PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 4, April 1998, Pages 983–987 S 0002-9939(98)04170-7

FINITISTIC DIMENSION AND ZIEGLER SPECTRUM

HENNING KRAUSE

(Communicated by Ken Goodearl)

Abstract. Given a two-sided artinian ring Λ, it is shown that the Ziegler spectrum of Λ forms a test class for certain homological properties of Λ. We discuss the finitistic dimension of Λ, Nunke’s condition, and also the relation between the big and the little finitistic dimension.

Let Λ be a two-sided artinian ring. Denote by Mod Λ the category of (right) Λ-modules and by mod Λ the full subcategory of all finitely presented Λ-modules. Given a Λ-module M, we denote by pd M its projective dimension, and the finitistic dimension Fin. dim Λ of Λ is the supremum of the projective dimensions of the Λ-modules with finite projective dimension. The Ziegler spectrum Zsp Λ of Λ is by definition the set of isomorphism classes of indecomposable pure-injective Λmodules [14].

The aim of this note is to show that the Ziegler spectrum forms a test class for certain homological properties of Λ. Furthermore, the Ziegler spectrum carries a topology, and we shall use its compactness to obtain an equivalent formulation of Nunke’s condition. In the final part of this note we study the relation between Fin. dim Λ and the little finitistic dimension fin. dim Λ of Λ.

Our first result is easily stated as follows.

Theorem 1. Fin. dim Λ = sup{pd M | M ∈ Zsp Λ and pd M < ∞}.

We postpone the proof and first discuss some related homological properties of Λ. Suppose that Λ has a self-duality D between mod Λ and mod Λ^{op}. We are interested in Nunke’s condition [9]:

Nd M < ∞ for every Λ-module M ≠ 0,

where Nd M denotes the smallest integer n (or ∞ if such an integer does not exist) such that Ext_{Λ}^{n}(DΛ, M) ≠ 0. It is convenient to define the Nunke dimension of Λ as follows:

Nun. dim Λ = sup{Nd M | M ∈ Mod Λ and M ≠ 0}.

The following relation between Nunke dimension and finitistic dimension is essentially due to Jans [10] (see also [5]).

Proposition 2. Nun. dim Λ ≤ Fin. dim Λ.

Proof. We use the fact that the functor

Inj Λ ⟶ Proj Λ, X ↦ Hom_{Λ}(DΛ, X)
\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors June 7, 1996 and, in revised form, September 25, 1996.

1991 Mathematics Subject Classification. Primary 16E10, 16G10.

is an equivalence between the full subcategories of injective and projective Λmodules. Suppose now that Nun. dim Λ ≥ n and choose a non-zero Λ-module M with ∐_{i=0}^{n−1} Ext_{Λ}^{i}(DΛ, M) = 0. Take an injective coresolution

0 ⟶ M ⟶ I_{0} ⟶^{ψ_{0}} I_{1} ⟶^{ψ_{1}} I_{2} ⟶^{ψ_{2}} …

and put N = Coker Hom_{Λ}(DΛ, ψ_{n−1}). It is easily checked that pd N = n. Thus Fin. dim Λ ≥ n.

Remark. Given any n ∈ ℕ_{0}, there is a finite dimensional algebra Λ over a field (with rad^{2} Λ = 0 and Nun. dim Λ = 2) such that Fin. dim Λ − Nun. dim Λ = n [7].

Theorem 3. Nun. dim Λ = sup{Nd M | M ∈ Zsp Λ}, and Nunke’s condition holds if and only if Nun. dim Λ < ∞.

Some further definitions are needed in order to give the proof of both theorems. An additive functor F: Mod Λ → Ab into the category of abelian groups is finitely presented if there is a presentation

Hom_{Λ}(Y, ) ⟶ Hom_{Λ}(X, ) ⟶ F ⟶ 0

with X and Y in mod Λ. We denote by Ker F the full subcategory of Λ-modules M with F (M) = 0. A full subcategory X of Mod Λ is definable if there is a family F_{i}, i ∈ I, of finitely presented functors such that X = ⋂_{i∈I} Ker F_{i}. A definable subcategory is automatically closed under direct limits, products and pure submodules, and it can be shown that this property characterizes a definable subcategory [12]. The notion of a definable subcategory allows us to describe the topology on Zsp Λ which was first introduced by Ziegler in model theoretic terms [14].

Proposition 4. (1) The assignment X ↦ X ∩ Zsp Λ defines a bijection between the definable subcategories of Mod Λ and the closed subsets of Zsp Λ.

(2) Zsp Λ is a quasi-compact space.

Proof. (1) This result is due to Crawley-Boevey [3], and he relies on work of Herzog [6] and Ziegler [14]. We refer to [12] for a proof which uses the localization theory for locally coherent categories developed in [11]. However, the inverse of the assignment X ↦ X ∩ Zsp Λ is easily constructed; it sends a closed subset U of Zsp Λ to ⋂_{F (U)=0} Ker F.

(2) See [14] or [11].

The next lemma shows that some relevant subcategories of Mod Λ are definable.

Lemma 5. Let n ∈ ℕ_{0}.

(1) The full subcategory P_{n} of Λ-modules M with pd M ≤ n is definable.

(2) The full subcategory Q_{n} of Λ-modules M with Nd M ≥ n is definable.

Proof. Projective dimension and flat dimension coincide for any module over an artinian ring, and therefore P_{n} = ⋂_{X∈mod Λ^{op}} Ker Tor_{n+1}^{Λ}( , X) since Tor_{n+1}^{Λ}(M, ) commutes with direct limits and every module can be written as a direct limit of finitely presented modules. Therefore P_{n} is definable, and Q_{n} is definable since Q_{n} = Ker ∐_{i=0}^{n−1} Ext_{Λ}^{i}(DΛ, ).

An easy consequence of the preceding lemma is the following.

Proposition 6. Let U be a subset of Zsp Λ. Then pd M ≤ sup{pd N | N ∈ U} for every module M in the closure of U.

Proof of Theorem 1. Suppose that P_{n} is properly contained in P_{n+1}. The first part of Proposition 4 then implies there is M ∈ Zsp Λ with M ∈ P_{n+1} \ P_{n}, since P_{n} and P_{n+1} are definable. Thus the assertion follows.

Proof of Theorem 3. The proof of the first part of the assertion is analogous to that of Theorem 1. Suppose now that Nunke’s condition holds. Thus ⋂_{n≥0} Q_{n} = 0, and the compactness of Zsp Λ then implies the existence of d ∈ ℕ_{0} with ⋂_{n=0}^{d} Q_{n} = 0. We conclude that Nun. dim Λ < d.

Remark. The finitistic dimension of Λ is finite if and only if the full subcategory of all Λ-modules with finite projective dimension is definable. Therefore one might conjecture that Fin. dim Λ < ∞ if and only if the modules in the Ziegler spectrum having finite projective dimension form a closed subset of Zsp Λ.

The rest of this paper is devoted to studying the relation between the finitistic dimension Fin. dim Λ and the little finitistic dimension fin. dim Λ, which is the supremum of the projective dimensions of the finitely presented Λ-modules with finite projective dimension. Note that there are examples of finite dimensional algebras where fin. dim Λ is different from Fin. dim Λ [15]. Our results are motivated by recent work of Huisgen-Zimmermann and Smalø, who have shown that both dimensions coincide for an interesting class of Artin algebras [8].

In order to state the next result we recall from [2] that for any subcategory C of Mod Λ a morphism M → M′ is a left C-approximation of M if M′ lies in C and the induced map Hom_{Λ}(M′, X) → Hom_{Λ}(M, X) is surjective for every X in C. We denote by P_{∞} the full subcategory of all Λ-modules with finite projective dimension.

Theorem 7. The following conditions are equivalent.

(1) A Λ-module has finite projective dimension if and only if it is a direct limit of finitely presented modules having finite projective dimension.

(2) The little finitistic dimension fin. dim Λ is finite, and every finitely presented Λ-module has a left P_{∞}-approximation which is finitely presented.

Moreover, if (1)–(2) hold, then fin. dim Λ = Fin. dim Λ < ∞.

Examples of rings which satisfy the conditions in this theorem are Artin algebras Λ where every finitely presented Λ-modules has a right P_{∞} ∩ mod Λ-approximation [1, 8]. There are also examples of Artin algebras Λ which show that in condition (2) the term “P_{∞}-approximation” cannot be replaced by “P_{∞} ∩mod Λ-approximation” [13]. The proof of Theorem 7 is based on the following lemma.

Lemma 8. Let C be any additive subcategory of mod Λ. Then a Λ-module M is a direct limit of objects in C iff every morphism X → M with X ∈ mod Λ factors through an object in C.

Proof. See [4, Lemma 4.1].

Proof of Theorem 7. (1) ⇒ (2) In order to show that fin. dim Λ < ∞ assume there is a family M_{i}, i ∈ ℕ, in mod Λ with pd M_{i} < pd M_{i+1} < ∞ for all i. Then ∐_{i∈ℕ} M_{i} is a direct limit of finitely presented objects in P_{∞}, but pd ∐_{i∈ℕ} M_{i} = ∞. Thus (1) implies fin. dim Λ = n < ∞ and therefore Fin. dim Λ = n, since P_{n} is closed under direct limits by Lemma 5. To show the second part of the assertion we use an idea from [4]. Let M be in mod Λ and choose a representative set φ_{i}: M → N_{i}, i ∈ I, of morphisms in mod Λ such that the domain has finite projective dimension. Then N = ∏_{i∈I} N_{i} lies in P_{∞}, since P_{∞} = P_{n} is closed under products by Lemma 5. Thus the induced morphism φ: M → N factors through a finitely presented object M′ in P_{∞} by the preceding lemma. It is easily checked that the corresponding morphism M → M′ is a left P_{∞}-approximation of M.

(2) ⇒ (1) It follows immediately from the preceding lemma that every module in P_{∞} is a direct limit of finitely presented modules in P_{∞}. Conversely, the projective dimension of a direct limit of finitely presented Λ-modules in P_{∞} is bounded by fin. dim Λ = n, since P_{n} is closed under direct limits by Lemma 5.

If the ring Λ has the property that every finitely presented Λ-module is pureinjective, for instance if Λ is an Artin algebra, then condition (1) in the preceding theorem implies that the finitely presented modules form a dense subset of {M ∈ Zsp Λ | pd M < ∞}. We point out that this condition is already sufficient for the equality fin. dim Λ = Fin. dim Λ.

Corollary 9. Let n ∈ ℕ_{0} and suppose that the finitely presented modules M ∈ Zsp Λ with pd M ≤ n form a dense subset of {M ∈ Zsp Λ | pd M < ∞}. Then fin. dim Λ = Fin. dim Λ ≤ n.

Proof. Let U_{n} = {M ∈ Zsp Λ | pd M ≤ n and M ∈ mod Λ}, and we may assume that n ∈ ℕ_{0} is the minimal choice with \{M ∈ Zsp Λ | pd M < ∞\} ⊆ \overline{U_{n}}. It follows from Proposition 6 and Theorem 1 that Fin. dim Λ ≤ n, and we therefore obtain n ≤ fin. dim Λ ≤ Fin. dim Λ ≤ n. Thus fin. dim Λ = Fin. dim Λ = n.

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Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

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