Lukas Tilting Module in Tame Algebras

• Lukas tilting module is an infinite-dimensional tilting module over tame hereditary algebras, defined by its p-filtered structure and absence of indecomposable preprojective summands.
• It is constructed as a countably generated direct limit of preprojective modules satisfying tilting axioms, ensuring controlled module approximation and Ext-orthogonality.
• Its exceptional nonlocalizability, unique endomorphism ring properties, and central role in classifying infinite-dimensional tilting modules underscore its theoretical and practical significance.

The Lukas tilting module is a distinguished infinite-dimensional tilting module arising over tame hereditary algebras, most notably over the Kronecker algebra. It is characterized as the unique large tilting module whose tilting class consists of all modules without indecomposable preprojective summands. The construction, properties, and role of the Lukas tilting module establish it as a central object in the classification of infinite-dimensional tilting modules for tame hereditary algebras, particularly highlighting its exceptionality compared to modules constructed via universal localization (Hügel et al., 2010).

1. Tame Hereditary Algebras and the Kronecker Context

Let $R = kQ$ denote the path algebra of the Kronecker quiver $Q$—a quiver with two vertices and two parallel arrows from vertex 1 to 2—over an algebraically closed field $k$. This algebra exemplifies the class of tame hereditary algebras. The category $\mathrm{Mod}\,R$ of (right) $R$-modules admits an Auslander–Reiten (AR) component classification:

• $p$: indecomposable preprojective modules (finite length, defect $>0$),
• $t$: indecomposable regular modules (organized in tubes, defect $0$),
• $q$: indecomposable preinjective modules (defect $<0$).

Every finite-length indecomposable $R$-module belongs to exactly one of these classes. The regular components, or tubes, play a crucial role in the construction of universal localization tilting modules, while the preprojective class $p$ is intrinsic to the definition of the Lukas tilting module (Hügel et al., 2010).

2. Construction and Defining Properties of the Lukas Tilting Module

The Lukas tilting module $L$ is constructed as a countably generated $p$-filtered module. A module is $p$-filtered if it admits a filtration whose successive quotients belong to $p$. F. Lukas's original construction (as formalized by Kerner–Trlifaj) exhibits a module $L$ with the following tilting axioms:

• (T1) $\operatorname{proj\,dim} L \leq 1$,
• (T2) $\operatorname{Ext}^1_R(L,L^{(\kappa)}) = 0$ for any cardinal $\kappa$,
• (T3) There exists an exact sequence $0 \to R \to L_0 \to L_1 \to 0$ with $L_0, L_1 \in \operatorname{Add} L$.

Alternatively, $L$ can be explicitly realized as the direct limit of an ascending chain of preprojective modules

$P_0 \to P_1 \to P_2 \to \cdots$

with $L = \varinjlim P_n$, where each $P_n \to P_{n+1}$ is a minimal right $p$-approximation. The vanishing $\operatorname{Ext}^1_R(L,L) = 0$ requires the direct system to approximate all $p$-modules in a controlled manner, and the sequence obtained from cokernels of the maps yields axiom (T3) (Hügel et al., 2010).

3. The Tilting Class $\operatorname{Gen} L$ and Its Characterization

For any tilting module $T$, the tilting class is $$\operatorname{Gen}\,T = \{ X \in \mathrm{Mod}\,R \mid \operatorname{Ext}^1_R(T,X) = 0\}$$. In the Lukas setting, explicit calculation yields

$$\operatorname{Gen}\,L = p_- := \{ X \mid X\text{ has no indecomposable preprojective direct summands} \}.$$

If $X$ has a direct summand in $p$, then $$\operatorname{Ext}^1_R(L,X) \cong D \operatorname{Hom}_R(P,L) \neq 0$$ because $L$ is $p$-filtered and $\operatorname{Hom}_R(P,-)$ detects the top layers of that filtration. Conversely, modules without preprojective parts are Ext-orthogonal to $p$ and thus belong to $\operatorname{Gen}\,L$. Therefore, $p_-$ is the minimal (infinite dimensional) tilting class not admitting a finite-dimensional generator, and $L$ is its unique tilting module [(Hügel et al., 2010), Example 1.4].

4. Exceptionality and Non-localizability of the Lukas Module

According to Angeleri Hügel–Sánchez (Corollary 2.8 in (Hügel et al., 2010)), all large tilting modules over the Kronecker algebra are equivalent to either:

• a module of the form $T_U = R_U \oplus (R_U/R)$, where $R_U$ is a universal localization at a union of tubes $U$ and $R_U / R$ is a direct sum of corresponding Prüfer modules,
• or the Lukas tilting module $L$.

In this dichotomy, $L$ is the only large tilting module not arising via universal localization. For any nonempty $U$, the associated tilting class $$U_+ := \{X \mid \operatorname{Ext}^1_R(U,X)=0 \text{ for all } U \in U\}$$ contains preprojective summands unless $U = \emptyset$. In the trivial case $U = \emptyset$, $T_U = R$ is finite-dimensional. Thus, $L$ is the exceptional, non-localizable infinite-dimensional tilting module in this scheme (Hügel et al., 2010).

5. Exact Sequences and Endomorphism Ring Structure

Tilting modules arising from universal localization enjoy exact sequences

$0 \to R \to R_U \to R_U/R \to 0.$

Although no such localization exists for $L$, there remains an exact sequence of the form

$$0 \to R \to L_0 \to L_1 \to 0, \quad L_0, L_1 \in \operatorname{Add}\,L$$

by axiom (T3). The endomorphism ring $\operatorname{End}_R L$ is a serial noetherian ring such that the simple modules correspond to the $p$-composition factors of $L$. Furthermore, $L$ is endofinite—it has finite length as a module over its endomorphism ring—and is noetherian over $\operatorname{End}_R L$ [(Hügel et al., 2010), Corollary 9].

A summary of the key ring-theoretic properties:

Property	Statement	Reference
Endofinite	$L$ has finite length over $\operatorname{End}_R L$	(Hügel et al., 2010)
Noetherian	$\operatorname{End}_R L$ is noetherian; $L$ is noetherian as a module	(Hügel et al., 2010)
Serial	$\operatorname{End}_R L$ is a serial ring	(Hügel et al., 2010)
Ext-orthogonality	$\operatorname{Ext}^1_R(L,L^{(\kappa)})=0$; $L$ generates exactly $p_-$	(Hügel et al., 2010)

6. Role in the General Classification over Tame Hereditary Algebras

For an arbitrary tame hereditary algebra $R$, every large (infinite-dimensional) tilting module $T$ decomposes uniquely as

$T = Y \oplus T'$

where $Y$ is a finite-dimensional branch module from non-homogeneous tubes, and the torsion-free part $T'$ is associated to a universal localization $R \to R'$ of $R$. There are two possibilities for $T'$:

• $T' \cong L_{R'}$ is a Lukas tilting module over $R'$; it generates exactly the $p'_-$-class over $R'$
• or $T' \cong R'_U \oplus (R'_U/R')$, corresponding to the Schofield–Crawley-Boevey universal localization.

Thus, the Lukas tilting module (possibly after localization) constitutes the only genuinely exotic piece in the general classification of infinite-dimensional tilting modules for tame hereditary algebras [(Hügel et al., 2010), Theorems A,B].

7. Summary and Significance

The Lukas tilting module $L$ fundamentally distinguishes itself among large tilting modules for the Kronecker algebra by its unique tilting class $p_-$ and its construction as a $p$-filtered direct limit, satisfying the minimal possible tilting class condition. Its absence from the universal localization framework, coupled with its classified role in broader settings, situates it as the prototypical example of a large, nonlocalizable tilting module. The structure of its endomorphism ring and its relationship to modules filtered by preprojectives underscores its centrality in artin algebra tilting theory (Hügel et al., 2010).