ON THE LATTICE OF COTILTING MODULES

ASLAK BAKKE BUAN, HENNING KRAUSE, AND ØYVIND SOLBERG

1. The lattice

Let Λ be an associative ring. In this note we show that the collection of (not necessarily finitely generated) cotilting modules over Λ carries the structure of a lattice. We work in the category Mod Λ of (right) Λ-modules and denote by mod Λ the full subcategory of finitely presented Λ-modules. Changing slightly^{1} the definition in [1], we say that a Λ-module T is a cotilting module if

(T1) the injective dimension of T is finite;

(T2) Ext_{Λ}^{i}(T^{α}, T) = 0 for all i > 0 and every cardinal α;

(T3) there exists an injective cogenerator Q and a long exact sequence 0 → T_{n} →

⋯ → T_{1} → T_{0} → Q → 0 with T_{i} in Prod T for all i = 0, 1, …, n; (T4) T is pure-injective. Here, Prod T denotes the closure under products and direct factors of T. Two

cotilting modules T and T′ are equivalent if Prod T = Prod T′. Our first result is a consequence of the fact that the equivalence class of a cotilting module T is determined by

⊥T = {X ∈ Mod Λ | Ext_{Λ}^{i}(X, T) = 0 for all i ≥ 1}.

Theorem 1.1. The equivalence classes of Λ-cotilting modules form a set of cardinality at most 2^{κ} where κ = max(ℵ_{0}, card Λ).

Proof. Recall that a class X of Λ-modules is definable if X is closed under taking products, filtered colimits, and pure submodules. In this case

X = {X ∈ Mod Λ | Hom_{Λ}(ϕ, X) is surjective for all ϕ ∈ Φ}

where Φ is the set of all maps in mod Λ such that Hom_{Λ}(ϕ, X) is surjective for all X ∈ X; see [4, Section 2.3].

If T is a cotilting module, then ^{⊥}T is definable. This follows from Theorem 5.6 and Proposition 5.7 in [9]. The cardinality of the set of isomorphism classes of maps in mod Λ is bounded by κ, and therefore we have at most 2^{κ} equivalence classes of cotilting modules. □

We denote by Cotilt Λ the set of equivalence classes of Λ-cotilting modules and we have a natural partial ordering via

T ≤ T′ ⟺ ^{⊥}T ⊆ ^{⊥}T′

for T, T′ ∈ Cotilt Λ. For finite dimensional algebras, the collection of finitely generated (co)tilting modules has some interesting combinatorial structure which is closely related to this partial ordering [10, 11, 3]. Our aim is to show that Cotilt Λ is in fact a lattice. For this we need the concept of a cotorsion pair. We fix a pair (X , Y) of full subcategories of Mod Λ. Let

1(T4) is added to avoid set-theoretic problems. For instance, the classification of modules satisfying (T1) – (T3) over a fixed Dedekind domain R seems to depend on set-theoretic properties of R.

X^{⊥} = {Y ∈ Mod Λ | Ext_{Λ}^{i}(X, Y) = 0 for all i ≥ 1 and X ∈ X },
⊥Y = {X ∈ Mod Λ | Ext_{Λ}^{i}(X, Y) = 0 for all i ≥ 1 and Y ∈ Y}.

The pair (X , Y) is called a cotorsion pair for Mod Λ if the following conditions are satisfied:

(1) X = ^{⊥}Y and Y = X^{⊥};

(2) every A ∈ Mod Λ fits into exact sequences 0 → Y_{1} → X_{1} → A → 0 and

0 → A → Y_{2} → X_{2} → 0 with X_{i} ∈ X and Y_{i} ∈ Y. For n ∈ ℕ we write I_{n}(Λ) = {X ∈ Mod Λ | id X ≤ n} and let I(Λ) = ⋃_{n∈ℕ} I_{n}(Λ), where id X denotes the injective dimension of a module X. We need the following example:

Example 1.2. For all n ∈ ℕ there exists a cotorsion pair (^{⊥}I_{n}(Λ), I_{n}(Λ)). This follows from Theorem 10 in [6] since

I_{n}(Λ) = {Y ∈ Mod Λ | Ext_{Λ}^{1}(Ω^{n}(Λ/𝔞), Y) = 0 for all right ideals 𝔞 ⊆ Λ}

by Baer’s criterion.

We have a description of cotilting modules in terms of cotorsion pairs which follows directly from work of Angeleri Hügel and Coelho [1, Theorem 4.2], in combination with [9, Proposition 5.7].

Proposition 1.3. For a full subcategory X ⊆ Mod Λ the following are equivalent: (1) X = ^{⊥}T for some cotilting module T with id T ≤ n;

(2) X is definable and there is a cotorsion pair (X , X^{⊥}) with X^{⊥} ⊆ I_{n}(Λ). Moreover, in this case X ∩ X^{⊥} = Prod T.

Observe that Proposition 1.3 shows how to compute for a cotilting module T its injective dimension:

id T = inf{n ∈ ℕ | ^{⊥}I_{n}(Λ) ⊆ ^{⊥}T }.

The next result describes the infimum of a collection of cotilting modules in Cotilt Λ.

Proposition 1.4. Let (T_{i})_{i∈I} be a family of cotilting modules and suppose that sup{id T_{i} | i ∈ I} < ∞. Then there exists a cotilting module T such that

⊥T = ⋂_{i∈I} ^{⊥}T_{i}.

Moreover, id T = sup{id T_{i} | i ∈ I}. The module T is unique up to equivalence and is denoted by ⋀_{i∈I} T_{i}.

Proof. We apply Proposition 1.3. There exists a cotorsion pair (X , Y) with X = ⊥(∏_{i} T_{i}) since ∏_{i} T_{i} is pure-injective; see [5, Corollary 10]. Each ^{⊥}T_{i} is definable and contains ^{⊥}I_{n}(Λ) where n = sup{id T_{i} | i ∈ I}. Therefore X = ⋂_{i∈I} ^{⊥}T_{i} is definable and contains ^{⊥}I_{n}(Λ). Thus Y ⊆ I_{n}(Λ), and we obtain X = ^{⊥}T for some cotilting module T. □

Example 1.5 (Happel). Fix a field k and let Λ be the path algebra of the quiver \xymatrix{· \ar[r] \ar[rr] & · \ar[r] & ·} which is tame hereditary. Denote by S_{1} = (1, 0, 1) and S_{2} = (0, 1, 0) the quasi-simples from the unique exceptional tube of rank 2. Then there are cotilting modules

T_{1} = (1, 0, 1) ⊕ (2, 1, 1) ⊕ (1, 0, 0) and T_{2} = (0, 1, 0) ⊕ (2, 2, 1) ⊕ (1, 1, 0)

such that ^{⊥}T_{1} ∩^{⊥}T_{2} = ^{⊥}T for T = Ŝ_{1} ⨿ Ŝ_{2} ⨿ (∐_{S} S_{∞}) where S runs through the isomorphism classes of quasi-simples different from S_{1} and S_{2}. Here, S_{∞} denotes the Prüfer and Ŝ denotes the adic module corresponding to S. Moreover, no finite dimensional cotilting module is equivalent to T.

Corollary 1.6. The partially ordered set Cotilt Λ is a lattice. More precisely, for a family (T_{i})_{i∈I} in Cotilt Λ the following holds:

(1) The infimum inf{T_{i} | i ∈ I} of all T_{i} exists if and only if sup{id T_{i} | i ∈ I} < ∞. In this case inf{T_{i} | i ∈ I} = ⋀_{i∈I} T_{i}.

(2) The supremum sup{T_{i} | i ∈ I} of all T_{i} equals the infimum inf{T ∈ Cotilt Λ | T_{i} ≤ T for all i ∈ I}.

Corollary 1.7. The map (Cotilt Λ, ≤) ⟶ (ℕ, ≤) sending T to id T has the following properties:

(1) T ≤ T′ implies id T′ ≤ id T.

(2) id(inf{T_{i} | i ∈ I}) = sup{id T_{i} | i ∈ I} for every family (T_{i})_{i∈I}, provided

that sup{id T_{i} | i ∈ I} < ∞.
(3) id(sup{T_{i} | i ∈ I}) ≤ inf{id T_{i} | i ∈ I} for every family (T_{i})_{i∈I}.

2. Finitistic dimension

In this section we relate the finitistic dimension of Λ to the structure of Cotilt Λ^{op}. Recall that the finitistic dimension Fin. dim Λ is the supremum of all projective dimensions of Λ-modules having finite projective dimension. Restriction to finitely presented Λ-modules gives fin. dim Λ. The finitistic injective dimension of Λ is by definition

Fin. inj. dim Λ = sup{id X | X ∈ Mod Λ and id X < ∞}.

Observe that Fin. dim Λ = Fin. inj. dim Λ^{op} provided that Λ is artinian.

Proposition 2.1. Let Λ be an artin algebra and let C be a class of finitely presented Λ-modules. If id C = sup{id X | X ∈ C} < ∞, then there exists a cotilting module T such that ^{⊥}T = ^{⊥}C and id T = id C.

Proof. We apply Proposition 1.3 to obtain the cotilting module T satisfying ^{⊥}T = ⊥C. It follows from Theorem 2 in [9] that every definable and resolving subcategory X of Mod Λ induces a cotorsion pair (X , X^{⊥}). Recall that X is resolving if X is closed under extensions, kernels of epimorphisms, and contains all projectives. Clearly, ^{⊥}C is resolving. Using the fact that the modules in C are finitely presented, it is not difficult to check that ^{⊥}C is definable; see for example the proof of [9, Corollary 6.4]. Finally, we have ^{⊥}I_{n}(Λ) ⊆ ^{⊥}C if and only if C ⊆ I_{n}(Λ), because (^{⊥}I_{n}(Λ))^{⊥} = I_{n}(Λ). Therefore id T = id C. □

Corollary 2.2. Let Λ be an artin algebra. Then

Fin. dim Λ ≥ sup{id T | T ∈ Cotilt Λ^{op}} ≥ fin. dim Λ.

3. Minimal cotilting modules

If Fin. inj. dim Λ < ∞, then we define

T_{min} = ⋀_{T ∈Cotilt Λ} T

to be the (unique) minimal element in Cotilt Λ. We have always ^{⊥}I(Λ) ⊆ ^{⊥}T_{min} and in this section we ask when both subcategories are equal. To this end we introduce another module which is of potential interest.

Lemma 3.1. Let Λ be right noetherian and suppose that Fin. inj. dim Λ < ∞. Then there exists a Λ-module T such that

⊥I(Λ) ∩ I(Λ) = Add T.

Proof. We have a cotorsion pair (^{⊥}I(Λ), I(Λ)) since Fin. inj. dim Λ < ∞. Observe that I(Λ) and ^{⊥}I(Λ) both are closed under taking kernels of epimorphisms. Therefore every epimorphism in ^{⊥}I(Λ) ∩ I(Λ) splits. Now fix an exact sequence 0 → Λ → T → X → 0 with T ∈ I(Λ) and X ∈ ^{⊥}I(Λ). Clearly, T ∈ ^{⊥}I(Λ) ∩ I(Λ). Taking coproducts we get for each cardinal α an exact sequence 0 → Λ^{(α)} → T^{(α)} → X^{(α)} → 0 with T^{(α)} ∈ ^{⊥}I(Λ) ∩ I(Λ) and X^{(α)} ∈ ^{⊥}I(Λ), since I(Λ) is closed under coproducts. Thus every map ϕ: Λ^{(α)} → Y with Y ∈ I(Λ) factors through Λ^{(α)} → T^{(α)} via some map ϕ′: T^{(α)} → Y. If Y ∈ ^{⊥}I(Λ) ∩ I(Λ) and ϕ is an epi, then ϕ′ splits. Thus ^{⊥}I(Λ) ∩ I(Λ) = Add T. □

By abuse of notation we denote by T_{inj} a module satisfying ^{⊥}I(Λ) ∩ I(Λ) = Add T_{inj}.

Lemma 3.2. Let Λ be right noetherian and suppose that Fin. inj. dim Λ = n < ∞. Then a Λ-module C has finite injective dimension if and only if there is an exact sequence

(∗)  0 ⟶ T_{n+1} ⟶ ⋯ ⟶ T_{1} ⟶ T_{0} ⟶ C ⟶ 0

with T_{i} ∈ Add T_{inj} for all i.

Proof. We have a cotorsion pair (^{⊥}I(Λ), I(Λ)). Starting with Y_{0} = C ∈ I(Λ), we obtain exact sequences ε_{i}: 0 → Y_{i+1} → T_{i} → Y_{i} → 0 for each i ≥ 0, with Y_{i} ∈ I(Λ) and T_{i} ∈ Add T_{inj} for all i. Using dimension shift, one sees that ε_{n} splits. Thus Y_{n+1} ∈ Add T_{inj}, and splicing together the ε_{i} produces a sequence of the form (∗). Conversely, if C fits into a sequence (∗), then C ∈ I(Λ) since I(Λ) is closed under taking cokernels of monomorphisms. □

Recall that a module C is Σ-pure-injective if every coproduct C^{(α)} is pureinjective.

Theorem 3.3. Let Λ be right noetherian and suppose that Fin. inj. dim Λ < ∞. Then the following are equivalent:

(1) ^{⊥}I(Λ) = ^{⊥}T_{min};

(2) ^{⊥}I(Λ) is closed under taking products;

(3) T_{inj} is product complete, that is, Add T_{inj} = Prod T_{inj};

(4) T_{inj} is a Σ-pure-injective cotilting module. Moreover, in this case T_{min} and T_{inj} are equivalent cotilting modules.

Proof. (1) ⇒ (2): Clear, since ^{⊥}T_{min} is closed under products.

(2) ⇒ (3): If ^{⊥}I(Λ) is closed under products, then ^{⊥}I(Λ) ∩ I(Λ) is closed under products. Thus every product of copies of T_{inj} belongs to Add T_{inj}. It follows that T_{inj} is Σ-pure-injective and therefore Add T_{inj} ⊆ Prod T_{inj}; see [8]. Thus T_{inj} is product complete.

(3) ⇒ (4): A product complete module is Σ-pure-injective. For T_{inj}, the defining conditions of a cotilting module are obviously satisfied, except (T3) which follows from Lemma 3.2.

(4) ⇒ (1): First observe that Add T_{inj} ⊆ Prod T_{inj} since T_{inj} is Σ-pure-injective. The cotilting module T_{inj} induces a cotorsion pair (^{⊥}T_{inj}, (^{⊥}T_{inj})^{⊥}) by Proposition 1.3. We claim that I(Λ) = (^{⊥}T_{inj})^{⊥}. We need to check I(Λ) ⊆ (^{⊥}T_{inj})^{⊥} and this follows from Lemma 3.2 since Add T_{inj} ⊆ Prod T_{inj}. Thus ^{⊥}I(Λ) = ^{⊥}T_{inj} and therefore T_{inj} is equivalent to the minimal cotilting module T_{min}. □

Remark 3.4. A cotilting module T is Σ-pure-injective if and only if (^{⊥}T)^{⊥} is closed under coproducts. In this case let T′ be the coproduct of a representative set of indecomposable modules in Prod T. Then T′ is a product complete cotilting module which is equivalent to T.

It seems to be an interesting project to describe the minimal cotilting module for a given algebra. For example, T_{min} = Λ if Λ is a Gorenstein artin algebra.

In fact, there is a more general result which discribes when T_{min} is finitely presented. This is inspired by a result about modules of finite projective dimension by Huisgen-Zimmermann and Smalø [7].

Proposition 3.5. Let Λ be an artin algebra. Then there exists a finitely presented minimal cotilting module if and only if the modules of finite injective dimension form a covariantly finite subcategory of mod Λ. Moreover, in this case the equivalent conditions of Theorem 3.3 are satisfied.

A similar result has been obtained independently by Happel and Unger for the category of finitely presented Λ-modules.

We do not give the complete proof but sketch the argument. Suppose first that I(mod Λ) = {X ∈ mod Λ | id X < ∞} is covariantly finite. Using the correspondence for cotilting modules in mod Λ, there exists a cotilting module T such that ⊥T = ^{⊥}I(mod Λ) in mod Λ; see [2]. The assumption implies that every module of finite injective dimension is a filtered colimit of modules in I(mod Λ). Using this, one proves that T is minimal. Conversely, if T_{min} is finitely presented, then one can use Proposition 2.1 to show that I(mod Λ) is covariantly finite in mod Λ.

References

[1] L. Angeleri Hügel and F. U. Coelho: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13 (2001), 239–250.

[2] M. Auslander and I. Reiten: Applications of contravariantly finite subcategories. Adv. Math. 86 (1991), 111–152.

[3] A. B. Buan and Ø. Solberg: Limits of pure-injective cotilting modules. Preprint (2001).

[4] W. W. Crawley-Boevey: Infinite-dimensional modules. In: The representation theory of finite-dimensional algebras. Algebras and modules, I (eds. I. Reiten, S. O. Smalø, and Ø. Solberg), CMS Conf. Proc. 23 (1998), 29–54.

[5] P. Eklof and J. Trlifaj: Covers induced by Ext. J. Algebra 231 (2000), 640–651.

[6] P. Eklof and J. Trlifaj: How to make Ext vanish. Bull. London Math. Soc. 33 (2001), 41–51.

[7] B. Huisgen-Zimmermann and S. O. Smalø: A homological bridge between finite- and infinite-dimensional representations of algebras. Algebr. Represent. Theory 1 (1998), 169– 188.

[8] H. Krause and M. Saorín: On minimal approximations of modules. In: Trends in the representation theory of finite dimensional algebras (eds. E. Green and B. Huisgen-Zimmermann), Contemp. Math. 229 (1998), 227–236.

[9] H. Krause and Ø. Solberg: Filtering modules of finite projective dimension. To appear in Forum Math.

[10] C. Riedtmann and A. Schofield: On a simplicial complex associated with tilting modules. Comment. Math. Helv. 66 (1991), 70–78.

[11] L. Unger: Shellability of simplicial complexes arising in representation theory. Adv. Math. 144 (1999), 211–246.

[AMA - Algebra Montpellier Announcements - 01-2002] [February 2002]
Received January 2002.

Aslak Bakke Buan, Øyvind Solberg, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

   E-mail address: aslakb@math.ntnu.no, oyvinso@math.ntnu.no

Henning Krause, Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany

   E-mail address: henning@mathematik.uni-bielefeld.de