, contradicting non-rigidity.

Technical Advances

The paper introduces and systematically analyzes numerically disjoint torsion classes and their polyhedral nature in the Grothendieck group. These geometric and arithmetic separation conditions are used to characterize functorially finite torsion classes in terms of bicompactness, further deepening the lattice-theoretic perspective.

The interplay between the wall-and-chamber combinatorics (g-fan completions), silting theory, and stability conditions in the sense of King and Baumann-Kamnitzer-Tingley underpins much of the technical apparatus.

Implications and Perspectives

The results enforce a strong identification of bicompact, compact, and functorially finite torsion classes in substantial settings, especially for tame and hereditary algebras. This characterization not only clarifies the role of silting and tilting theory in the module category but also has implications for the generative properties of torsion pairs, module lattices, and the geometry of brick objects.

Regarding brick infinite algebras, the logical implication between Demonet’s and Enomoto’s conjectures provides a hierarchy of finiteness conditions and points to future work on the combinatorics of walls, non-rigid points in Grothendieck group fans, and eventually settling the full scope of these conjectures for general finite-dimensional algebras.

The development offers tools for further exploration of wide subcategories, generic presentation spaces, and the construction of infinite semibricks via lattice points, paving the way for deeper connections between representation theory, polyhedral geometry, and homological algebra.

Conclusion

The work unifies several strands of torsion theory, silting theory, and brick combinatorics by providing necessary and sufficient conditions for bicompact torsion classes to be functorially finite under various algebraic conditions. The equivalences established solidify the connections between internal generation, functorial properties, and geometric data of the Grothendieck group. Further, the implications for brick infinite algebras and the conjectures of Enomoto and Demonet affirm the rich structure yet to be fully understood in module category lattices and stability conditions (2604.04505).