June 22, 2026
We classify torsion pairs in an essentially small abelian category through cosilting subsets of the Ziegler spectrum of the ind-completion of the abelian category. For Artin algebras, this classification is reformulated as an infinite analog of \(\tau\)-tilting theory, where torsion classes correspond to support \(\tau\)-tilting subsets of the Ziegler spectrum and torsion-free classes correspond to support \(\tau^-\)-tilting subsets. We further express the classification through ideals of the module category, thereby obtaining a formulation that involves finite length modules only. The developed theory is applied to study generic bricks and generic \(\tau^-\)-rigid modules, in particular for tame algebras, for which we show that these classes of modules coincide. We also recover a result of Bautista, Pérez and Salmerón stating that a tame algebra admits infinitely many bricks of a fixed dimension if and only if there exists a generic brick. Finally, we prove that every algebra whose Krull-Gabriel dimension is defined satisfies the brick version of the second Brauer-Thrall conjecture.
Torsion pairs are important objects in the representation theory of finite dimensional algebras (more generally, Artin algebras \(A\)). In the seminal work of Adachi, Iyama and Reiten [1], \(\tau\)-tilting theory was developed to study the functorially
finite torsion pairs inside the category \(\textrm{mod}\,A\) of finite length \(A\)-modules. We build on recent developments by Angeleri Hügel, Laking and Sentieri [2], [3], who make use of the Ziegler spectrum to study arbitrary torsion pairs in \(\textrm{mod}\,A\).
Originating in model theory [4], the Ziegler spectrum \(\textrm{Zsp}\,A\) is a topological space that has become an
effective tool within representation theory. The points of \(\textrm{Zsp}\,A\) are the indecomposable pure-injective \(A\)-modules, which include all indecomposable finite length \(A\)-modules as discrete points. Our main contribution is to show that the Auslander-Reiten translation \(\tau\), which lies at the heart of \(\tau\)-tilting
theory, can also be used to study arbitrary torsion pairs through the Ziegler spectrum, leading to what we call infinite \(\tau\)-tilting theory.
The Auslander-Reiten translation \(\tau\) is usually considered only for finite length \(A\)-modules, but it was shown by Krause [5] that \(\tau\) naturally extends to infinite length \(A\)-modules and induces a homeomorphism \(\tau \colon \textrm{Zsp}\,A \setminus \mathrm{Proj}\,A \rightarrow \textrm{Zsp}\,A \setminus \textrm{Inj}\,A\). Building on this extension, we introduce the following notions, which may be viewed as infinite analogs of the corresponding concepts in classical \(\tau\)-tilting theory.
A subset \(U \subseteq \textrm{Zsp}\,A\) is \(\tau\)-rigid if \(\textrm{Hom}_A^\mathrm{fin}(U, \tau U) = 0\). That is, there are no non-zero morphisms \(X \to \tau Y\) with finite length image for every \(X,Y \in U\).
We call \((U,V)\) a \(\tau\)-rigid pair if \(U \subseteq \textrm{Zsp}\,A\) is \(\tau\)-rigid and \(V \subseteq \mathrm{Proj}\,A\) fulfills \(\textrm{Hom}_A(V, U )= 0\). There is a natural order on \(\tau\)-rigid pairs given by \((U,V)\leq (U',V')\) if \(U \subseteq U'\) and \(V\subseteq V'\).
A subset \(U \subseteq \textrm{Zsp}\,A\) is support \(\tau\)-tilting if there exists a maximal \(\tau\)-rigid pair \((U,V)\).
The point of \(\tau\)-rigid subsets \(U \subseteq \textrm{Zsp}\,A\) is that they give rise to torsion classes inside \(\textrm{mod}\,A\). More precisely,
the collection \(\textrm{gen}\,U\) of all finite length quotients of direct sums of modules in \(U\) is a torsion class. One may define dual notions, which are related to torsion-free
classes, using the inverse Auslander-Reiten translation \(\tau^-\) instead. This leads to the following classifications.
Theorem A. (18, 32) There are one-to-one correspondences \[\begin{align} \begin{Bmatrix} \text{torsion classes}\\ \mathcal{T}\subseteq \mathrm{mod}\,A \end{Bmatrix} &\longleftrightarrow \begin{Bmatrix} \text{support \tau-tilting subsets}\\ {U}\subseteq \textrm{Zsp}\,A \end{Bmatrix}\\
\begin{Bmatrix} \text{torsion-free classes}\\ \mathcal{F}\subseteq \mathrm{mod}\,A \end{Bmatrix} &\longleftrightarrow \begin{Bmatrix} \text{support \tau^--tilting subsets}\\ {U}'\subseteq \textrm{Zsp}\,A \end{Bmatrix}
\end{align}\] given by \(U \mapsto \textrm{gen}\,U\) and \(U' \mapsto \textrm{cogen}\,U'\). Moreover, support \(\tau\)-tilting subsets and
support \(\tau^-\)-tilting subsets are closed sets of \(\textrm{Zsp}\,A\).
The theorem above involves modules that are possibly of infinite length, which makes the situation more complicated than in classical \(\tau\)-tilting theory. To address this difficulty, we consider ideals of \(\mathrm{mod}\,A\), that is, collections of morphisms that satisfy closure properties similar to ideals of rings. For \(U \subseteq \textrm{Zsp}\,A\) let \(\langle U
\rangle\) be the collection of all morphisms in \(\textrm{mod}\,A\) factoring through a product of modules in \(U\). Krause showed that the assignment \(U
\mapsto \langle U \rangle\) induces a one-to-one correspondence between closed sets \(U \subseteq \textrm{Zsp}\,A\) and certain ideals \(\mathcal{I}\) of \(\textrm{mod}\,A\), see [6]. Now \(U\) is \(\tau\)-rigid if and
only if the associated ideal \(\mathcal{I}\) satisfies \(\tau \mathcal{I} \circ \mathcal{I} = 0\), where \(\tau \mathcal{I}\) may be defined via the
Auslander-Reiten translation on morphisms. Such an ideal is called \(\tau\)-rigid. This naturally gives rise to the notions of support \(\tau\)-tilting ideals and
support \(\tau^{-}\)-tilting ideals, leading to the following correspondences.
Theorem B. (24, 29) There are one-to-one correspondences \[\begin{align} \begin{Bmatrix} \text{torsion classes}\\ \mathcal{T}\subseteq \mathrm{mod}\,A \end{Bmatrix} &\longleftrightarrow \begin{Bmatrix} \text{support \tau-tilting ideals}\\ \mathcal{I}\text{ of } \textrm{mod}\,A
\end{Bmatrix}\\ \begin{Bmatrix} \text{torsion-free classes}\\ \mathcal{F}\subseteq \mathrm{mod}\,A \end{Bmatrix} &\longleftrightarrow \begin{Bmatrix} \text{support \tau^--tilting ideals}\\ \mathcal{I}'\text{ of } \textrm{mod}\,A \end{Bmatrix}
\end{align}\] given by \(\mathcal{I} \mapsto \textrm{gen}\,\mathcal{I}\) and \(\mathcal{I}' \mapsto \textrm{cogen}\,\mathcal{I}'\).
Here \(\textrm{gen}\,\mathcal{I}\) denotes the collection of all \(X\in \textrm{mod}\,A\) that admit an epimorphism \(Y\to X\) in \(\mathcal{I}\) and \(\textrm{cogen}\,\mathcal{I}'\) is defined dually. The following example illustrates the usefulness of the ideal approach. If \(U \subseteq
\textrm{Zsp}\,A\) is an infinite collection of finite length \(A\)-modules with closure \(\bar{U}\), then the ideal \(\langle \bar{U}\setminus
U\rangle\) equals the collection of all morphisms that factor through a direct sum of modules in \(U\setminus V\) for all finite subsets \(V\subseteq U\). If \(U\) is, in some sense, generically \(\tau\)-rigid, the ideal can easily be shown to be \(\tau\)-rigid and so \(\bar{U}\setminus
U\) is a non-empty \(\tau\)-rigid set consisting of infinite length modules.
Another advantage of the ideal approach is that we can apply duality, as we are only dealing with finite length \(A\)-modules. This is not directly possible for infinite length \(A\)-modules. In fact, we first prove the \(\tau^-\)-part of Theorem A to deduce the \(\tau^-\)-part of Theorem B, implying the \(\tau\)-part of Theorem B via duality, which is used to show the \(\tau\)-part of Theorem A. Thus, the mentioned results are based on the \(\tau^-\)-part of
Theorem A, which is a special instance of a more general classification. Namely, for an essentially small abelian category \(\mathcal{C}\) we characterize torsion pairs through cosilting subsets of the Ziegler spectrum
\(\textrm{Zsp}\,\bar{\mathcal{C}}\) of the ind-completion \(\bar{\mathcal{C}}\). For the definition of cosilting subsets, see Section 2.
Theorem C. (10) Let \(\mathcal{C}\) be an essentially small abelian category. There is a one-to-one correspondence \[\begin{Bmatrix} \text{torsion-free classes}\\ \mathcal{F}\subseteq \mathcal{C} \end{Bmatrix} \longleftrightarrow \begin{Bmatrix} \text{cosilting subsets}\\ {U}\subseteq \textrm{Zsp}\,\bar{\mathcal{C}} \end{Bmatrix}\] given by
\(\mathcal{F} \mapsto \textrm{Inj}\,\bar{\mathcal{F}}\) and \(U \mapsto \textrm{cogen}\,U\). Moreover, cosilting subsets are closed.
For the module category of an Artinian ring, a similar classification as above was established by Angeleri Hügel, Laking and Sentieri [2]. Their approach is based on
2-term complexes of injective objects and the Ziegler spectrum of a derived category, while ours relies on a connection between exact structures and purity established in [7]. We naturally consider a torsion-free class \(\mathcal{F}\subseteq \mathcal{C}\) as well as its direct limit closure \(\bar{\mathcal{F}} =\varinjlim
\mathcal{F}\) as an exact category, since they are extension-closed subcategories. The collection of indecomposable injective objects in the exact category \(\bar{\mathcal{F}}\) then forms the associated cosilting
subset \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq \bar{\mathcal{C}}\).
Let us return to Artin algebras \(A\), where the cosilting subsets of \(\textrm{Zsp}\,A\) coincide with the support \(\tau^-\)-tilting subsets. This gives
support \(\tau^{-}\)-tilting subsets a natural interpretation in terms of injective objects in an exact category, an interpretation that is not available for support \(\tau\)-tilting
subsets. Thus, the \(\tau^{-}\)-perspective is more natural when studying modules of infinite length. We focus on generic modules, that is, indecomposable \(A\)-modules \(X\) of infinite length that have finite length over \(\mathrm{End}_A(X)\). Generic modules form closed points \(\{X\} \subseteq \textrm{Zsp}\,A\) and we
investigate them in terms of \(\tau^-\)-rigidity. This property is related to bricks, which are \(A\)-modules whose endomorphism ring is a division algebra. We apply the developed
theory of infinite \(\tau\)-tilting theory to show the following result.
Theorem D. (40, 42) Let \(A\) be a finite dimensional
algebra over an algebraically closed field. Consider the following statements.
There are infinitely many non-isomorphic finite dimensional bricks.
There are infinitely many non-isomorphic finite dimensional bricks of the same dimension.
There exists a generic brick.
If \(A\) is tame, then (2) and (3) are equivalent, and if the Krull-Gabriel dimension of \(A\) is defined, then all of the statements are equivalent.
The equivalences of the statements in the above theorem were conjectured by Mousavand and Paquette [8], and the equivalence of (2) and (3) has already been proven for
tame algebras by Bautista, Pérez and Salmerón [9] using matrix reduction techniques. The Krull-Gabriel dimension, introduced by Geigle [10], is a measure for the complexity of the module category. A conjecture of Prest states that the Krull-Gabriel dimension of \(A\) is defined if
and only if \(A\) is domestic [11], which is known for several classes of algebras, see [12] for a detailed discussion. Lastly, we show the following surprising connection between \(\tau^-\)-rigidity and bricks over tame algebras.
Theorem E. (43) For a tame finite dimensional algebra \(A\) over an algebraically closed field, a generic module \(X\) is a brick if and only if \(\{X\}\) is \(\tau^-\)-rigid.
Acknowledgments. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the Walter Benjamin Programme – Project number 580964112.
We will later use the fact that torsion-free classes naturally form exact categories and consider big objects in order to classify them. A main tool to achieve this is the theory of purity for finitely accessible categories with products and its connection with exact structures established in [7].
An additive category \(\mathcal{A}\) is called finitely accessible if \(\mathcal{A}\) has filtered colimits, the subcategory \(\textrm{fp}\,\mathcal{A}\) of finitely presented objects is essentially small, and every object in \(\mathcal{A}\) is a filtered colimit of finitely presented objects. Recall that \(X\in \mathcal{A}\) is finitely presented if \(\textrm{Hom}_{\mathcal{A}}(X,-)\) commutes with filtered colimits. For an essentially small additive category \(\mathcal{C}\) its ind-completion \(\bar{\mathcal{C}}\) is a finitely accessible category. This assignment induces the following bijection due to Crawley-Boevey.
Theorem 1. [13] There exists a one-to-one correspondence, up to equivalence, between finitely accessible categories \(\mathcal{A}\) and essentially small additive categories \(\mathcal{C}\) with split idempotents, given by \(\mathcal{A} \mapsto \textrm{fp}\,\mathcal{A}\) and \(\mathcal{C} \mapsto \bar{\mathcal{C}}\).
Let \(\mathcal{A}\) be a finitely accessible category. A filtered colimit of split exact sequences in \(\mathcal{A}\) is a pure-exact sequence. The pure-exact sequences form an exact structure and the injective objects are called pure-injective. We are particularly interested in the case when \(\mathcal{A}\) has products. Then, the isomorphism classes of pure-injective indecomposable objects form a topological space \(\textrm{Zsp}\,\mathcal{A}\), the Ziegler spectrum of \(\mathcal{A}\). For more details, see for example [14].
For an additive category \(\mathcal{C}\) and a full additive subcategory \(\mathcal{D} \subseteq \mathcal{C}\) a morphism \(f\colon X \to C_X\) in \(\mathcal{C}\) is a left \(\mathcal{D}\)-approximation of \(X\) if \(C_X \in \mathcal{D}\) and \(\textrm{Hom}_{\mathcal{D}}(f,Y)\) is surjective for all \(Y\in \mathcal{D}\). Moreover \(\mathcal{D}\) is covariantly finite if every \(X\in \mathcal{C}\) admits a left \(\mathcal{D}\)-approximation.
Proposition 2. [14] Let \(\mathcal{A}\) be a finitely accessible category with products, \(\mathcal{D} \subseteq \textrm{fp}\,\mathcal{A}\) a covariantly finite subcategory, and \(\mathcal{B} = \varinjlim \mathcal{D}\) the closure of \(\mathcal{D}\) in \(\mathcal{A}\) under filtered colimits. Then \(\mathcal{B}\) is a finitely accessible category with products such that \(\textrm{fp}\,\mathcal{B} = \mathcal{D}\) and \(\textrm{Zsp}\,\mathcal{B} \subseteq \textrm{Zsp}\,\mathcal{A}\) is a closed subset.
Next, we discuss the connection with exact structures. For the definition of exact structures and basic properties we refer to the exposition of Bühler [15]. We use the terminology of admissible short exact sequences and admissible monomorphisms.
Following Positselski [16], an exact structure on a finitely accessible category \(\mathcal{A}\) is locally coherent if every admissible short exact sequence is a filtered colimit of admissible short exact sequences \(0 \to X \to Y \to Z \to 0\) with \(X,Y,Z \in \textrm{fp}\,\mathcal{A}\). For example, the pure-exact sequences form a locally coherent exact structure. The following connects locally coherent exact structures with the Ziegler spectrum.
Theorem 3. [7] Let \(\mathcal{A}\) be a finitely accessible category with products and consider a locally coherent exact structure on \(\mathcal{A}\). Every \(X\in \mathcal{A}\) admits an admissible monomorphism \(X\to Q\) where \(Q\) is a product of indecomposable injectives. Moreover, the isomorphism classes of indecomposable injectives form a closed set in \(\textrm{Zsp}\,\mathcal{A}\).
Let \(\mathcal{C}\) be an essentially small abelian category. A torsion pair \((\mathcal{T}, \mathcal{F})\) of \(\mathcal{C}\) is a pair of full subcategories such that \(\textrm{Hom}_{\mathcal{C}}(\mathcal{T}, \mathcal{F}) = 0\) and every \(X\in \mathcal{C}\) admits a short exact sequence \(0 \to X' \to X \to X'' \to 0\) with \(X' \in \mathcal{T}\) and \(X'' \in \mathcal{F}\). Moreover \(\mathcal{T}\) is a torsion class and \(\mathcal{F}\) a torsion-free class. Recall that a torsion pair is uniquely determined by the torsion-free class. For our classification, we restrict to torsion-free classes and use the following characterization by Gentle and Todorov.
Proposition 4. [17] Let \(\mathcal{C}\) be an abelian category and \(\mathcal{F}\subseteq \mathcal{C}\) a full subcategory. Then \(\mathcal{F}\) is a torsion-free class if and only if \(\mathcal{F}\) is covariantly finite and closed under extensions as well as subobjects.
From now on we fix an essentially small abelian category \(\mathcal{C}\). By 1 the ind-completion \(\bar{\mathcal{C}}\) is finitely accessible and we may identify \(\textrm{fp}\,\bar{\mathcal{C}} = \mathcal{C}\). In fact \(\bar{\mathcal{C}}\) is a Grothendieck category [18] so it has products. In particular, the Ziegler spectrum \(\textrm{Zsp}\,\bar{\mathcal{C}}\) is defined.
For a torsion-free class \({\mathcal{F}} \subseteq {\mathcal{C}}\) let \(\bar{\mathcal{F}} = \varinjlim \mathcal{F}\) be the closure of \(\mathcal{F}\) in \(\bar{\mathcal{C}}\) under filtered colimits. Since \(\mathcal{F}\) is covariantly finite in \(\mathcal{C}\) it follows from 2 that \(\bar{\mathcal{F}}\) is a finitely accessible category with products and \(\textrm{fp}\,\bar{\mathcal{F}} = \mathcal{F}\). Moreover, the Ziegler spectrum \(\textrm{Zsp}\,\bar{\mathcal{F}}\) is a closed subset of \(\textrm{Zsp}\,\bar{\mathcal{C}}\). Note that again \(\bar{\mathcal{F}}\subseteq \bar{\mathcal{C}}\) is a torsion-free class [13] and in particular \(\bar{\mathcal{F}}\) is closed under extensions. Thus, there is a canonical exact structure on \(\bar{\mathcal{F}}\) given by all short exact sequences \(0\to X\to Y \to Z\to 0\) in \(\bar{\mathcal{C}}\) with \(X,Y,Z\in\bar{ \mathcal{F}}\), see for example [15]. In this sense, we regard \(\bar{\mathcal{F}}\) as an exact category.
Lemma 5. Let \(\mathcal{F}\) be a torsion-free class in \(\mathcal{C}\). Then \(\bar{\mathcal{F}}\) is a locally coherent exact category.
Proof. Let \(0 \to X \to Y \to Z \to 0\) be a short exact sequence in \(\bar{\mathcal{C}}\) with \(X,Y,Z \in \bar{\mathcal{F}}\). Since \(\textrm{fp}\,\bar{\mathcal{F}} = \mathcal{F}\) there exists a filtered colimit \(Z = \varinjlim Z_i\) with \(Z_i \in \mathcal{F}\). Consider the commutative diagram of short exact sequences \[\begin{tikzcd} 0 \arrow[r] & X \arrow[r] & Y \arrow[r] & Z \arrow[r] & 0 \\ 0 \arrow[r] & X \arrow[u, equals] \arrow[r] & P_i \arrow[u] \arrow[r] & Z_i \arrow[r]\arrow[u] & 0 \end{tikzcd}\] where the right square is a pullback diagram. Then \(P_i \in \bar{\mathcal{F}}\) and there are filtered colimits \(P_i = \varinjlim P_{ij}\) with \(P_{ij} \in \mathcal{F}\). Let \(K_{ij} \in \mathcal{F}\) be the kernel and \(I_{ij} \in \mathcal{F}\) the image of the composition \(P_{ij} \to P_i \to Z_i\). Again, there are commutative diagrams of short exact sequences \[\begin{tikzcd} 0 \arrow[r] & X \arrow[r] & P_{i} \arrow[r] & Z_i \arrow[r] & 0 \\ 0 \arrow[r] & K_{ij} \arrow[u] \arrow[r] & P_{ij} \arrow[u] \arrow[r] & I_{ij} \arrow[r]\arrow[u] & 0. \end{tikzcd}\] Since filtered colimits are exact inside the Grothendieck category \(\bar{\mathcal{C}}\) it follows that the filtered colimit over \(j\) of the short exact sequences in the second row recovers the short exact sequence in the first row. Taking the filtered colimit over \(i\) we then recover the original short exact sequence. Note that \(0 \to K_{ij} \to P_{ij} \to I_{ij} \to 0\) is an admissible short exact sequence inside the exact category \(\bar{\mathcal{F}}\) that only involves finitely presented objects. Thus \(\bar{\mathcal{F}}\) is locally coherent. ◻
For a torsion-free class \(\mathcal{F}\subseteq \mathcal{C}\) let \(\textrm{Inj}\,\bar{\mathcal{F}}\) denote the set of isomorphism classes of indecomposable injective objects inside the locally coherent exact category \(\bar{\mathcal{F}}\). By 3 it follows that \(\textrm{Inj}\,\bar{\mathcal{F}}\) is a closed subset of \(\textrm{Zsp}\,\bar{\mathcal{F}}\) and we already observed that \(\textrm{Zsp}\,\bar{\mathcal{F}}\) is a closed subset of \(\textrm{Zsp}\,\bar{\mathcal{C}}\). In combination, we obtain the following result, which may be seen as a generalization of [2].
Corollary 6. Let \(\mathcal{F}\) be a torsion-free class in \(\mathcal{C}\). Then \(\textrm{Inj}\,\bar{\mathcal{F}}\) is a closed subset of \(\textrm{Zsp}\,\bar{\mathcal{C}}\).
The goal will be to classify torsion-free classes \(\mathcal{F} \subseteq \mathcal{C}\) via the associated closed subset \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\). This requires the following definition. (Note that for the mo- dule category of an Artinian ring, cogen-rigid pairs coincide with rigid pairs in [2].)
Definition 7. For a set \(U\) of objects in \(\bar{\mathcal{C}}\) we denote by \(\textrm{cogen}\,U \subseteq \mathcal{C}\) the full subcategory of all \(X\in \mathcal{C}\) that admit a monomorphism \(X \to Y\) in \(\bar{\mathcal{C}}\) where \(Y\) is a product of objects in \(U\).
A subset \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) is cogen-rigid* if \(\mathrm{Ext}_{\bar{\mathcal{C}}}^1 (\textrm{cogen}\,U, U) = 0\), every \(X\in U\) is a filtered colimit of objects in \(\textrm{cogen}\,U\), and \(\textrm{cogen}\,U\) is covariantly finite.*
We call \((U,V)\) a cogen-rigid* pair if \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) is cogen-rigid and \(V\subseteq
\textrm{Inj}\,\bar{\mathcal{C}}\) fulfills \(\textrm{Hom}_{\bar{\mathcal{C}}}(X, V) = 0\) for all products \(X\) of objects in \(U\). There is a
partial
order on cogen-rigid pairs given by \((U,V)\leq (U',V')\) if \(U \subseteq U'\) and \(V \subseteq V'\).*
A subset \(U\subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) is cosilting if there is a maximal cogen-rigid pair \((U,V)\).
Remark 8. Some of the properties for a subset \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) to be cogen-rigid hold automatically under suitable finiteness conditions on \(\mathcal{C}\).
Suppose that \(\mathcal{C}\) is Noetherian. That is, the lattice of subobjects of any object in \(\mathcal{C}\) fulfills the ascending chain condition. Then every \(X \in U\) is a filtered colimit of objects in \(\textrm{cogen}\,U\). Namely \(X = \varinjlim Y\) where the colimit goes over all finitely generated subobjects \(Y \subseteq X\). Being Noetherian implies that \(Y\) is finitely presented [19] and so each \(Y\) is contained in \(\textrm{cogen}\,U\).
Suppose that \(\mathcal{C}\) is Artinian. That is, the lattice of subobjects of any object in \(\mathcal{C}\) fulfills the descending chain condition. Then \(\textrm{cogen}\,U\) is covariantly finite. For \(X\in \mathcal{C}\) a left \(\textrm{cogen}\,U\)-approximation \(X\to C_X\) is given by the image of the canonical map \(f\colon X \to \prod Y\) where the product goes over all \(g\colon X \to Y\) with \(Y \in \textrm{cogen}\,U\). Being Artinian implies that \(\ker f = \bigcap \ker g\) for a finite intersection and so \(\textrm{Im}\,f \in \textrm{cogen}\,U\).
If \(\mathcal{C}\) is a length category, equivalently \(\mathcal{C}\) is Noetherian and Artinian, then by the previous observations a subset \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) is cogen-rigid if and only if \(\mathrm{Ext}_{\bar{\mathcal{C}}}^1 (\textrm{cogen}\,U, U) = 0\). Moreover, in this case \((U,V)\) is a cogen-rigid pair if and only if \(\textrm{Hom}_{\bar{\mathcal{C}}}(U,V) = 0\) with \(V\subseteq \textrm{Inj}\,\bar{\mathcal{C}}\). This follows from the fact that any product \(\prod X\) in \(\bar{\mathcal{C}}\) equals the union of its finite length subobjects, which are isomorphic to subobjects of finite direct sums \(\bigoplus X\leq \prod X\).
The definition of cogen-rigid subsets \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) is essentially made such that they give rise to torsion-free classes, see the following proposition.
Proposition 9. Let \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) be cogen-rigid and \(\mathcal{F} \subseteq \mathcal{C}\) a torsion-free class.
The subcategory \(\textrm{cogen}\,U \subseteq \mathcal{C}\) is a torsion-free class and \(U \subseteq \textrm{Inj}\,\overline{\textrm{cogen}\,U}\).
The subset \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) is cogen-rigid and \(\textrm{cogen}\,\textrm{Inj}\,\bar{\mathcal{F}} = \mathcal{F}\).
**Proof.* (1) Clearly \(\textrm{cogen}\,U\) is closed under subobjects. By definition of cogen-rigid subsets \(\textrm{cogen}\,U\) is covariantly finite. We are left to show that \(\textrm{cogen}\,U\) is closed under extensions to be a torsion-free class, see 4. Consider a short exact sequence \(0 \to X \to Y \to Z \to 0\) in \(\mathcal{C}\) with \(X,Z \in \textrm{cogen}\,U\). Let \(X \to \prod A\) be a monomorphism in \(\bar{\mathcal{C}}\) where the product only involves objects in \(U\). Taking a pushout \(P\) yields a commutative diagram \[\begin{tikzcd} 0 \arrow[r] & \prod A \arrow[r] & P \arrow[r] & Z \arrow[r] & 0\\ 0\arrow[r] & X \arrow[r] \arrow[u] & Y\arrow[u] \arrow[r] & Z \arrow[r] \arrow[u, equals] & 0 \end{tikzcd}\] with exact rows. Note that the canonical map \(\textrm{Ext}_{\bar{\mathcal{C}}}^1 (Z, \prod A) \to \prod \textrm{Ext}_{\bar{\mathcal{C}}}^1 (Z, A)\) is always a monomorphism. Because \(U\) is cogen-rigid, \(\mathrm{Ext}_{\bar{\mathcal{C}}}^1 (Z,A) = 0\) and so the short exact sequence in the first row splits. It follows that \(P \cong Z \oplus \prod A\) and since \(Z \in \textrm{cogen}\,U\) also \(Y \in \textrm{cogen}\,U\).*
Next, we prove that \(U \subseteq \textrm{Inj}\,\bar{\mathcal{G}}\) for \(\mathcal{G} = \textrm{cogen}\,U\). By definition of a cogen-rigid subset \(U\) is contained in \(\bar{\mathcal{G}}\). It is left to show that every \(X \in U\) is injective in the exact category \(\bar{\mathcal{G}}\). Let \(0 \to X \to Y \to Z \to 0\) be a short exact sequence in \(\bar{\mathcal{C}}\) with \(Z \in \bar{\mathcal{G}}\) and write \(Z = \varinjlim Z_i\) for \(Z_i \in \textrm{cogen}\,U\). Taking pullbacks \(P_i\) yields commutative diagrams \[\begin{tikzcd} 0 \arrow[r] & X \arrow[r] & Y \arrow[r] & Z \arrow[r] & 0\\ 0\arrow[r] & X \arrow[r] \arrow[u, equals] & P_i \arrow[u] \arrow[r] & Z_i \arrow[r] \arrow[u] & 0 \end{tikzcd}\] such that the short exact sequence in the first row is a filtered colimit of the short exact sequences in the second row, which split since \(\textrm{Ext}_{\bar{\mathcal{C}}}^1 (Z_i, X ) = 0\). It follows that \(0 \to X \to Y \to Z \to 0\) is pure-exact so it also splits as \(X\) is pure-injective. Thus \(X\) is injective in \(\bar{\mathcal{G}}\).
(2) First \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) by 6 and \(\mathcal{F} \subseteq \textrm{cogen}\,\textrm{Inj}\,\bar{\mathcal{F}}\) by 3. The second inclusion is an equality because \(\mathcal{F} = \bar{\mathcal{F}} \cap \mathcal{C} \supseteq \textrm{cogen}\,\textrm{Inj}\,\bar{\mathcal{F}}\). Using this equality, it follows that \(\textrm{Inj}\,\bar{\mathcal{F}}\) is cogen-rigid since clearly \(\textrm{Ext}^1_{\bar{\mathcal{C}}}(\mathcal{F},\textrm{Inj}\,\bar{\mathcal{F}}) = 0\), \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq \bar{\mathcal{F}}\) and \(\mathcal{F}\) is covariantly finite by 4. ◻
In order to assign a cogen-rigid subset \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) to a torsion-free class \(\mathcal{F}\subseteq \mathcal{C}\) in a unique way, we must restrict to cosilting subsets. With the following theorem we achieve our goal and obtain a classification of torsion pairs for essentially small abelian categories. Note that a similar correspondence has been established by Angeleri Hügel, Laking and Sentieri [2] for the case \(\mathcal{C} = \textrm{mod}\,R\) where \(R\) is an Artinian ring. Whereas our approach relies on the connection between purity and exact structures, theirs is based on 2-term complexes of injective objects and the Ziegler spectrum of the derived category.
Theorem 10. Let \(\mathcal{C}\) be an essentially small abelian category. There is a one-to-one correspondence \[\begin{Bmatrix} \text{torsion-free classes}\\ \mathcal{F}\subseteq \mathcal{C} \end{Bmatrix} \longleftrightarrow \begin{Bmatrix} \text{cosilting subsets}\\ {U}\subseteq \textrm{Zsp}\,\bar{\mathcal{C}} \end{Bmatrix}\] given by \(\mathcal{F} \mapsto \textrm{Inj}\,\bar{\mathcal{F}}\) and \(U \mapsto \textrm{cogen}\,U\). Moreover, cosilting subsets are closed.
Proof. By 9 the assignments yield \(\mathcal{F} \mapsto \textrm{Inj}\,\bar{\mathcal{F}} \mapsto \textrm{cogen}\,\textrm{Inj}\,\bar{\mathcal{F}} = \mathcal{F}\) for every torsion-free class \(\mathcal{F} \subseteq \mathcal{C}\) and \(U \mapsto \textrm{cogen}\,U \mapsto \textrm{Inj}\,\overline{\textrm{cogen}\,U} \supseteq U\) for every cosilting subset \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\). Further \(\textrm{Inj}\,\bar{\mathcal{F}}\subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) is closed by 6. It is left to show that \(\textrm{Inj}\,\bar{\mathcal{F}}\) is cosilting and not only cogen-rigid, and \(U = \textrm{Inj}\,\overline{\textrm{cogen}\,U}\). We begin with the latter.
Let \(V\) be the collection of all \(Y\in \textrm{Inj}\,\bar{\mathcal{C}}\) with \(\textrm{Hom}_{\bar{\mathcal{C}}}(X,Y) = 0\) for every product \(X\) of objects in \(U\). By definition of being cosilting \((U,V)\) is a maximal cogen-rigid pair. One easily checks that \(\textrm{Hom}_{\bar{\mathcal{C}}}(\textrm{cogen}\,U, V) = 0\) which implies \(\textrm{Hom}_{\bar{\mathcal{C}}}(\overline{\textrm{cogen}\,U}, V) = 0\). It follows that \((U,V) \leq (\textrm{Inj}\,\overline{\textrm{cogen}\,U}, V)\) as cogen-rigid pairs and so \(U = \textrm{Inj}\,\overline{\textrm{cogen}\,U}\) by the maximality condition.
We know that \(\textrm{Inj}\,\bar{\mathcal{F}}\) is cogen-rigid and must show that \((\textrm{Inj}\,\bar{\mathcal{F}}, V)\) is a maximal cogen-rigid pair, where \(V\) is the collection of all \(Y\in \textrm{Inj}\,\bar{\mathcal{C}}\) with \(\textrm{Hom}_{\bar{\mathcal{C}}}(X,Y) = 0\) for all products \(X\) of objects in \(\textrm{Inj}\,\bar{\mathcal{F}}\) or equivalently \(\textrm{Hom}_{\bar{\mathcal{C}}}(\mathcal{F}, Y) = 0\). Let \((U',V')\) be a cogen-rigid pair with \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq U'\) and \(V \subseteq V'\). Then \(\mathcal{F}' = \textrm{cogen}\,U'\) is a torsion-free class with \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq \textrm{Inj}\,\bar{\mathcal{F}'}\) by 9. By [20] the condition \(V \subseteq V'\) implies that the smallest Serre subcategory \(\mathcal{S} \subseteq \mathcal{C}\) containing \(\mathcal{F}\) also contains \(\mathcal{F}'\). Note that \(\mathcal{S}\) consists of all objects in \(\mathcal{C}\) that are filtered by quotients of objects in \(\mathcal{F}\). It follows that \(\mathcal{F}' = \bigcup_{n\geq 0} \mathcal{F}'_n\) where \(\mathcal{F}'_n \subseteq \mathcal{F}'\) is the full subcategory of all objects filtered by \(n\)-many quotients of objects in \(\mathcal{F}\). We will show that \(\mathcal{F}'_n \subseteq \mathcal{F}\) for all \(n\) but before we make the following observation.
For \(X\in \bar{\mathcal{C}}\) consider the short exact sequence \(0 \to tX \to X \to X/tX \to 0\) where \(tX\) is contained in the torsion class corresponding to \(\bar{\mathcal{F}}\) and \(X/tX \in \bar{\mathcal{F}}\). Then \(X\to X/tX\) is a left \(\bar{\mathcal{F}}\)-approximation and \(L\colon \bar{\mathcal{F}'}\to \bar{\mathcal{F}}, X \mapsto X/tX\) is a left adjoint of the inclusion \(\iota \colon \bar{\mathcal{F}} \to \bar{\mathcal{F}'}\). The condition \(\textrm{Inj}\,\bar{\mathcal{F}} \subseteq \textrm{Inj}\,\bar{\mathcal{F}'}\) implies that \(L\) is exact by 11. It follows that for every epimorphism \(f\colon Y\to X\) with \(X \in \mathcal{F}'\) and \(Y\in \mathcal{F}\) already \(X\in \mathcal{F}\). Indeed, applying the exact functor \(L\) to the short exact sequence \(0 \to \ker f \to Y\to X\to 0\) shows that \(LX = X\).
Now, for \(X \in \mathcal{F}_n'\) there is a short exact sequence \(0 \to X' \to X \to X'' \to 0\) with \(X' \in \mathcal{F}'_{n-1}\) and \(X''\) is the quotient of some \(Y \in \mathcal{F}\). By induction we may assume that \(X' \in \mathcal{F}\). Taking a pullback \(P\) yields a commutative diagram \[\begin{tikzcd} 0 \arrow[r] & X' \arrow[d, equals] \arrow[r] & P \arrow[r] \arrow[d] & Y \arrow[r] \arrow[d] & 0 \\ 0 \arrow[r] & X' \arrow[r] & X \arrow[r] & X'' \arrow[r] & 0 \end{tikzcd}\] with exact rows. Since \(P \to X\) is epic and \(P \in \mathcal{F}\) it follows from the previous observation that \(X\in \mathcal{F}\). In total, we have shown that \(\mathcal{F}' = \bigcup_{n\geq 0 }\mathcal{F}'_n = \mathcal{F}\) and in particular \(\textrm{Inj}\,\bar{\mathcal{F}} = \textrm{Inj}\,\bar{\mathcal{F}'}\). Hence \(\textrm{Inj}\,\bar{\mathcal{F}}\) is cosilting. ◻
The proof of 10 is based on viewing torsion-free classes naturally as exact categories. It uses the following lemma, which is well-known in the abelian case and generalizes to the exact context.
Lemma 11. Let \(\mathcal{A}, \mathcal{B}\) be exact categories and \(L : \mathcal{A} \rightleftarrows \mathcal{B} : R\) an adjoint pair \(L \dashv R\) with the following property. There is a collection \(U\) of injectives in \(\mathcal{B}\) such that every \(X\in \mathcal{B}\) admits an admissible monomorphism \(X\to Y\) where \(Y\) is a product of objects in \(U\), and \(RX\) is injective in \(\mathcal{A}\) for all \(X \in U\). Then \(L\) is exact.
Proof. Let \(0 \to X \to Y \to Z \to 0\) be an admissible short exact sequence in \(\mathcal{A}\). Then \(LX \to LY\) is a morphism in \(\mathcal{B}\) with cokernel \(LZ\) since \(L\) commutes with colimits as a left adjoint. Consider an admissible monomorphism \(LX \to
Q\) where \(Q\) is a product of objects in \(U\). This morphism corresponds, via adjointness, to a morphism \(X \to RQ\) where \(RQ\) is a product of injectives in \(\mathcal{A}\) by assumption
(\(R\) commutes with products as a right adjoint). By injectivity \(X\to RQ\) factors through \(X\to Y\) and so \(LX \to Q\)
factors through \(LX \to LY\). It follows from the Obscure axiom, see [15], that \(LX \to
LY\) is an admissible monomorphism and so \(0\to LX \to LY \to LZ \to 0\) is an admissible short exact sequence. Hence \(L\) is exact. ◻
Corollary 12. Let \(U \subseteq \textrm{Zsp}\,\bar{\mathcal{C}}\) be a cogen-rigid subset.
There is a cosilting subset of \(\textrm{Zsp}\,\bar{\mathcal{C}}\) containing \(U\).
The closure of \(U\) in \(\textrm{Zsp}\,\bar{\mathcal{C}}\) is cogen-rigid.
Proof. By 9 and 10 the cogen-rigid subset \(U\) is contained in the cosilting subset \(U' = \textrm{Inj}\,\overline{\textrm{cogen}\,U}\) which is closed in \(\textrm{Zsp}\,\bar{\mathcal{C}}\). Thus, the closure \(\bar{U}\) of \(U\) in \(\textrm{Zsp}\,\bar{\mathcal{C}}\) is contained in \(U'\) and fulfills \(\textrm{cogen}\,U = \textrm{cogen}\,\bar{U} = \textrm{cogen}\,U'\). It follows that \(\bar{U}\) is cogen-rigid. ◻
Let \(A\) be an Artin algebra (for example a finite dimensional algebra), \(\textrm{Mod}\,A\) the category of (left) \(A\)-modules, \(\textrm{mod}\,A \subseteq \textrm{Mod}\,A\) the full subcategory of finite length modules, and set \(\textrm{Zsp}\,A = \textrm{Zsp}\,\textrm{Mod}\,A\). Torsion pairs in \(\textrm{mod}\,A\) can be classified via cosilting subsets \(U \subseteq \textrm{Zsp}\,A\) by 10. One goal will be to characterize cosilting subsets in terms of the (infinite) Auslander-Reiten translation in the spirit of \(\tau\)-tilting theory [1].
Definition 13. We follow the definition in [5] to deal with arbitrary \(A\)-modules. Let \(\underline{\mathrm{Mod}}\, A\) and \(\overline{\mathrm{Mod}}\,A\) be the projectively and injectively stable module categories, respectively. The Auslander-Reiten translation* is the functor \(\tau \colon \underline{\mathrm{Mod}}\, A \to \overline{\mathrm{Mod}}\, A\) defined by the exact sequences \[\begin{align} P_1 \longrightarrow P_0 \longrightarrow X \longrightarrow 0, \qquad 0 \longrightarrow \tau X \longrightarrow DA \otimes_A P_1 \to DA\otimes_A P_0 \end{align}\] for \(X\in \textrm{Mod}\,A\). Here \(D\) is the standard duality and the first exact sequence is a minimal projective presentation of \(X\). The definition of \(\tau\) on morphisms is as usual given by lifting morphisms along projective covers.*
The main connection between the Auslander-Reiten translation and the Ziegler spectrum is the following.
Proposition 14. [5] The Auslander-Reiten translation \(\tau\colon \underline{\mathrm{Mod}}\, A \to \overline{\mathrm{Mod}}\, A\) is an equivalence and induces a homeomorphism \(\textrm{Zsp}\,A \setminus \mathrm{Proj}\,A \to \textrm{Zsp}\,A \setminus \textrm{Inj}\,A\) where \(\mathrm{Proj}\,A\) and \(\mathrm{Inj}\,A\) are the sets of indecomposable projective and injective \(A\)-modules, respectively.
The inverse of \(\tau\) is denoted by \(\tau^{-}\) and is defined by the exact sequences \[\begin{align} 0 \longrightarrow X \longrightarrow I_0 \longrightarrow I_1, \qquad \textrm{Hom}_{A}(DA,I_0) \to \textrm{Hom}_{A}(DA, I_1) \longrightarrow \tau^- X \longrightarrow 0 \end{align}\] for \(X\in \textrm{Mod}\,A\) where the first exact sequence is a minimal injective presentation. We introduce the following notions analogous to classical \(\tau\)-tilting theory.
Definition 15. For \(X,Y\in \textrm{Mod}\,A\) let \(\textrm{Hom}_A(X,Y)^\mathrm{fin} \subseteq \textrm{Hom}_A(X,Y)\) be the subgroup of morphisms with finite length image.
A subset \(U\subseteq \textrm{Zsp}\,A\) is \(\tau^-\)-rigid* if \(\textrm{Hom}_A^\mathrm{fin}(\tau^- U, U) = 0\).*
We call \((U, V)\) a \(\tau ^-\)-rigid* pair if \(U \subseteq \textrm{Zsp}\,A\) is \(\tau^-\)-rigid and \(V\subseteq \mathrm{Inj}\,A\) fulfills \(\mathrm{Hom}_A(U,V) = 0\). There is a partial order on \(\tau^-\)-rigid pairs given by \((U, V) \leq (U', V')\) if \(U\subseteq U'\) and \(V\subseteq V'\).*
A subset \(U\subseteq \textrm{Zsp}\,A\) is support \(\tau^-\)-tilting if there exists a maximal \(\tau^-\)-rigid pair \((U, V)\).
Next, we show that \(\tau^-\)-rigid subsets coincide with cogen-rigid subsets of \(\textrm{Zsp}\,A\). The key ingredient is the following Auslander-Reiten formula.
Lemma 16. [21] For \(X\in \textrm{mod}\,A\) and \(Y\in \textrm{Mod}\,A\) there exists an isomorphism \[\begin{align} D\,\mathrm{Ext}^1_{A}(X, Y) \cong \overline{\mathrm{Hom}}_A(Y,\tau X) \end{align}\] functorial in \(X\) and \(Y\).
Theorem 17. Let \(U \subseteq \textrm{Zsp}\,A\). Then \(U\) is cogen-rigid if and only if \(U\) is \(\tau^-\)-rigid.
Proof. We will show that for all \(X,Y \in \textrm{Mod}\,A\) the following are equivalent: \[\begin{align} \mathrm{(1)}\quad \mathrm{Ext}_A^1 (\textrm{cogen}\,X, Y) = 0, \qquad \mathrm{(2)}\quad \textrm{Hom}_A^\mathrm{fin}(\tau^- Y, X) = 0. \end{align}\] Here \(\textrm{cogen}\,X\) denotes all finite length submodules of arbitrary products of \(X\). This implies the statement of the theorem by 8 as \(\textrm{mod}\,A\) is a length category. For \(Z \in \textrm{mod}\,A\) we obtain the Auslander-Reiten formula \[\begin{align} D\,\mathrm{Ext}^1_{A}(Z, Y) \cong \overline{\mathrm{Hom}}_A(Y,\tau Z) \cong \underline{\mathrm{Hom}}_A(\tau^- Y,Z) \end{align}\] by 16 and 14.
(2)\(\implies\)(1) If \(\mathrm{Ext}^1_A(\textrm{cogen}\,X, Y) \neq 0\) then by the formula above, there exists a non-zero morphism \(\tau^{-}Y \to Z\) for \(Z\in \textrm{cogen}\,X\). Further, there is a monomorphism \(Z\to \prod X\) and the composition \(\tau^{-}Y \to Z \to \prod X\) is non-zero. But then composing with a suitable projection \(\prod X \to X\) yields a non-zero morphism \(\tau^- Y \to X\) with a finite length image contradicting (2).
(1)\(\implies\)(2) Let \(\tau^- Y \to X\) be a morphism with finite length image \(Z\). By (1) and the Auslander-Reiten formula, the canonical epimorphism \(\tau^{-} Y \to Z\) factors through a projective \(A\)-module \(P\) as \(\tau^-{} Y \to P \to Z\). Without loss of generality we assume that \(P \to Z\) is a projective cover. Now \(P \to Z\) factors through the epimorphism \(\tau^{-}Y \to Z\). In total \(P \to Z\) factors as \(P \to \tau^{-}Y \to P \to Z\) and it follows that \(P \to \tau^{-}Y\) is a split monomorphism. But \(\tau^- Y\) has no non-zero projective summands and so \(P = 0\). Hence \(\tau^- Y \to X\) is a zero morphism. ◻
Corollary 18. There is a one-to-one correspondence \[\begin{Bmatrix} \text{torsion-free classes}\\ \mathcal{F}\subseteq \mathrm{mod}\,A \end{Bmatrix} \longleftrightarrow \begin{Bmatrix} \text{support \tau^--tilting subsets}\\ {U}\subseteq \textrm{Zsp}\,A \end{Bmatrix}\] given by \(\mathcal{F} \mapsto \textrm{Inj}\,\bar{\mathcal{F}}\) and \(U \mapsto \textrm{cogen}\,U\). Moreover, support \(\tau^-\)-tilting subsets are closed.
Proof. By 17 a subset \(U \subseteq \textrm{Zsp}\,A\) is support \(\tau^-\)-tilting if and only if \(U\) is cosilting. Now apply 10. ◻
In the above classification we are possibly dealing with infinite length modules and the next goal is to reduce everything to finite length modules. The trade-off is that we have to consider ideals of the module category. A non-empty collection \(\mathcal{I}\) of morphisms in \(\textrm{mod}\,A\) is an ideal if for all \(\varphi , \psi \in \mathcal{I}\) and arbitrary morphisms \(\alpha, \beta\) in \(\textrm{mod}\,A\) we have \(\varphi+ \psi \in \mathcal{I}\) and \(\beta \varphi \alpha \in \mathcal{I}\) if the expressions are defined. We are interested in certain ideals introduced by Krause [6].
Definition 19. An ideal \(\mathcal{I}\) of \(\textrm{mod}\,A\) is fp-idempotent* if the full subcategory of finitely presented functors \(F\colon \textrm{mod}\,A \to \textrm{Ab}\) with \(F(\varphi) = 0\) for all \(\varphi\in \mathcal{I}\) is closed under extensions. Recall that \(F\) is finitely presented if there exist \(X,Y \in \textrm{mod}\,A\) and a short exact sequence \(\textrm{Hom}_A(Y,-) \to \textrm{Hom}_A(X,-) \to F \to 0\).*
For \(U \subseteq \textrm{Zsp}\,A\) let \(\langle U\rangle\) be the ideal of morphisms in \(\textrm{mod}\,A\) that factor through a product of modules in \(U\). The next result, due to Krause, is the main connection between fp-idempotent ideals and the Ziegler spectrum.
Theorem 20. [6] There is a one-to-one correspondence between closed sets \(U \subseteq \textrm{Zsp}\,A\) and fp-idempotent ideals \(\mathcal{I}\) of \(\textrm{mod}\,A\) given by \(U \mapsto \langle U \rangle\).
The following result by Prest offers a nice description of fp-idempotent ideals.
Proposition 21. [22] Let \(U \subseteq \textrm{Zsp}\,A\) and \(\bar{U}\) its closure. The ideal \(\langle \bar{U}\rangle\) consists of all morphisms in \(\textrm{mod}\,A\) that factor through a finite direct sum of objects in \(U\). In particular \(\langle U\rangle = \langle \bar{U} \rangle\) is always fp-idempotent.
We will express \(\tau^-\)-rigidity of subsets \(U \subseteq \textrm{Zsp}\,A\) in terms of the corresponding fp-idempotent ideal \(\mathcal{I} = \langle U \rangle\). We define \(\tau \mathcal{I} = \langle \tau U\rangle\) and \(\tau^- \mathcal{I} = \langle \tau^- {U}\rangle\). These expressions are independent of the choice of \(U\) by 21 since \(\tau\) is a homeomorphism, see 14. Set \[\begin{align} \textrm{cogen}\,\mathcal{I} &= \{X \in \textrm{mod}\,A \mid \text{there is a mono }X\to Y\text{ in }\mathcal{I}\} \\ &= \{X\in \textrm{mod}\,A \mid \textrm{Hom}_A(X,DA) = \mathcal{I}(X,DA)\}. \end{align}\] The following notions are variants of 15 on the level of ideals.
Definition 22. Let \(\mathcal{I}\) be an ideal of \(\textrm{mod}\,A\).
The ideal \(\mathcal{I}\) is \(\tau^-\)-rigid* if \(\mathcal{I}\) is fp-idempotent and \(\mathcal{I}\circ \tau^- \mathcal{I} = 0\).*
We call \((\mathcal{I}, V)\) a \(\tau^-\)-rigid* pair if \(\mathcal{I}\) is \(\tau^-\)-rigid and \(V\subseteq \textrm{Inj}\,A\) fulfills \(\mathcal{I}(X,V) = 0\) for all \(X\in \textrm{mod}\,A\). There is a partial order on \(\tau^-\)-rigid pairs given by \((\mathcal{I}, V) \leq (\mathcal{I}', V')\) if \(\mathcal{I} \subseteq \mathcal{I}'\) and \(V\subseteq V'\).*
The ideal \(\mathcal{I}\) is support \(\tau^-\)-tilting* if there is a maximal \(\tau^-\)-rigid pair \((\mathcal{I}, V)\).*
Proposition 23. Let \(U \subseteq \textrm{Zsp}\,A, V \subseteq \textrm{Inj}\,A\) and \(\mathcal{I} = \langle U\rangle\). Then \((U,V)\) is \(\tau^-\)-rigid if and only if \((\mathcal{I}, V)\) is \(\tau^-\)-rigid and in this case \(\textrm{cogen}\,\mathcal{I} = \textrm{cogen}\,U\) is a torsion-free class.
Proof. First, we show that \(U\) is \(\tau^-\)-rigid if and only if the ideal \(\mathcal{I}\) is \(\tau^-\)-rigid. By 21 every morphism in \(\mathcal{I} \circ \tau^- \mathcal{I}\) factors as \[\begin{align} X\longrightarrow \bigoplus_{i=1}^n \tau^{-} Y_i \longrightarrow X' \longrightarrow \bigoplus_{j=1}^m Y_j' \longrightarrow X'' \end{align}\] with \(X,X',X'' \in \textrm{mod}\,A\) and \(Y_i, Y_j' \in U\). If \(U\) is \(\tau^-\)-rigid, then the composition \(\bigoplus \tau^- Y_i \to X' \to \bigoplus Y_j'\) is zero as \(\textrm{Hom}_A^\mathrm{fin}(\tau^- U, U) = 0\). Thus \(\mathcal{I} \circ \tau^- \mathcal{I} = 0\).
For \(X, Y \in U\) and a morphism \(\tau^-Y \to X\) with finite length image \(Z\) consider an epimorphism \(\bigoplus A \to \tau^- Y\) and a monomorphism \(X \to \prod DA\). If the ideal \(\mathcal{I}\) is \(\tau^-\)-rigid then each component \(A \to DA\) of the composition \[\begin{align} \bigoplus A \longrightarrow \tau^- Y \longrightarrow Z \longrightarrow X\longrightarrow \prod DA \end{align}\] is zero as \(\mathcal{I}\circ \tau^- \mathcal{I} = 0\). Thus \(U\) is \(\tau^-\)-rigid.
Next, we show that \(\textrm{Hom}_A(U, Q) = 0\) if and only if \(\mathcal{I}(-,Q) = 0\) for \(Q\in \textrm{Inj}\,A\). The only if part is trivial. Suppose that \(\mathcal{I}(-,Q)= 0\). For \(X\in U\) let \(\bigoplus A \to X\) be an epimorphism. Then, for \(X\to Q\) every component \(A\to Q\) of the composition \(\bigoplus A \to X \to Q\) is contained in \(\mathcal{I}(A,Q) = 0\). Thus \(X\to Q\) equals zero and \(\textrm{Hom}_A(U, Q) = 0\).
Lastly, we prove that \(\textrm{cogen}\,\mathcal{I} = \textrm{cogen}\,U\). Note that \(U\) being \(\tau^-\)-rigid implies that \(U\) is cogen-rigid by 17 and so \(\textrm{cogen}\,U\) is a torsion-free class, see 9. The inclusion \(\textrm{cogen}\,\mathcal{I }\subseteq \textrm{cogen}\,U\) is trivial. For \(X\) in \(\textrm{cogen}\,U\) there is a monomorphism \(X\to \prod Y\) where the product goes over some \(Y\in U\). The injective hull \(f\colon X\to Q\) factors through \(X\to \prod Y\) and so \(f\in \mathcal{I}\). Hence \(X\in \textrm{cogen}\,\mathcal{I}\) and \(\textrm{cogen}\,\mathcal{I} = \textrm{cogen}\,U\). ◻
Corollary 24. There is a one-to-one correspondence \[\begin{Bmatrix} \text{torsion-free classes}\\ \mathcal{F}\subseteq \textrm{mod}\,A \end{Bmatrix} \longleftrightarrow \begin{Bmatrix} \text{support \tau^--tilting ideals}\\ \mathcal{I}\text{ of } \textrm{mod}\,A \end{Bmatrix}\] given by \(\mathcal{F} \mapsto \langle \textrm{Inj}\,\bar{\mathcal{F}} \rangle\) and \(\mathcal{I} \mapsto \textrm{cogen}\,\mathcal{I}\).
Proof. The assignment \(U \mapsto \langle U \rangle\) induces a one-to-one correspondence between support \(\tau^-\)-tilting subsets \(U \subseteq \textrm{Zsp}\,A\) and support \(\tau^-\)-tilting ideals \(\mathcal{I}\) of \(\textrm{mod}\,A\) such that \(\textrm{cogen}\,\mathcal{I} = \textrm{cogen}\,U\) by 23. Combining this correspondence with the one in 18 shows the desired result. ◻
Example 25. Consider the Kronecker algebra \[\begin{tikzcd} A =k\,(\bullet \arrow[r, shift left] \arrow[r, shift right] & \bullet) \end{tikzcd}\] over an algebraically closed field \(k\). The indecomposable modules in \(\textrm{mod}\,A\) can be divided into three parts: The preprojective modules \(\mathcal{P} = \{P_1, P_2, \dots \}\), the preinjective modules \(\mathcal{Q} = \{Q_1, Q_2, \dots \}\) and the regular modules \(\mathcal{R}\), which further divide into tubes \(\mathcal{R}^\lambda = \{R^\lambda_1, R^\lambda_2, \dots\}\) with \(\lambda\in k\cup \{\infty \}\). The Auslander-Reiten quiver of \(\textrm{mod}\,A\) can be visualized as follows. =[circle,draw=black!50,fill=black!100,thick, inner sep=0pt,minimum size=1mm] \[\begin{align} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/lqrhzgdc.png}\label{bcqnxsjy}\end{figure} \end{align}\qquad{(1)}\]
For all \(\lambda\) the Prüfer module \(R_\infty^\lambda\) equals a filtered colimit of monomorphisms in \(\mathcal{R}^\lambda\) and the adic module
\(\hat{R}^\lambda\) is given by an inverse limit of epimorphisms in \(\mathcal{R}^\lambda\).
The Ziegler spectrum \(\textrm{Zsp}\,A\) consists of all indecomposable modules in \(\textrm{mod}\,A\), the Prüfer modules \(R_\infty^\lambda\), the adic
modules \(\hat{R}^\lambda\), and the unique generic module \(G\), see for example [19].
Moreover, a subset \(U \subseteq \textrm{Zsp}\,A\) is closed if and only if the following holds.
If \(U\) contains infinitely many modules in \(\mathcal{R}_\lambda\) then \(U\) contains \(R_\infty^\lambda\) and \(\hat{R}^\lambda.\)
If \(U\) contains infinitely many modules in \(\mathcal{P}\) then \(U\) contains \(\hat{R}^\lambda\) for all \(\lambda\).
If \(U\) contains infinitely many modules in \(\mathcal{Q}\) then \(U\) contains \({R}^\lambda_\infty\) for all \(\lambda\).
If \(U\) contains infinitely many modules or a module of infinite length, then \(U\) contains \(G\).
From the topology of the Ziegler spectrum and the behavior of \(\tau\) on finite length modules, it follows that \(\tau X \cong X\) for every infinite length module \(X\in \textrm{Zsp}\,A\) since \(\tau \colon \textrm{Zsp}\,A \setminus \{P_1, P_2\} \to \textrm{Zsp}\,A \setminus \{Q_1, Q_2\}\) is a homeomorphism, see 14. This will be used to determine all \(\tau^-\)-rigid subsets \(U \subseteq \textrm{Zsp}\,A\) that consist of infinite length modules only. For our calculations, we apply fp-idempotent ideals and make use of the fact that there are only zero morphisms from the right to the left in the displayed Auslander-Reiten quiver.
The fp-idempotent ideal \(\langle G\rangle\) is generated by all morphisms from \(\mathcal{P}\) to \(\mathcal{Q}\). For \(\lambda \in k\cup \{\infty\}\) the fp-idempotent ideal \(\langle R^\lambda_\infty \rangle\) is generated by all morphisms from \(\mathcal{R}^\lambda\) to \(\mathcal{Q}\), and \(\langle \hat{R}^\lambda\rangle\) is generated by all morphisms from \(\mathcal{P}\) to \(\mathcal{R}^\lambda\). The only non-zero compositions, using these ideals, are given by \(\langle R_\infty ^\lambda \rangle \circ \langle \hat{R}^\lambda \rangle \neq 0\). Since \(\tau X \cong X\) for infinite length modules \(X\in \textrm{Zsp}\,A\) it follows from 23 that a subset \(U \subseteq \textrm{Zsp}\,A \setminus \textrm{mod}\,A\) is \(\tau^-\)-rigid if and only if \(R_\infty^\lambda \in U\) implies \(\hat{R}^\lambda \notin U\). Assume that \(U\) is such a \(\tau^-\)-rigid non-empty set. Then the torsion-free class \(\mathcal{F}= \textrm{cogen}\,U\) associated to \(U\) is given by the additive closure of \(\mathcal{P}\) as well as all \(\mathcal{R}^\lambda\) with \({R}_\infty^\lambda \in U\). Now \(U\) is support \(\tau^-\)-tilting if and only if \[\begin{align} U = \{R_\infty^\lambda, \hat{R}^\mu , G\mid \lambda\in S, \mu \in T\} \end{align}\] with \(S\dot{\cup } T = k \cup \{\infty\}\). The corresponding support \(\tau^-\)-tilting ideal \(\langle U \rangle\) is generated by all morphisms from \(\mathcal{R}^\lambda\) to \(\mathcal{Q}\), and \(\mathcal{P}\) to \(\mathcal{R}^\mu\) with \(\lambda \in S\) and \(\mu \in T\).
Let \(A\) be an Artin algebra. The aim of this section is to present dual counterparts of the notions and results developed in the previous section. The difficulty lies in dealing with infinite length \(A\)-modules for which duality is not well behaved. We overcome this by working on the level of ideals first. The following definition is dual to 22.
Definition 26. Let \(\mathcal{I}\) be an ideal of \(\textrm{mod}\,A\).
The ideal \(\mathcal{I}\) is \(\tau\)-rigid* if \(\mathcal{I}\) is fp-idempotent and \(\tau \mathcal{I}\circ \;\mathcal{I} = 0\).*
We call \((\mathcal{I}, V)\) a \(\tau\)-rigid* pair if \(\mathcal{I}\) is \(\tau\)-rigid and \(V\subseteq \mathrm{Proj}\, A\) fulfills
\(\mathcal{I}(V,X) = 0\) for all \(X\in \textrm{mod}\,A\). There is a partial order on \(\tau\)-rigid pairs given by \((\mathcal{I}, V) \leq (\mathcal{I}', V')\) if \(\mathcal{I} \subseteq \mathcal{I}'\) and \(V\subseteq V'\).*
The ideal \(\mathcal{I}\) is support \(\tau\)-tilting* if there is a maximal \(\tau\)-rigid pair \((\mathcal{I}, V)\).*
One issue is that these notions still make reference to infinite length \(A\)-modules, namely through the expression \(\tau\mathcal{I}\). We will show that \(\tau\mathcal{I}\) can instead be described using the Auslander-Reiten translation on morphisms. For an ideal \(\mathcal{I}\) of \(\mathrm{mod}\,A\) let \(\underline{\mathcal{I}}\) and \(\overline{\mathcal{I}}\) be the induced ideals in \(\underline{\mathrm{mod}}\,A\) and \(\overline{\mathrm{mod}}\,A\) respectively. Applying \(\tau\) and \(\tau^-\) on morphisms naturally yields an ideal \(\tau \underline{\mathcal{I}}\) of \(\overline{\mathrm{mod}}\,A\) and an ideal \(\tau^- \overline{ \mathcal{I}}\) of \(\underline{\mathrm{mod}}\,A\). The following relates \(\tau \mathcal{I}\) and \(\tau \underline{\mathcal{I}}\) as well as \(\tau^- \mathcal{I}\) and \(\tau^- \overline{\mathcal{I}}\).
Lemma 27. Let \(\mathcal{I}\) be an fp-idempotent ideal of \(\textrm{mod}\,A\).
The ideal \(\tau \mathcal{I}\) equals the unique fp-idempotent ideal \(\mathcal{J}\) of \(\textrm{mod}\,A\) containing no \(1_Q\) with \(Q\in \textrm{Inj}\,A\) such that \(\tau \underline{\mathcal{I}} = \overline{\mathcal{J}}\).
The ideal \(\tau^- \mathcal{I}\) equals the unique fp-idempotent ideal \(\mathcal{J}\) of \(\textrm{mod}\,A\) containing no \(1_P\) with \(P\in \mathrm{Proj}\,A\) such that \(\tau^- \overline{\mathcal{I}} = \underline{\mathcal{J}}\).
Proof. We only show (1) as (2) is similar. Let \(\mathcal{I} = \langle U \rangle\) for a closed set \(U \subseteq \textrm{Zsp}\,A\). Since the Auslander-Reiten translation \(\tau \colon \underline{\mathrm{Mod}}\, A \to \overline{\mathrm{Mod}}\, A\) is an equivalence by 14 it follows that \(\tau \underline{\mathcal{I}} = \overline{\tau \mathcal{I}}\). Moreover \(Q\notin \tau U\) implies that \(1_Q \notin \tau \mathcal{I}\) for \(Q\in \textrm{Inj}\,A\). Let \(\mathcal{J} = \langle U'\rangle\) have the properties stated with \(U'\subseteq \textrm{Zsp}\,A\) closed. Then \(\overline{\mathcal{J}} = \overline{\tau \mathcal{I}}\) implies that \(\mathcal{J} + \langle \textrm{Inj}\,A \rangle = \tau \mathcal{I}+\langle \textrm{Inj}\,A \rangle\) and so \(U' \cup \textrm{Inj}\,A = \tau U \cup \textrm{Inj}\,A\) by 20. By assumption \(U' \cap \textrm{Inj}\,A = \emptyset\) and thus \(U ' = \tau {U}\). We conclude that \(\mathcal{J} = \tau \mathcal{I}\). ◻
For an ideal \(\mathcal{I}\) and an additive subcategory \(\mathcal{C}\) of \(\textrm{mod}\,A\) the duality \(D\) naturally induces an ideal \(D \mathcal{I}\) and an additive subcategory \(D\mathcal{C}\) of \(\textrm{mod}\,A ^\textrm{op}\). Let \(\textrm{gen}\,\mathcal{I}\) be the collection of all \(X\in \textrm{mod}\,A\) that admit an epimorphism \(Y \to X\) in \(\mathcal{I}\).
Proposition 28. Let \(\mathcal{I}\) be an ideal of \(\textrm{mod}\,A\) and \(V\subseteq \mathrm{Proj}\, A\). Then \((\mathcal{I}, V)\) is \(\tau\)-rigid if and only if \((D \mathcal{I}, DV)\) is \(\tau^-\)-rigid. Moreover \(\textrm{gen}\,\mathcal{I} = D\,\textrm{cogen}\,D\mathcal{I}\).
Proof. By [7] the ideal \(\mathcal{I}\) is fp-idempotent if and only if \(D\mathcal{I}\) is fp-idempotent. Thus, the definition of being \(\tau\)-rigid is completely dual to being \(\tau^-\)-rigid and may only be expressed in terms of finite length modules by 27. It follows that \((\mathcal{I}, V)\) is \(\tau\)-rigid if and only if \((D \mathcal{I}, DV)\) is \(\tau^-\)-rigid. Moreover, any epimorphism \(X\to Y\) in \(\mathcal{I}\) corresponds to a monomorphism \(DY \to DX\) in \(D\mathcal{I}\) and so \(\textrm{gen}\,\mathcal{I} = D \, \textrm{cogen}\,D\mathcal{I}\). ◻
Corollary 29. There is a one-to-one correspondence \[\begin{Bmatrix} \text{torsion classes}\\ \mathcal{T}\subseteq \textrm{mod}\,A \end{Bmatrix} \longleftrightarrow \begin{Bmatrix} \text{support \tau-tilting ideals}\\ \mathcal{I}\text{ of } \textrm{mod}\,A \end{Bmatrix}\] given by \(\mathcal{I} \mapsto \textrm{gen}\,\mathcal{I}\).
We also want to establish a dual result of 18 for the module category of an Artin algebra. This requires the following definition.
Definition 30.
A subset \(U\subseteq \textrm{Zsp}\,A\) is \(\tau\)-rigid* if \(\textrm{Hom}_A^\mathrm{fin}(U, \tau U) = 0\).*
We call \((U, V)\) a \(\tau\)-rigid* pair if \(U \subseteq \textrm{Zsp}\,A\) is \(\tau\)-rigid and \(V\subseteq \mathrm{Proj}\,A\) fulfills \(\mathrm{Hom}_A(V,U) = 0\). There is a partial order on \(\tau\)-rigid pairs given by \((U, V) \leq (U', V')\) if \(U\subseteq U'\) and \(V\subseteq V'\).*
A subset \(U\subseteq \textrm{Zsp}\,A\) is support \(\tau\)-tilting* if there exists a maximal \(\tau\)-rigid pair \((U, V)\).*
For \(U \subseteq \textrm{Zsp}\,A\) let \(\textrm{gen}\,U\) be the collection of all \(X\in \textrm{mod}\,A\) that admit an epimorphism \(\bigoplus_{i=1}^n Y_i\to X\) with \(Y_i\in U\). The following is precisely the dual version of 23.
Proposition 31. Let \(U \subseteq \textrm{Zsp}\,A, V \subseteq \mathrm{Proj}\,A\) and \(\mathcal{I} = \langle U\rangle\). Then \((U,V)\) is \(\tau\)-rigid if and only if \((\mathcal{I}, V)\) is \(\tau\)-rigid and in this case \(\textrm{gen}\,\mathcal{I} = \textrm{gen}\,U\) is a torsion class.
Proof. The fact that \((U,V)\) is \(\tau\)-rigid if and only if \((\mathcal{I},V)\) is \(\tau\)-rigid can be shown as in the proof of 23. Moreover \(\textrm{gen}\,\mathcal{I} = D\,\textrm{cogen}\,D\mathcal{I}\) is a torsion class in this case by duality, see 28. ◻
Corollary 32. There is a one-to-one correspondence \[\begin{Bmatrix} \text{torsion classes}\\ \mathcal{T}\subseteq \textrm{mod}\,A \end{Bmatrix} \longleftrightarrow \begin{Bmatrix} \text{support \tau-tilting subsets}\\ {U}\subseteq \textrm{Zsp}\,A \end{Bmatrix}\] given by \(U \mapsto \textrm{gen}\,U\). Moreover, support \(\tau\)-tilting subsets are closed.
Proof. By 31 it follows that support \(\tau\)-tilting subsets \(U \subseteq \textrm{Zsp}\,A\) are one to one with support \(\tau\)-tilting ideals \(\mathcal{I}\) and \(\textrm{gen}\,U = \textrm{gen}\,\mathcal{I}\). Now, the desired correspondence follows from 29. The equality \(\langle U\rangle = \langle \bar{U} \rangle\) for \(U \subseteq \textrm{Zsp}\,A\), see 21, shows that support \(\tau\)-tilting subsets are closed. ◻
The above is a dual variant of the correspondence in 18. Note that for torsion-free classes \(\mathcal{F}\subseteq \textrm{mod}\,A\) the associated support \(\tau^-\)-tilting subset may be recovered via \(\mathcal{F}\mapsto \textrm{Inj}\,\bar{\mathcal{F}}\). We do not know how to directly recover the support \(\tau\)-tilting subset corresponding to a torsion class. However, this will be achieved indirectly by investigating the relation between the support \(\tau\)-tilting subset and the support \(\tau^-\)-tilting subset corresponding to a torsion pair. Recall that every indecomposable projective \(P\) corresponds to a simple \(S\) which corresponds to an injective \(Q\). We set \(P_* = Q\) and \(Q_* = P\).
Proposition 33. Let \((\mathcal{T}, \mathcal{F})\) be a torsion pair in \(\textrm{mod}\,A\). Moreover, let \((U,V)\) be the maximal \(\tau\)-rigid pair corresponding to \(\mathcal{T}\) and \((U',V')\) the maximal \(\tau^-\)-rigid pair corresponding to \(\mathcal{F}\) with \(U,U'\subseteq \textrm{Zsp}\,A\). Then \(U' = \tau U \cup V_*\) and \(U = \tau^-U'\cup V'_*\).
Proof. First, we show that \(\tau U \cup V_* \subseteq U' = \textrm{Inj}\,\bar{\mathcal{F}}\). The equality \(\textrm{Hom}_A(V, U) =0\) implies \(\textrm{Hom}_A( U, V_*) =0\) and so \(\textrm{Hom}_A(\mathcal{T}, V_*) =\textrm{Hom}_A(\textrm{gen}\,U , V_*) = 0\). Thus \(\mathcal{F}\) contains \(V_*\) and even \(V_* \subseteq \textrm{Inj}\,\bar{\mathcal{F}}\) since \(V_*\) consists of injective \(A\)-modules. Moreover \(\textrm{Hom}_A^\mathrm{fin}(U,\tau U) = 0\) shows that \(\textrm{Hom}_A(\mathcal{T}, \textrm{cogen}\,\tau U) = \textrm{Hom}_A(\textrm{gen}\,U, \textrm{cogen}\,\tau U) = 0\) and so \(\textrm{cogen}\,\tau U \subseteq \mathcal{F}\). Hence \(\tau U\subseteq \bar{\mathcal{F}}\). Applying the Auslander-Reiten formula in [21] yields \[\begin{align} D\textrm{Ext}_A^1(Y, \tau X ) \cong \overline{\mathrm{Hom}}_A(\tau X, \tau Y) \cong \underline{\mathrm{Hom}}_A(X,Y) = 0 \end{align}\] for \(X\in U\) and \(Y\in \mathcal{F}\) since \(\textrm{Hom}_A(\textrm{gen}\,U , \mathcal{F}) = \textrm{Hom}_A(\mathcal{T},\mathcal{F}) = 0\). It follows that \(\textrm{Ext}_A^1(\mathcal{F}, \tau U) = 0\) and so \(\tau U \subseteq \textrm{Inj}\,\bar{\mathcal{F}}\).
The inclusion \(\tau U \cup V_* \subseteq U'\) implies \(\tau \mathcal{I} + \langle V_*\rangle \subseteq \mathcal{I}'\) where \(\mathcal{I}\) is the support \(\tau\)-tilting ideal and \(\mathcal{I}'\) the support \(\tau^-\)-tilting ideal corresponding to \(\mathcal{T}\) and \(\mathcal{F}\), respectively. On the level of ideals we can apply duality, see 28, to deduce that \(\tau^- \mathcal{I}'+\langle V'_* \rangle \subseteq \mathcal{I}\). On the level of the Ziegler spectrum we obtain \(\tau^- U'\cup V'_* \subseteq U\). Thus \[\begin{align} U'\supseteq \tau U \cup V_* \supseteq \tau (\tau^- U' \cup V'_*) \cup V_*= (U'\setminus \mathrm{Inj}\,A) \cup V_*. \end{align}\] Clearly \(U' \cap \mathrm{Inj}\,A = V_*\) consists of all \(Q\in \textrm{Inj}\,A\) that fulfill \(\textrm{Hom}_A(\mathcal{T}, Q) = 0\) or equivalently \(\textrm{Hom}_A(U, Q ) = 0\). It follows that the above inclusions are equalities and so \(U' = \tau U \cup V_*\). Similarly \(U = \tau^- U'\cup V'_*\). ◻
Example 34. Let \(A\) be the Kronecker algebra as in 25. We already determined the support \(\tau^-\)-tilting subsets \(U\subseteq \textrm{Zsp}\,A\) that consist only of infinite length modules as \[\begin{align} U = \{R_\infty^\lambda, \hat{R}^\mu , G\mid \lambda\in S, \mu \in T\} \end{align}\] for \(S\dot{\cup } T = k \cup \{\infty\}\). The corresponding torsion-free class \(\mathcal{F}\) is given by the additive closure of \(\mathcal{P}\) and all \(\mathcal{R}^\lambda\) with \(\lambda \in S\). Let \(\mathcal{T}\) be the associated torsion class. By 33 we know that \(\tau U\) is the support \(\tau\)-tilting subset corresponding to \(\mathcal{T}\). Since the infinite length \(A\)-modules are \(\tau\)-invariant here, it follows that \(\mathcal{T} = \textrm{gen}\,U\) which is the additive closure of \(\mathcal{Q}\) and all \(\mathcal{R}^\mu\) with \(\mu \in T\).
We will apply the developed theory to study special points inside the Ziegler spectrum of an Artin algebra \(A\). Recall that an \(A\)-module \(X\) is a brick if \(\mathrm{End}_A(X)\) is a division ring, and \(X\) is generic if \(X\) is indecomposable, of infinite length, and has finite length over \(\mathrm{End}_A(X)\). Generic modules are closed points inside \(\textrm{Zsp}\,A\), see [6]. We are particularly interested in generic bricks because of the following conjecture formulated by Mousavand and Paquette.
Conjecture 35. [8] For a finite dimensional algebra \(A\) over an algebraically closed field, the following are equivalent.
There are infinitely many non-isomorphic finite dimensional bricks.
There are infinitely many non-isomorphic finite dimensional bricks of the same dimension.
There exists a generic brick.
The equivalence between (2) and (3) of the conjecture has been shown for tame algebras by Bautista, Pérez and Salmerón [9]. We provide an alternative proof for tame algebras using infinite \(\tau\)-tilting theory. Moreover, we will prove the full conjecture under the assumption that the Krull-Gabriel dimension of \(A\) is defined. A conjecture of Prest states that the Krull-Gabriel dimension of \(A\) is defined if and only if the algebra \(A\) is domestic [11], which is known for several classes of algebras, see [12] for a detailed discussion.
For now, we are still working over an arbitrary Artin algebra \(A\). We say that \(X\in \textrm{Zsp}\,A\) is \(\tau^-\)-rigid if \(\{X\} \subseteq \textrm{Zsp}\,A\) is \(\tau^-\)-rigid. These modules coincide with the notion of grains in [2]. The \(\tau^-\)-perspective is preferred to the dual perspective here, because it admits a more natural interpretation of the support \(\tau^-\)-tilting subset corresponding to a torsion-free class.
Lemma 36. Let \(X\) be a generic \(\tau^-\)-rigid \(A\)-module. We also view \(X\) as a module over \(\mathrm{End}_A(X)\) and denote by \(S = \mathrm{soc}_{\mathrm{End}_A(X)}X\) the socle of \(X\) over \(\mathrm{End}_A(X)\).
The \(A\)-module \(S\) is a generic brick and there is a short exact sequence \[\begin{align} 0 \longrightarrow S \longrightarrow X \longrightarrow Y \longrightarrow 0 \end{align}\] inside \(\bar{\mathcal{F}}\) for the torsion-free class \(\mathcal{F} = \textrm{cogen}\,X\).
Every torsion-free class \(\mathcal{G} \subseteq \textrm{Mod}\,A\) containing \(S\) also contains \(X\).
Proof. (1) Consider the canonical map \(X\to \prod X\) of \(A\)-modules where the product goes over all radical morphisms \(X\to X\) and let \(Y\) be its image. There is a short exact sequence \(0 \to S \to X \to Y \to 0\) and \(Y\) is contained inside the torsion-free class \(\bar{\mathcal{F}}\subseteq \textrm{Mod}\,A\) by 9. Note that this short exact sequence is exactly the reject sequence in [2].
Because \(X\) is injective in \(\bar{\mathcal{F}}\) it follows that for all \(f\colon S \to S\) there is \(g\colon X \to X\) making the diagram \[\begin{tikzcd} 0 \arrow[r] & S \arrow[r] \arrow[d, "f", swap] & X \arrow[r] \arrow[d, "g"] & Y \arrow[d] \arrow[r] & 0\\ 0 \arrow[r] & S \arrow[r] & X \arrow[r] & Y \arrow[r] & 0 \end{tikzcd}\] commute. Conversely, every \(g\colon X \to X\) gives rise to such a commutative diagram. In particular, the length of \(S\) over its endomorphism ring coincides with the length of \(S\) over \(\mathrm{End}_A(X)\) which is bounded by the length of \(X\) over \(\mathrm{End}_A(X)\). Now, if \(g\) is a radical morphism, then \(f = 0\) and otherwise both \(g\) and \(f\) are isomorphisms. It follows that \(\mathrm{End}_A(S)\) coincides with \(D= \mathrm{End}_A(X)/ \mathrm{rad}\,\mathrm{End}_A(X)\) which is a division algebra of infinite length over the center of \(A\) since \(X\) is generic. Thus \(S\) is a generic brick.
(2) Consider the socle sequence \[\begin{align} S = S_1 \subsetneq S_2\subsetneq \dots \subsetneq S_n = X \end{align}\] where \(S_i\) equals the intersection of kernels of morphisms in \(M_i = (\mathrm{rad}\, \mathrm{End}_A(X))^i\) and \(n\) is the smallest natural number with the property \(M_n = 0\). Such \(n\) exists by the Nakayama lemma as \(X\) is of finite length over its endomorphism ring. For \(i>1\) there is a natural monomorphism \[\begin{align} S_i/S_{i-1} \longrightarrow \prod S,\quad x+S_{i-1}\mapsto (\varphi(x))_\varphi \end{align}\] where the product goes over all morphisms \(\varphi \in M_{i-1}\). Hence, every torsion-free class \(\mathcal{G} \subseteq \textrm{Mod}\,A\) containing \(S\) also contains \(X\) since \(\mathcal{G}\) is closed under products, submodules and extensions. ◻
Proposition 37. There is, up to isomorphism, an injective assignment \[\begin{Bmatrix} \text{generic \tau^--rigid A-modules X} \end{Bmatrix}\longrightarrow \begin{Bmatrix} \text{generic bricks }S\text{ over }A \end{Bmatrix}\] given by \(X \mapsto \mathrm{soc}_{\mathrm{End}_A(X)} X\).
Proof. Let \(X,X'\) be generic \(\tau^-\)-rigid \(A\)-modules such that \(S = \mathrm{soc}_{\mathrm{End}_A(X)} X\) is isomorphic to \(S' = \mathrm{soc}_{\mathrm{End}_A(X')} X'\). The modules \(S\) and \(S'\) are generic bricks over \(A\) by 36 (1) and there are short exact sequences \[\begin{align} 0 \longrightarrow S \longrightarrow X \longrightarrow Y \longrightarrow 0, \qquad 0 \longrightarrow S' \longrightarrow X' \longrightarrow Y' \longrightarrow 0 \end{align}\] where the first one is inside \(\overline{\textrm{cogen}\,X}\) and the second inside \(\overline{\textrm{cogen}\,X'}\). Since \(S\cong S'\) is contained in both torsion-free classes, it follows that \(\mathcal{G}:=\overline{\textrm{cogen}\,X'} = \overline{\textrm{cogen}\,X}\) by 36 (2). Now \(X\) and \(X'\) are injective in \(\mathcal{G}\) by 9. Thus, the isomorphism \(S\cong S'\) induces a commutative diagram \[\begin{tikzcd} 0 \arrow[r] & S \arrow[d, "\rotatebox{90}{\sim}", swap] \arrow[r] & X \arrow[d, "\alpha"] \arrow[r] & Y \arrow[d] \arrow[r] & 0 \\ 0 \arrow[r] & S' \arrow[d, "\rotatebox{90}{\sim}", swap] \arrow[r] & X' \arrow[d, "\beta"] \arrow[r] & Y' \arrow[d] \arrow[r] & 0\\ 0 \arrow[r] & S \arrow[r] & X \arrow[r] & Y \arrow[r] & 0. \end{tikzcd}\] In particular \((\beta \alpha)^n \neq 0\) for all \(n\in \mathbb{N}\). Because \(X\) has finite length over \(\mathrm{End}_A(X)\) there is \(m \in \mathbb{N}\) with \(\gamma^m = 0\) for every radical morphism \(\gamma\colon X\to X\) by the Nakayama lemma. It follows that \(\beta \alpha\) is an isomorphism and so \(\alpha\) is a split monomorphism. Since \(X,X'\) are indecomposable, we conclude that \(X\cong X'\). ◻
The previous proposition has been independently proven in the work of Angeleri Hügel, Laking, and Pfeifer [23]. We show how to obtain a generic \(\tau^-\)-rigid module from a collection of finite length modules, based on the following lemma.
Lemma 38. Let \(U \subseteq \textrm{Zsp}\,A\) be a collection of finite length modules. The ideal \(\mathcal{I}= \langle \bar{U} \setminus U\rangle\) consists of all morphisms in \(\textrm{mod}\,A\) that factor through a finite direct sum of modules in \(U \setminus V\) for all finite subsets \(V\subseteq U\).
Proof. For every finite subset \(V \subseteq U\) consider the fp-idempotent ideal \(\mathcal{I}_V = \langle U \setminus V\rangle\). By 21 the ideal \(\mathcal{I}_V\) corresponds, under 20, to the closure of \(U\setminus V\) in \(\textrm{Zsp}\,A\) which equals \(\bar{U} \setminus V\) since the finite length modules are open points in \(\textrm{Zsp}\,A\), see [6]. Let \(\mathcal{J} = \bigcap \mathcal{I}_V\) where the intersection goes over all finite subsets \(V \subseteq U\). Then \(\mathcal{J}\) is fp-idempotent as the intersection is directed, see [7]. Now, under 20, the ideal \(\mathcal{J}\) corresponds to the closed set \(\bigcap \bar{U} \setminus V = \bar{U} \setminus U\). Hence \(\mathcal{I} = \mathcal{J}\) consists of all morphisms in \(\textrm{mod}\,A\) that factor through a finite direct sum of modules in \(U\setminus V\) for all finite subsets \(V \subseteq U\). ◻
Proposition 39. Let \(U \subseteq \textrm{Zsp}\,A\) be an infinite collection of finite length modules such that
for all but finitely many \(X\in U\) we have \(\textrm{Hom}_A(\tau^- Y , X ) = 0\) for all but finitely many \(Y \in U\), or
for all but finitely many \(Y\in U\) we have \(\textrm{Hom}_A(\tau^- Y , X) = 0\) for all but finitely many \(X \in U\).
Then \(\bar{U} \setminus U \subseteq \textrm{Zsp}\,A\) is a non-empty \(\tau^-\)-rigid set and the corresponding torsion-free class \(\mathcal{F} = \textrm{cogen}\,\bar{U}\setminus U\) is given by \[\begin{align} \mathcal{F} = \bigcap_{\substack{V\subseteq U\\ \text{finite}}} \textrm{cogen}\,U\setminus V. \end{align}\] Moreover, if the length of \(X\) over \(\mathrm{End}_A(X)\) is bounded for \(X\in U\), then \(\bar{U}\setminus U\) contains a generic \(\tau^-\)-rigid module.
Proof. We only prove the case (1) as (2) is similar. By 23 the set \(\bar{U}\setminus U\) is \(\tau^-\)-rigid if and only if \(\mathcal{I} = \langle \bar{U} \setminus U \rangle\) is \(\tau^-\)-rigid. By 14 the ideal \(\tau^- \mathcal{I}\) agrees with \(\langle \tau^- \bar{U} \setminus \tau^- U \rangle\). With 38 it follows that \(\mathcal{I}\) consists of all morphisms that factor through a finite direct sum of modules in \(U \setminus V\) for all finite subsets \(V\subseteq U\), and \(\tau^- \mathcal{I}\) consists of all morphisms that factor through a finite direct sum of modules in \(\tau^- (U \setminus W)\) for all finite subsets \(W \subseteq U\). Consider a composition \(\varphi \psi\) with \(\varphi\in \mathcal{I}\) and \(\psi \in \tau^- \mathcal{I}\). By assumption, there is a finite subset \(V \subseteq U\) such that for all \(X \in U \setminus V\) we have \(\textrm{Hom}_{A}(\tau^- Y, X) = 0\) for all but finitely many \(Y \in U\). Now \(\varphi\) factors through a finite direct sum \(\bigoplus_{i=1}^n X_i\) with \(X_i \in U \setminus V\). For each \(i\) let \(W_i\) be the finite set of all modules \(Y\in U\) with \(\textrm{Hom}_{A}(\tau^- Y, X_i)\neq 0\) and set \(W = \bigcup_{i=1}^n W_i\). Then \(\psi\) factors through a finite direct sum \(\bigoplus_{j=1}^m \tau^- Y_j\) with \(Y_j \in U \setminus W\). By construction \(\textrm{Hom}_A (\tau^- Y_j, X_i) = 0\) for all \(i,j\) and thus \(\varphi \psi = 0\). It follows that \(\mathcal{I} \circ \tau^- \mathcal{I} = 0\) and \(\bar{U} \setminus U\) is \(\tau^-\)-rigid.
To deduce the desired description of \(\mathcal{F}\) we use that \(\textrm{cogen}\,\bar{U} \setminus U = \textrm{cogen}\,\mathcal{I}\), see 23, as well as the description of \(\mathcal{I}\). For \(Z\in \textrm{mod}\,A\) the injective hull \(Z\to Q\) is contained in \(\mathcal{I}\) if and only if \(Z\in \textrm{cogen}\,U \setminus V\) for all finite subsets \(V\subseteq U\) and so \(\mathcal{F} = \textrm{cogen}\,\bar{U}\setminus U = \textrm{cogen}\,\mathcal{I} = \bigcap_V \textrm{cogen}\,U \setminus V\).
Next, we show that \(\bar{U} \setminus U\) is non-empty. Since \({U}\) is an infinite set and \(\textrm{Zsp}\,A\) is compact [4] the set \(\bar{U}\) contains an accumulation point, which cannot be a finite length module as finite length modules are isolated points, see
[6]. Finally, if the length of \(X\) over \(\mathrm{End}_A(X)\) is bounded by some \(n\in \mathbb{N}\) for \(X\in U\) then the length of \(Y \in \bar{U}\setminus U\) over \(\mathrm{End}_A(Y)\) is bounded by \(n\), see [24].
Thus, in this case every \(Y \in \bar{U} \setminus U\) is a generic \(\tau^-\)-rigid module. ◻
We are now ready to show the equivalence of (2) and (3) in 35 over tame algebras, which was first proven in [9] using matrix reduction techniques.
Theorem 40. For a tame finite dimensional algebra \(A\) over an algebraically closed field, the following are equivalent.
There are infinitely many non-isomorphic finite dimensional bricks of the same dimension.
There exists a generic brick.
Proof. (1)\(\implies\)(2) By [25] we may assume that there is an infinite hom-orthogonal collection \(U\) of non-isomorphic bricks of the same dimension \(n\). That is \(\textrm{Hom}_A(X,Y) = 0\) for all \(X\neq Y\) in \(U\). Over a tame algebra, all but finitely many of these modules are \(\tau\)-invariant [26]. It follows that \(U\) fulfills the assumptions in 39 and so \(\bar{U}\setminus U\) contains a generic \(\tau^-\)-rigid module, as the length of \(X\in U\) over \(\mathrm{End}_A(X)\) is bounded by \(n\). By 36 the existence of a generic \(\tau^-\)-rigid module implies the existence of a generic brick.
(2)\(\implies\)(1) Let \(k\) be the ground field. By [27] every generic \(A\)-module \(X\) is of the form \(X\cong {}_AM_{k[t]_f} \otimes_{k[t]_f} k(t)\) such that
\(M\) is free over \(k[t]_f\) of finite rank,
\(\mathrm{End}_A(X)/ \mathrm{rad}\, \mathrm{End}_A(X) \cong k(t)\), and
\(M\otimes_{k[t]_f} k[t]/(t-\lambda)\) for \(f(\lambda) \neq 0\) are all non-isomorphic \(A\)-modules.
Here \(k[t]_f\) is the ring of polynomials in one variable over \(k\) localized at \(f\in k[t]\). Assume that \(X\) is a brick, so \(\mathrm{End}_A(X) \cong k(t)\). Since \(M\) is free over \(k[t]_f\) of finite rank, it follows that after a finite localization \(M_g\) with \(g \in k[t]_f\) base change commutes with taking the endomorphism ring. Thus \(\mathcal{E} = \mathrm{End}_{A\otimes_k k[t]_{f,g}} (M_{g})\) fulfills \(\mathcal{E} \otimes_{k[t]_{f,g}} k(t) \cong \mathrm{End}_{A\otimes_k k(t)} (X) \cong k(t)\) and so \[\begin{align} \mathrm{End}_{A} (M \otimes_{k[t]_{f,g}} k [t]/(t-\lambda)) \cong \mathcal{E} \otimes_{k[t]_{f,g}} k[t]/(t-\lambda) \cong k \end{align}\] for all but finitely many \(\lambda \in k\). This yields the desired family of infinitely many non-isomorphic bricks. ◻
The Krull-Gabriel dimension, introduced by Geigle [10], is a measure for the complexity of the module category. We show the full 35 under the assumption that the Krull-Gabriel dimension of \(A\) is defined. In this case, the Ziegler spectrum has the following nice property.
Lemma 41. If the Krull-Gabriel dimension of \(A\) is defined, then every closed subset \(U \subseteq \textrm{Zsp}\,A\) that contains infinitely many finite length modules or an infinite length module must contain a generic module.
Proof. See the proof of [28]. ◻
Theorem 42. Let \(A\) be a finite dimensional algebra over an algebraically closed field whose Krull-Gabriel dimension is defined. Then \(A\) fulfills 35.
Proof. Clearly (2) implies (1) in 35. Under the assumptions \(A\) is tame, see for example [6]. It follows that (2) and (3) are equivalent by 40. In fact, we only need to use that (3) implies (2). We are left to show that (1) implies (3).
By [29] the existence of infinitely many non-isomorphic bricks implies that there exists a torsion-free class \(\mathcal{F}\) that is not functorially finite. Let \(U\subseteq \textrm{Zsp}\,A\) be the associated support \(\tau^-\)-tilting subset. Then \(U\) contains infinitely many finite length modules or an infinite length module, as otherwise \(\mathcal{F} = \textrm{cogen}\,U\) would be functorially finite, see [30]. By 41 the subset \(U\) contains a generic module, which is necessarily \(\tau^-\)-rigid. It follows from 36 that \(A\) admits a generic brick. ◻
So far, we have often considered generic bricks and generic \(\tau^-\)-rigid modules over tame algebras. It is surprising that these classes of modules coincide in this case.
Theorem 43. Let \(A\) be a tame finite dimensional algebra over an algebraically closed field. Then, a generic \(A\)-module \(X\) is \(\tau^-\)-rigid if and only if \(X\) is a brick.
Proof. Over a tame algebra, every generic \(A\)-module \(X\) is the unique accumulation point of infinitely many \(\tau\)-stable finite dimensional \(A\)-modules inside \(\textrm{Zsp}\,A\) and so \(X\) is \(\tau\)-stable by 14. This was already observed in [5]. Now, if \(X\) is also a brick, then every morphism \(X\to X\) with a finite dimensional image must be a non-isomorphism and hence zero. Thus \(X\) is \(\tau^-\)-rigid.
Suppose that \(X\) is a generic \(\tau^-\)-rigid \(A\)-module and let \(S = \mathrm{soc}_{\mathrm{End}_A(X)} X\) be the associated generic brick from the injective assignment in 37. Then \(S\) is again \(\tau^-\)-rigid and the corresponding brick equals \(\mathrm{soc}_{\mathrm{End}_A(S)} S = S\) since \(\mathrm{End}_A(S)\) is a division algebra. By injectivity \(X\cong S\) and so \(X\) is a brick. ◻
Remark 44. Let \(A\) be a tame finite dimensional algebra over an algebraically closed field \(k\). In the proof of (2)\(\implies\)(1) of 40 we have seen that if a generic \(A\)-module \(X\) is a brick, then \(M(\lambda)\) is a brick for almost all \(\lambda\in k\) where \(M\) is the geometric realization of \(X\). Slightly adapting the proof of (1)\(\implies\)(2) shows that if \(M(\lambda)\) is a brick for infinitely many \(\lambda \in k\) then \(X\) is \(\tau^-\)-rigid and hence a brick by 43. This recovers the main result of [9]. One has to use that \(X\) is an accumulation point of the \(M(\lambda)\) in \(\textrm{Zsp}\,A\), see [31].