Projectively Wakamatsu Tilting Modules over One-Point Extensions

Abstract: Let $Γ= Λ[M]$ be the one-point extension of an algebra $Λ$ by a $Λ$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{mod}\,Λ$ to $\mathrm{mod}\,Γ$ by adding the new projective module, and prove that this lifting process perfectly preserves mutation relations under certain homological conditions. Furthermore, for source point extensions of representation-finite algebras, we obtain a complete classification of PWT $Γ$-modules in terms of those over $Λ$. In particular, we establish a bijection [ \mathrm{PWT}(Γ) \longleftrightarrow \mathrm{PWT}(Λ) \coprod \mathrm{RPWT}(Λ, S_i). ] which yields the counting formula about $|\mathrm{PWT}(Γ)|$.

• The paper presents an explicit method for lifting projectively Wakamatsu tilting modules under one-point extensions, ensuring homological consistency.
• It demonstrates that mutation properties, including left mutations and minimal approximations, are preserved between the base and extended algebras.
• The classification yields a precise counting formula unifying PWT modules from the original algebra with additional summands introduced at the extension vertex.

Projectively Wakamatsu Tilting Modules and One-Point Extensions

Introduction

Tilting theory serves as a central tool in the representation theory of finite-dimensional algebras, with broad ramifications for module categories, homological algebra, and connections to derived and cluster categories. Among generalizations of tilting modules, Wakamatsu tilting modules, introduced to facilitate the study of derived equivalences and Gorenstein homological structures, have recently been refined by Enomoto's notion of projectively Wakamatsu tilting (PWT) modules. This paper, "Projectively Wakamatsu Tilting Modules over One-Point Extensions" (2604.10118), provides a rigorous analysis of the behavior and classification of PWT modules under one-point extensions of algebras, establishing both explicit lifting techniques and precise combinatorial correspondences.

Preliminaries

A $\Lambda$-module $T$ is self-orthogonal if $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$. Given such $T$, the right perpendicular category $T^{\perp_\Lambda}$ consists of all modules $X$ with $\operatorname{Ext}^i_\Lambda(T, X)=0$ for all $i>0$. A module $T$ is projectively Wakamatsu tilting (PWT) if it is self-orthogonal and an Ext-progenerator of $T^{\perp_\Lambda}$, i.e., every $T$0 admits an epimorphism from a direct sum of $T$1 with kernel again in $T$2.

The PWT property interpolates between classical tilting and general Wakamatsu tilting. In representation-finite settings, PWT, maximal self-orthogonal, and Wakamatsu tilting modules coincide. A fundamental algebraic construction studied in this context is the one-point extension: Given a finite-dimensional algebra $T$3 and a module $T$4, the one-point extension $T$5 adds a new vertex and module structure organized via a matrix algebra. Module categories between $T$6 and $T$7 are connected by exact extension and restriction functors, which are critical in describing homological correspondences.

Lifting and Mutation Preservation of PWT Modules

The main contribution is an explicit lifting method for PWT modules under one-point extensions. Given a PWT module $T$8 over $T$9, if the extension module $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$0, the algebraic sum $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$1—where $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$2 is the new projective at the extension vertex and $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$3 is the extension functor—constructs a PWT module for $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$4. This approach utilizes the adjointness and exactness properties of $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$5 and the adjunction isomorphisms for $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$6, providing sufficient and necessary homological conditions for liftability.

A salient feature of $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$7-tilting and PWT frameworks is the mutation procedure, analogous to simple transpositions in tilting module quivers. This paper establishes that, under suitable homological constraints—specifically, $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$8 must lie in the intersection $\operatorname{Ext}_\Lambda^{>0}(T,T) = 0$9—the mutation relations among PWT modules in $T$0 are perfectly preserved after lifting. Thus, left mutations in $T$1 are mirrored as left mutations in $T$2, with clear control over the approximation and exact sequence structures. The proof leverages conditions on the vanishing of certain $T$3 spaces ensuring the surjectivity required by left-approximation properties after applying the extension functor.

Classification for Source-Point Extensions

A major result concerns source-point extensions—one-point extensions $T$4 where $T$5 is a simple injective module at a source vertex. The paper achieves a complete classification of PWT modules for $T$6 in terms of PWT modules over $T$7: $T$8 where $T$9 denotes modules in $T^{\perp_\Lambda}$0 with $T^{\perp_\Lambda}$1 removed as a direct summand. This yields the counting formula

$T^{\perp_\Lambda}$2

This combinatorial identity clarifies the exact increase in the number of isomorphism classes of basic PWT modules under source-point extensions and settles the relationship between new projective and injective summands introduced by the extension vertex.

The classification technique relies on structural features: all projective-injective modules must appear as summands in maximal self-orthogonal (hence PWT) modules, and the homological vanishing conditions are carefully traced through the functors and canonical exact sequences associated to $T^{\perp_\Lambda}$3. A dichotomy, according to whether the new simple module $T^{\perp_\Lambda}$4 is or is not a summand, stratifies the construction to ensure bijectivity.

There is a sharp contrast in behavior with classical tilting modules: for tilting, the number does not increase under source point extension, reflected by the fact that the new simple typically has large projective dimension and so does not participate in basic tilting configurations.

Examples and Explicit Calculations

To support these theoretical developments, the paper provides explicit computations with Nakayama and path algebras of small quivers, demonstrating:

• Enumeration of basic PWT modules over $T^{\perp_\Lambda}$5 and after one-point extension $T^{\perp_\Lambda}$6;
• Concrete realization of the bijection, with modules written out explicitly;
• Verification of mutation commutativity after extension, evidenced by the persistence of minimal left approximation property for summands and the invariance of cokernel structure.

Such examples illustrate the abstract lifting and classification process in settings where direct calculation is feasible and support the claim of perfect preservation of mutation quivers for PWT modules under these extensions.

Theoretical and Practical Implications

The main results establish essential functorial and numerical properties for PWT modules vis-à-vis one-point extensions, solidifying the relationship between module categories of $T^{\perp_\Lambda}$7 and $T^{\perp_\Lambda}$8. This unifies and generalizes several strands: the inductive construction of representation-finite algebras, the role of projective-injective summands, and the ladder of generalizations from tilting through Wakamatsu tilting to PWT modules.

From a theoretical perspective, this classification resolves, for the PWT setting, the change in mutation quivers and module configurations under these algebra extensions, which is meaningful for both the structure theory and numerical invariants of module categories. Practically, the methodology supplies a constructive means for generating new PWT modules in larger algebras, with predictable relationships to the corresponding modules in the base algebra.

Prospective developments could include:

• Extension of lifting methods to infinite or wild representation types, possibly under restrictions;
• Exploration of the interaction with derived categories, stable equivalences, or the corresponding endomorphism algebras;
• Analysis of how these correspondences impact support $T^{\perp_\Lambda}$9-tilting modules, silting objects, or the combinatorial realization of cluster structures in broader settings.

Conclusion

The paper provides a complete, homologically explicit framework for lifting, mutating, and classifying PWT modules over one-point extensions of representation-finite algebras. Numerical counting formulas and bijections reveal a precise structure in the passage from $X$0 to $X$1, and the work advances understanding of how the intricate homological properties of modules behave under algebra extensions intimately tied to quiver data. This positions PWT theory as a robust tool for both explicit construction and theoretical analysis in the representation theory of finite-dimensional algebras.