Canadian Mathematical Society Conference Proceeding Volume 00, 0000

Repetitive resolutions over classical orders and finite dimensional algebras

K. R. GOODEARL AND B. HUISGEN-ZIMMERMANN

I. Introduction

It has long been known that a great deal of homological information can be gleaned from repetitions occurring in projective and injective resolutions. A fundamental – and ancient – example is the (ℤ/p^{n}ℤ)-module ℤ/pℤ, whose successive syzygies alternate between pℤ/p^{n}ℤ and ℤ/pℤ; from this phenomenon one immediately deduces that the module has infinite projective dimension. Numerous authors have systematized various kinds of repetitive homological behavior, e.g., with the notion of an ‘ultimately closed’ projective resolution which is due to Jans [13], or that of a resolution ‘with a strongly redundant image’ which was introduced by Colby and Fuller [4] (see also [6] and [7]). The consequences of various types of repetitive homology have also been studied by Igusa-Zacharia [12], Kirkman-Kuzmanovich-Small [19], and Wilson [24], to name a few other instances among many.

In the first part of this note, we introduce some variations on the theme of repetitiveness and show that the corresponding invariants lead to very good – and quite accessible – upper bounds on various finitistic dimensions in terms of individual modules. These invariants are the ‘repetition index’ and the ‘syzygy type’ of a module M over an artinian ring Λ. The repetition index, rep(M), measures the degree of repetitiveness among non-projective direct summands of the syzygies of M; to say that rep(M) = n < ∞, means tightening somewhat the Colby-Fuller concept of a ‘projective resolution with a strongly redundant image from n’. The syzygy type of M measures the number of indecomposable modules among direct summands of the syzygies of M; in case the module M is finitely generated, finiteness of its syzygy type is equivalent to the existence of

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

0000-0000/00 $1.00 + $.25 per page

The research of both authors was partially supported by National Science Foundation grants.

an ultimately closed projective resolution in the sense of Jans. We prove that if T is a right Λ-module which contains an isomorphic copy of Λ/ rad(Λ), then the left big finitistic dimension of Λ is bounded above by rep(T), which in turn is bounded above by the syzygy type of T. On one hand, this point of view leads to several methods for bounding finitistic dimensions; on the other, it frequently allows us to reduce the computation of a host of projective dimensions to the task of resolving a single module.

Our second goal here is to present a new class of algebras Λ for which the global repetition index (the supremum of the repetition indices of all Λ-modules) is finite. These are the finite dimensional K-algebras Λ = O/πO, where O is a classical order over a discrete valuation ring D with uniformizing parameter π and residue class field K. The homological properties of O are to a great extent determined by those of Λ, as was observed in [22], [15], and [8], while the latter algebra is substantially easier to handle. We are indebted to E. Kirkman and J. Kuzmanovich for bringing the crucial role played by these algebras to our attention. From our main result (Theorem 4.3) it follows that, if gl dim O = d < ∞, then l gl rep Λ = d − 1 and all finitely generated Λ-modules have finite syzygy type. To illustrate the transport of information between O and Λ, we present an example in which the left and right finitistic dimensions of O differ, and one in which O has infinite global dimension while Λ is Gorenstein; in particular, l fin dim O = r fin dim O in this latter example.

Finally, we wish to advertize the algebras O/πO within the representation theory community, as constituting a very interesting class of finite dimensional algebras deserving further study, both for their own sake and for their impact on the theory of classical orders.

Terminology. Given an artinian ring Λ, we denote by Λ-Mod and Λ-mod the category of all left Λ-modules and the full subcategory of finitely generated left Λ-modules, respectively. Moreover, given a class A of left Λ-modules, the notation add A will stand for the full subcategory of Λ-Mod having as objects all finite direct sums of copies of direct summands of objects from A.

In addition to the global dimensions, we will consider the big and little finitistic dimensions, l Fin dim Λ and l fin dim Λ; the former is the supremum of all the \underline{finite} projective dimensions attained on objects in Λ-Mod, while the latter is the analogous supremum in Λ-mod.

Throughout, J will be the Jacobson radical of Λ.

We next present a few known results which will be used in the sequel.

Proposition 1.1. If Λ is left noetherian, then l Fin dim Λ ≤ i dim_{Λ}Λ.

Proof. [3], Proposition 4.3. □

Proposition 1.2. If Λ is noetherian and i dim_{Λ}Λ and i dim Λ_{Λ} are both finite, then l Fin dim Λ = r Fin dim Λ = i dim_{Λ}Λ = i dim Λ_{Λ}.

Proof. [25], Lemma A and [19], Proposition 2.1. □

In case Λ is a finite dimensional algebra, there is an easier proof of the above result, which, moreover, yields some additional information:

Proposition 1.3. Let Λ be a finite dimensional algebra over a field K, with Jacobson radical J. If i dim_{Λ}Λ and i dim Λ_{Λ} are both finite, then

l Fin dim Λ = r Fin dim Λ = l fin dim Λ = r fin dim Λ

= i dim_{Λ}Λ = i dim Λ_{Λ} = p dim E(_{Λ}(Λ/J)) = p dim E((Λ/J)_{Λ}).

Proof. If i dim_{Λ}Λ = m and i dim Λ_{Λ} = n, then l Fin dim Λ ≤ m and r Fin dim Λ ≤ n by Proposition 1. The duality Hom_{K}(−, K) carries _{Λ}Λ to the right minimal injective cogenerator E((Λ/J)_{Λ}), and so

m = p dim E((Λ/J)_{Λ}) ≤ r fin dim Λ ≤ r Fin dim Λ ≤ n

because E((Λ/J)_{Λ}) is finitely generated. Similarly,

n = p dim E(_{Λ}(Λ/J)) ≤ l fin dim Λ ≤ l Fin dim Λ ≤ m. □

Graphing conventions. For the convenience of the reader, we include a sketch of our graphing conventions for modules over path algebras modulo relations; these may differ from standard ones in that they take the J-layering of the modules into account. Suppose that Λ = KΓ/I is a path algebra modulo relations over a field K, where Γ is a quiver and I an admissible ideal in KΓ. We will label the vertices of Γ by integers, and denote the primitive idempotent in Λ corresponding to a vertex i by e_{i}; further, we will often write S_{i} for the simple left Λ-module Λe_{i}/Je_{i}.

For example, suppose that Γ is the quiver

[figure: 1; 3; α; 2; τ; 5; σ; δ; ρ; 6; β; 4; γ; ϵ]

That a left Λ-module M has graph

[figure: 4; γ; ϵ; 5; ρ; σ; δ; 6; β; 2; τ; 3; α; 1]

is to communicate the following information: First,

M/JM ≅ S_{4} ⊕ S_{5}; JM/J^{2}M ≅ S_{6} ⊕ S_{2} ⊕ S_{3}; J^{2}M ≅ S_{1}; J^{3}M = 0.

Second, there exist elements x = e_{4}x and y = e_{5}y in M \ JM such that γx, ρy ∈ JM \ J^{2}M and these last two elements differ only by a nonzero scalar factor modulo J^{2}M; analogously, ϵx and σy belong to JM \ J^{2}M and differ only by a nonzero scalar modulo J^{2}M. Finally, βγx, τϵx, αδy are to differ only by nonzero scalar factors, that is, Λβγx = Λτϵx = Λαδy = J^{2}M in this case. (It follows from the information in the previous ‘layer’ that βρy and τσy are automatically included in this list.)

Note that, in general, a module M will have more than one graph, since the latter may depend on the choice of the ‘reference elements’ in M \ JM (the elements x and y in the example above). In case there is just one arrow from a vertex i to a vertex j, we will often omit naming this arrow. For additional information, we refer to [11].

Finally, observe that right Λ-modules correspond to left modules over a quotient of the path algebra of the ‘dual’ of the quiver Γ, that is, of the quiver with the same vertices as Γ but reversed arrows. Thus in the graph of a \underline{right} Λ-module, an edge leading from a vertex j down to a vertex i corresponds to an arrow i → j, rather than to an arrow j → i.

2. Syzygy type and repetition index

Throughout this section, let Λ be an artinian ring. There are analogs of many of the results of this section for finitely generated modules over a noetherian semiperfect ring – see Remark 2.10.

Definitions 2.1. Let M and A be left Λ-modules.

(1) Let n be a nonnegative integer. We shall say that the minimal projective resolution of M is repetitive at degree n if there exists a decomposition Ω^{n}(M) = P ⊕ ⨁_{i∈I} A_{i} such that P is projective and each A_{i} occurs as a direct summand of infinitely many Ω^{j}(M). In case Ω^{n}(M) is finitely generated, this is the same as to say that each non-projective indecomposable direct summand of Ω^{n}(M) occurs as a direct summand of infinitely many Ω^{j}(M).

(2) The repetition index of M, denoted rep(M), is the least nonnegative integer k such that the minimal projective resolution of M is repetitive at degree k (if such a k exists) or ∞ (otherwise). The corresponding globalized indices are

l gl rep(Λ) = sup{rep(M) | M ∈ Λ-Mod}
l Fin rep(Λ) = sup{rep(M) | M ∈ Λ-Mod and rep(M) < ∞}
l fin rep(Λ) = sup{rep(M) | M ∈ Λ-mod and rep(M) < ∞},

which we shall call the left global repetition index, the left big finitistic repetition index, and the left little finitistic repetition index of Λ, respectively.

(3) Next, we set σ_{M}(A) = −1 if A is not isomorphic to a direct summand of any syzygy Ω^{j}(M), j ≥ 0, and

σ_{M}(A) = sup{j ≥ 0 | A ≅ a direct summand of Ω^{j}(M)}

otherwise. We call σ_{M}(A) the (homological) contingency of A to M.

Clearly rep(M) equals the minimum of those k ≥ 0 such that Ω^{k}(M) is a direct sum of a projective module and modules with infinite contingency to M.

(4) The syzygy category of M is the full subcategory add({Ω^{j}(M) | j ≥ 0}) of Λ-Mod. If this category is a Krull-Schmidt category of finite representation type, then we say that M has finite syzygy type, and we define the syzygy type of M to be the number of isomorphism types of nonzero indecomposable objects in the syzygy category. Otherwise, we say that the syzygy type of M is ∞.

(5) For any nonnegative integer τ, let Ω^{τ} (Λ-Mod) denote the full subcategory

add({Ω^{j}(M) | j ≥ τ; M ∈ Λ-Mod})

of Λ-Mod. As in (4), if Ω^{τ} (Λ-Mod) is a Krull-Schmidt category of finite representation type, then we say that Λ-Mod has finite syzygy type from degree τ, and we define the syzygy type of Λ-Mod from degree τ to be the number of isomorphism types of nonzero indecomposable objects in the category Ω^{τ} (Λ-Mod).

Comments 2.2. (a) The earliest consideration of these conditions goes back to Jans [13], who said that a Λ-module M has an ultimately closed projective resolution in case some syzygy Ω^{n}(M), n ≥ 1, has a decomposition Ω^{n}(M) = ⨁_{i∈I} A_{i} such that each A_{i} already occurred as a direct summand of a previous syzygy Ω^{j(i)}(M) with j(i) < n. It is easily seen that, for a finitely generated module M, this is the same as to say that M has finite syzygy type. Subsequently, Colby-Fuller [4] and Fuller-Wang [7] relaxed this condition slightly as follows: M is said to have a projective resolution with a strongly redundant image from an integer n ≥ 1 in case the n-th syzygy has a decomposition Ω^{n}(M) = ⨁_{i∈I} A_{i} such that each A_{i} is a direct summand of a \underline{later} syzygy Ω^{j(i)}(M) with j(i) > n. In case Ω^{n}(M) has a decomposition into modules with local endomorphism rings, this latter condition clearly implies rep(M) ≤ n. Note that strict inequality may occur, because we exclude projective direct summands of the Ω^{j}(M)’s from consideration in rep(M). We point to [4] for an example of a projective resolution with a strongly redundant image which fails to be ultimately closed.

(b) Observe that the restriction to non-projective direct summands in our definition of ‘repetition index’ allows us to compute the repetition index of a finitely generated module M by looking at \underline{any} projective resolution, not just a minimal one.

(c) If Λ-Mod has finite syzygy type from some degree, then each left Λ-module has finite repetition index. In fact, l gl rep(Λ) < ∞ in that case – see Lemma 2.4(b).

(d) Note that p dim(M) = rep(M) for any module M with finite projective dimension. Hence, we have the inequalities l fin dim Λ ≤ l fin rep Λ ≤ l gl rep Λ and l Fin dim Λ ≤ l Fin rep Λ ≤ l gl rep Λ. Strict inequalities may occur in all places; for an instance where even l Fin dim Λ < l fin rep Λ, see Example 2.3(d).

Examples 2.3. (a) The module category Λ-Mod of any finite dimensional monomial relation algebra Λ = KΓ/I has finite syzygy type from degree 2, by [9], Theorem A; in fact, the syzygy type of Λ-Mod from degree 2 is bounded above by the number of paths in KΓ whose residue classes are nontrivial in Λ.

(b) If gl dim Λ = m < ∞, then Λ has finite syzygy type from degree m.

(c) If Λ = K[X, Y]/(X^{2}, Y^{2}) for a field K, then Λ has infinite repetition index and, a fortiori, infinite syzygy type; in fact, rep(Λ/J) = ∞. On the other hand, Fin dim Λ = rep(E(Λ/J)) = 0.

(d) Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α; 2; β; 3; γ]

and I is generated by all paths of length 3. It is easy to see that l fin dim Λ = l Fin dim Λ = 0 in this case (see [2], Theorem 6.3 and [10], Corollary 8). On

the other hand, l fin rep(Λ) ≥ rep(S_{1}) = 2, since Ω^{1}(S_{1}) has graph {2 \atop 3} , while

Ω^{2}(S_{1}) ≅ Ω^{2n}(S_{1}) ≅ S_{3} for n ≥ 1, and Ω^{3}(S_{1}) ≅ Ω^{2n+1}(S_{1}) has graph {3 \atop 3} for

n ≥ 1. Thus indeed, l Fin dim Λ < l gl rep Λ.

Finally, observe that the syzygy category of S_{1} is the additively closed category generated by the indecomposable left Λ-modules Ω^{i}(S_{1}), i = 0, 1, 2, 3, in this example. □

The principal connections among syzygy type, repetition index, and finitistic dimensions are as follows. Note that, in essence, most of the arguments are familiar – see, e.g., [12], [7]. In fact, for T = Λ/J, where Λ is an artin algebra, the first statement of Theorem 2.6 is implicit in [12].

Lemma 2.4. (a) The repetition index of any Λ-module M is bounded above by the syzygy type of M.

(b) If Λ-Mod has syzygy type s from degree τ, then l gl rep(Λ) ≤ τ + s.

Proof. (a) Assume that the syzygy type of M is s < ∞. If s = 0, then M = 0 and hence rep(M) = 0.

Now assume that s > 0, and let B = B_{s} be an indecomposable direct summand of Ω^{s}(M). Since M has finite syzygy type, Ω^{s−1}(M) is a direct sum of indecomposable modules I_{k}, whence Ω^{s}(M) ≅ ⨁_{k} Ω^{1}(I_{k}), and so B_{s} is isomorphic to a direct summand of some Ω^{1}(I_{k}). (Here we are using the assumption that the syzygy category of M is a Krull-Schmidt category.) Label this I_{k} as B_{s−1}. Continuing by induction, we obtain indecomposable modules B_{s}, B_{s−1}, …, B_{0} such that each B_{j} is a direct summand of Ω^{j}(M) and B_{j} is isomorphic to a direct summand of Ω^{1}(B_{j−1}) for j > 0.

By definition of s, there exist indices p, q with 0 ≤ p < q ≤ s such that B_{p} ≅ B_{q}. Thus B_{p} is isomorphic to a direct summand of Ω^{q−p}(B_{p}), and hence isomorphic to a direct summand of Ω^{m(q−p)}(B_{p}) for all m ≥ 0. Since B is isomorphic to a direct summand of Ω^{s−p}(B_{p}) and B_{p} is isomorphic to a direct summand of Ω^{p}(M), it follows that B is isomorphic to a direct summand of Ω^{s+m(q−p)}(M) for all m ≥ 0. Therefore rep(M) ≤ s, because B was an arbitrary indecomposable direct summand of Ω^{s}(M).

(b) It follows from our assumptions that for any M ∈ Λ-Mod, the module Ω^{τ} (M) has syzygy type at most s. Thus rep(Ω^{τ} (M)) ≤ s by part (a), and therefore rep(M) ≤ τ + s. □

Observation 2.5. Whenever T is a right Λ-module such that Λ/J, viewed as a right Λ-module, embeds into T,

l Fin dim Λ ≤ rep(T).

Proof. Assume that rep(T) = r < ∞, and write Ω^{r}(T) = P ⊕ ⨁_{i∈I} A_{i} where P is projective and σ_{T} (A_{i}) = ∞ for all i. Consider a left Λ-module M with p dim M = d < ∞, and note that Tor_{d}^{Λ}(Λ/J, M) ≠ 0. By assumption, there exists a short exact sequence of right Λ-modules 0 → Λ/J → T → C → 0. Since Tor_{d+1}^{Λ}(C, M) = 0, we must have Tor_{d}^{Λ}(T, M) ≠ 0.

If d > r, then Tor_{d−r}^{Λ}(Ω^{r}(T), M) ≠ 0, and so Tor_{d−r}^{Λ}(A_{i}, M) ≠ 0 for some i. Since σ_{T} (A_{i}) = ∞, it follows that Tor_{d−r}^{Λ}(Ω^{j}(T), M) ≠ 0 for infinitely many j, and consequently Tor_{d−r+j}^{Λ}(T, M) ≠ 0 for infinitely many j. However, this is impossible for j > r. Therefore d ≤ r.

Since M was an arbitrary left Λ-module of finite projective dimension, this proves that l Fin dim Λ ≤ r. □

Theorem 2.6. Suppose there exists a right Λ-module T of finite syzygy type s such that Λ/J, viewed as a right Λ-module, embeds into T. Moreover, let A_{1}, …, A_{s} be representatives for the isomorphism types of the indecomposable objects in the syzygy category of T. Then

(a) l Fin dim Λ ≤ rep(T) ≤ s.

(b) If M is a left Λ-module of finite projective dimension and

μ = max{σ_{T} (A_{i}) | 1 ≤ i ≤ s and Tor_{1}^{Λ}(A_{i}, M) ≠ 0},

then p dim(M) = μ + 1. (Let μ = −1 if the above set is empty.)

Proof. (a) follows from Lemma 2.4 and Observation 2.5.

(b) Set d = p dim M < ∞, and note that Tor_{d}^{Λ}(T, M) ≠ 0, as in the previous proof.

Consider one of the A_{i} for which Tor_{1}^{Λ}(A_{i}, M) ≠ 0, and choose an index j such that A_{i} is isomorphic to a direct summand of Ω^{j}(T). Then Tor_{j+1}^{Λ}(T, M) ≅ Tor_{1}^{Λ}(Ω^{j}(T), M) ≠ 0, and so j + 1 ≤ d. Thus σ_{T} (A_{i}) + 1 ≤ d for all A_{i} such that Tor_{1}^{Λ}(A_{i}, M) ≠ 0, that is, μ + 1 ≤ d.

It remains to prove the reverse inequality. If d = 0, then Tor_{1}^{Λ}(−, M) ≡ 0, whence μ = −1 and d = μ+1. If d > 0, then Tor_{1}^{Λ}(Ω^{d−1}(T), M) ≅ Tor_{d}^{Λ}(T, M) ≠ 0, and so Tor_{1}^{Λ}(A_{i}, M) ≠ 0 for some A_{i} which is isomorphic to a direct summand of Ω^{d−1}(T). In this case, d − 1 ≤ σ_{T} (A_{i}) ≤ μ, and therefore d ≤ μ + 1. □

Corollary 2.7. Let T and A_{1}, …, A_{s} be right Λ-modules as in the hypotheses of Theorem 2.6. Then

l Fin dim Λ ≤ 1 + max_{1≤i≤s}{σ_{T} (A_{i}) | A_{i} non-projective and σ_{T} (A_{i}) < ∞}. □

Of course, the module T in (2.5–2.7) may be chosen to be either the right module Λ/J or the minimal injective cogenerator E for the category of right Λmodules. In [7], certain related bounds on the finitistic dimensions of noetherian rings are obtained. For the case where the base ring is an artin algebra, our Observation 2.5 provides a mild tightening of these bounds: For Theorem 3 of [7], take T = E; for Theorem 9, take T = Λ/J.

For reasons that are not transparent as yet, the choice T = E appears to systematically yield better bounds on l Fin dim Λ. In particular, while it is easy to construct algebras with rep(Λ/J) = ∞ (see Example 2.3(c) above), examples where rep(E) = ∞ are not immediate.

The following example will illustrate the methods just introduced; further, it will be relevant to Example 4.6.

Example 2.8. Let Λ = KΓ/I be a binomial relation algebra (i.e., I can be generated by paths in KΓ and by differences p − kq with k ∈ K, where p and q are paths) with six vertices such that the indecomposable projective right Λ-modules P_{1}, …, P_{6} have the following graphs:

[figure: 1; 2; 3; 4; 5; 6; 2; 3; 1; 5; 5; 1; 4; 2; 6; 6; 2; 3; 5; 1; 4; 5; 4; 3; 6; 6; 2; 5; 1; 1; 4; 3; 2; 6; 4; 3]

Then the indecomposable injective right Λ-modules E_{i} are E_{3} = P_{4}, E_{4} = P_{2}, E_{6} = P_{1}, while E_{1}, E_{2}, E_{5} have graphs

[figure: 4; 5; 3; 6; 1; 4; 6; 2; 3; 5; 1; 4; 3; 2; 6; 1; 2; 5]

respectively. Set E = E_{1} ⊕ ⋯ ⊕ E_{6}. Clearly rep E_{3} = rep E_{4} = rep E_{6} = 0. To compute rep E, we determine the minimal projective resolutions of E_{1}, E_{2}, E_{5}:

[figure: Ω^{1}(E_{1}); Ω^{2}(E_{1}); Ω^{3}(E_{1}); 6; 2; 5; 1; 3; 2; 4; 5; 1; 6; 3; 2; 1; 4; 5; 3; 4; 6]

and Ω^{4}(E_{1}) ≅ Ω^{2}(E_{1}). Moreover, Ω^{1}(E_{2}) has graph

[figure: 5; 1; 4; 3; 6; 2]

and Ω^{2}(E_{2}) ≅ Ω^{1}(E_{1}) ⊕ Ω^{3}(E_{1}). Finally, Ω^{1}(E_{5}) has graph

[figure: 4; 2; 5; 1; 6; 3]

and Ω^{2}(E_{5}) ≅ Ω^{1}(E_{1}). It follows that rep E = rep(E_{1} ⊕E_{2} ⊕E_{5}) = 3, and hence Observation 2.5 yields l Fin dim Λ ≤ 3. We actually obtain equality, l Fin dim Λ = l fin dim Λ = 3, since the indecomposable injective left Λ-module E(Λe_{3}/Je_{3}) has projective dimension 3. Note that, on the other hand, r Fin dim Λ = 0, since the left socle of Λ contains a copy of Λ/J (see [2], Theorem 6.3 and [10], Corollary 8). □

Often the following variant of Theorem 2.6 provides better bounds:

Corollary 2.9. Suppose there exists a right Λ-module T of finite syzygy type such that Λ/J, viewed as a right Λ-module, embeds into T. For m ≥ 0, let s_{m} be the syzygy type of Ω^{m}(T). Then l Fin dim Λ ≤ s_{m} + m for all m.

Proof. By Lemma 2.4, rep(Ω^{m}(T)) ≤ s_{m}, and so rep(T) ≤ s_{m} + m. The corollary thus follows immediately from part (a) of Theorem 2.6. □

Remark 2.10. Since the definitions and proofs in this section rely mainly on the existence and behavior of syzygies, the results carry over to the case when Λ is only assumed to be a noetherian semiperfect ring, provided we restrict attention to finitely generated modules. (Recall that semiperfect rings are precisely the rings over which every finitely generated module has a projective cover.) In particular, the argument of Lemma 2.4(a) shows that the repetition index of any finitely generated Λ-module is bounded above by its syzygy type, and the argument of Observation 2.5 shows that if T is a finitely generated right Λ-module containing an isomorphic copy of (Λ/J)_{Λ/J}, then

l fin dim Λ ≤ rep(T).

Similarly, there are analogs of Theorem 2.6(b) (giving a formula for the projective dimensions of finitely generated Λ-modules with finite projective dimension) and of Corollaries 2.7, 2.9 (giving additional upper bounds for l fin dim Λ). We leave the details to the interested reader.

3. Computation of individual projective dimensions in the case of finite syzygy type

We now specialize to the case where Λ is a finite dimensional algebra over a field K. Suppose there exists a finitely generated right Λ-module T of finite syzygy type s such that (Λ/J)_{Λ} embeds into T, and let A_{1}, …, A_{s} be representatives for the isomorphism types of the indecomposable objects in the syzygy category of T.

Guiding idea: If one knows the A_{i} and their first syzygies, the computation of projective dimensions of left Λ-modules M can be reduced to the calculation of the K-dimensions of the tensor products A_{i} ⊗_{Λ} M.

Indeed, by the choice of the A_{i}, there exists an s × s matrix B = (b_{ij}) of nonnegative integers such that

Ω^{1}(A_{i}) ≅ ⨁_{j=1}^{s} A_{j}^{b_{ij}}

for all i. Moreover, let e_{1}, …, e_{n} be a full set of orthogonal primitive idempotents in Λ, and let P_{i} denote the projective cover of A_{i}. Write

P_{i} ≅ ⨁_{l=1}^{n}(e_{l}Λ)^{p_{il}}

with p_{il} ≥ 0 for 1 ≤ i ≤ s.

Observe that since the entries of B are nonnegative integers, the product of B with any column vector of cardinal numbers is defined. (We use the convention that 0 · α = 0.)

Proposition 3.1. Let M be a left Λ-module, set τ_{i} = dim_{K} Tor_{1}^{Λ}(A_{i}, M) for i = 1, …, s, and set τ = τ(M) = (τ_{1}, τ_{2}, …, τ_{s})^{tr}.

(a) If M has finite projective dimension, then B^{m}τ = 0 for some m ≥ 0, and the least such m equals p dim M.

(b) Suppose that T = (Λ/J)_{Λ}. Then p dim M < ∞ if and only if B^{m}τ = 0 for some m.

(c) If M is finitely generated, the integers τ_{i}, 1 ≤ i ≤ s, can be computed as follows:

τ_{i} = dim_{K}(A_{i} ⊗_{Λ} M) + ∑_{j=1}^{s} b_{ij} dim_{K}(A_{j} ⊗_{Λ} M) − ∑_{l=1}^{n} p_{il} dim_{K}(e_{l}M).

Proof. For i = 1, …, s, we have

Tor_{1}^{Λ}(A_{i}, Ω^{1}(M)) ≅ Tor_{2}^{Λ}(A_{i}, M) ≅ Tor_{1}^{Λ}(Ω^{1}(A_{i}), M) ≅ ⨁_{j=1}^{s} Tor_{1}^{Λ}(A_{j}, M)^{b_{ij}},

whence dim_{K} Tor_{1}^{Λ}(A_{i}, Ω^{1}(M)) = ∑_{j=1}^{s} b_{ij}τ_{j}. This shows that τ(Ω^{1}(M)) = Bτ(M). It follows by induction that

B^{m}τ(M) = τ(Ω^{m}(M))

for all m ≥ 0.

(a) Suppose that p dim M = m < ∞. Then Ω^{m}(M) is projective, and consequently Tor_{1}^{Λ}(A_{i}, Ω^{m}(M)) = 0 for all i. Hence, B^{m}τ = τ(Ω^{m}(M)) = 0. We must also show that if m > 0, then B^{m−1}τ ≠ 0. As in the proof of Observation 2.5, Tor_{m}^{Λ}(T, M) ≠ 0, whence Tor_{1}^{Λ}(T, Ω^{m−1}(M)) ≠ 0. Since T is a direct sum of copies of the A_{i}, it follows that Tor_{1}^{Λ}(A_{i}, Ω^{m−1}(M)) ≠ 0 for some i, and therefore B^{m−1}τ = τ(Ω^{m−1}(M)) ≠ 0 as desired.

(b) Suppose that B^{m}τ = 0 for some m. Then τ(Ω^{m}(M)) = 0, and so we have Tor_{1}^{Λ}(A_{i}, Ω^{m}(M)) = 0 for all i. Consequently, Tor_{1}^{Λ}(Λ/J, Ω^{m}(M)) = 0, which implies that Ω^{m}(M) is projective, and thus p dim M ≤ m.

(c) From the resolutions 0 → Ω^{1}(A_{i}) → P_{i} → A_{i} → 0, we obtain long exact sequences

0 → Tor_{1}^{Λ}(A_{i}, M) → Ω^{1}(A_{i}) ⊗_{Λ} M → P_{i} ⊗_{Λ} M → A_{i} ⊗_{Λ} M → 0,

and consequently

τ_{i} = dim_{K}(A_{i} ⊗_{Λ} M) + dim_{K}(Ω^{1}(A_{i}) ⊗_{Λ} M) − dim_{K}(P_{i} ⊗_{Λ} M).

Since Ω^{1}(A_{i}) ⊗_{Λ} M ≅ ⨁_{j=1}^{s}(A_{j} ⊗_{Λ} M)^{b_{ij}} and P_{i} ⊗_{Λ} M ≅ ⨁_{l=1}^{n}(e_{l}M)^{p_{il}}, the desired formula is clear. □

Corollary 3.2. If d is the least nonnegative integer with the property that B^{d} ∈ M_{s}(ℚ)B^{d+1}, then l Fin dim Λ ≤ d.

Proof. Observe that B^{d} ∈ M_{s}(ℚ)B^{d+k} for all k > 0. Hence, there exist positive integers z_{k} and nonnegative integer matrices U_{k}, V_{k} such that z_{k}B^{d} = (U_{k} − V_{k})B^{d+k}.

Given any left Λ-module M with p dim M = m < ∞, we have B^{m}τ(M) = 0 by Proposition 3.1. If m > d, multiply both sides of the equation z_{m−d}B^{d} + V_{m−d}B^{m} = U_{m−d}B^{m} on the right by τ(M) to obtain z_{m−d}B^{d}τ(M) = 0. But then B^{d}τ(M) = 0, contradicting Proposition 3.1. Therefore p dim M ≤ d. □

We conclude this section with an example illustrating the use of Proposition 3.1.

Example 3.3. Let Λ = KΓ/I be the monomial relation algebra with quiver

[figure: Γ :; 1; 2; 3; 4; ϵ; γ; δ; 5; α; β]

such that the graphs of the indecomposable projective left Λ-modules are

[figure: 1; •; 2; 3; 4; γ; ϵ; δ; 5; α; β; 1; 2; 4; 3; 4; 4; γ; 4; δ; 4; 4]

Then the indecomposable projective right Λ-modules have graphs

[figure: 1; 2; 3; 4; α; γ; δ; β; 5; •; 2; 3; 4; 5; 4; α; 4; β; 5; 5; 5]

Let e_{1}, …, e_{5} be the primitive idempotents of Λ corresponding to the vertices of Γ, set S_{i} = e_{i}Λ/e_{i}J, and let T be the right Λ-module Λ/J ≅ ⨁_{i=1}^{5} S_{i}. One computes that

Ω^{1}(S_{1}) ≅ S_{2};  Ω^{1}(S_{2}) ≅ S_{3};  Ω^{1}(S_{3}) ≅ S_{4};  Ω^{1}(S_{5}) = 0;

while

Ω^{1}(S_{4}) ≅ S_{5}^{2} ⊕ 4_{α}5 ⊕ 4_{β}5  and  Ω^{2}(S_{4}) ≅ S_{5}^{2} ⊕ ( 4_{α}5 )^{2} ⊕ ( 4_{β}5 )^{2} .

Thus rep((Λ/J)_{Λ}) = 4, which by Observation 2.5 yields l Fin dim Λ ≤ 4. (Since the left Λ-module Λ/Λ(α + β) has projective dimension 4, we actually obtain equality.)

Moreover, we observe that T has finite syzygy type, the syzygy category of T

being equal to add(A_{1}, …, A_{7}) with A_{i} ≅ S_{i} for i = 1, …, 5 while A_{6} = 4_{α}5

and A_{7} = 4_{β}5 . Using the notation introduced above, we thus obtain s = 7 and

the integer 7 × 7 matrix

B = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \end{matrix} ) .

The projective covers of the A_{i} being P_{i} for i ≤ 5 and P_{6} ≅ P_{7} ≅ e_{4}Λ, we finally see that p_{ii} = 1 for i ≤ 5 and p_{64} = p_{74} = 1, while all the other exponents p_{il} are zero.

We will use Proposition 3.1 to determine the projective dimension of the left Λ-module

M = (Λe_{4} ⊕ Λe_{5}) / (Λ(γ, α) + Λ(β, δ) + Λ(ϵ, 0)).

Clearly, dim_{K}(A_{i} ⊗_{Λ} M) = 0 for i = 1, 2, 3 and

dim_{K}(A_{4} ⊗_{Λ} M) = dim_{K}(e_{4}M/e_{4}JM) = 1 = dim_{K}(A_{5} ⊗_{Λ} M).

Moreover, we compute that

dim_{K}(A_{6} ⊗_{Λ} M) = dim_{K}(e_{4}M/(γΛ + δΛ + βΛ)M) = 1,

and analogously, dim_{K}(A_{7} ⊗_{Λ} M) = 1. In view of 3.1(c), this yields

τ(M) = τ = (0, 0, 1, 2, 0, 1, 1)^{tr}.

Since the last three rows of each power B^{m} are of the form

( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2^{m} & 2^{m} & 2^{m} \\ 0 & 0 & 0 & 0 & 2^{m} & 2^{m} & 2^{m} \end{matrix} )

for m ∈ ℕ, we deduce that B^{m}τ ≠ 0 for all m. Thus 3.1(b) tells us that p dim M = ∞. □

4. A new class of algebras of finite global repetition index

As we will see, any classical order O of finite global dimension d over a discrete valuation ring D has a finite dimensional satellite algebra Λ which has (left and right) global repetition index d − 1. This algebra, first studied by Tarsy [22] and Jategaonkar [15], and subsequently by Kirkman-Kuzmanovich [17], faithfully reflects the homological behavior of O while being structurally less involved.

We start with a synopsis of problems and results pertaining to classical orders. Let D be a discrete valuation ring with uniformizing parameter π (that is, πD is the maximal ideal of D), and quotient field Q. Moreover, let O be a classical order in some matrix ring M_{n}(Q), that is, O ⊆ M_{n}(Q) is a D-subalgebra which is finitely generated as a D-module on one hand and which generates M_{n}(Q) over Q on the other. We will identify D with the subring D · 1 of O.

In 1970, Tarsy [21] conjectured that the global dimension of any classical order O ⊆ M_{n}(Q), whenever finite, is bounded above by n − 1. At least in the special case where O is tiled, meaning that O contains a complete set of n primitive orthogonal idempotents of M_{n}(Q), this conjecture seemed to have some merit. Tarsy showed in [22] that there are only finitely many possible finite global dimensions for tiled classical orders in M_{n}(Q), thus guaranteeing a finite upper bound on these dimensions. Moreover, in [15] Jategaonkar proved that, up to conjugation, M_{n}(Q) contains only finitely many tiled classical orders of finite global dimension. Subsequently, Tarsy’s conjecture was confirmed in the triangular case (i.e., when O is conjugate to an order of the form

( \begin{matrix}D & D & ⋯ & D \\ (π^{λ_{2,1}}) & D & ⋯ & D \\ ⋮ & ... & ... & ⋮ \\ (π^{λ_{n,1}}) & ⋯ & (π^{λ_{n,n−1}}) & D \end{matrix} )

with λ_{i,j} ≥ 0) in [15], and for tiled orders in M_{n}(D) containing the ideal M_{n}(πD) in [17]. However, shortly afterwards, Fujita [5] constructed a class of tiled examples O_{n} ⊆ M_{n}(K((π))) for n ≥ 6, where K is a field, such that gl dim O_{n} = n, thus refuting the conjecture even in the tiled case. Moreover, in view of an example of Rump ([20], §7, Example 3) and a class of examples exhibited by Jansen and Odenthal in [14], raising ‘n − 1’ to ‘n’ (or even ‘n + 1’) in Tarsy’s conjecture will not suffice to provide a valid bound either.

The satellite algebra of O which we will consider here is the factor Λ = O/πO. Clearly, Λ is a finite dimensional algebra over the residue field K = D/πD. For the case where O is tiled and basic, Kirkman and Kuzmanovich assembled a list of distinguished properties of Λ, including the following [18]:

• Λ is a split basic algebra. More precisely, Λ ≅ KΓ/I is a binomial relation algebra, where the ideal I has a generating set consisting of paths and differences of paths. Furthermore, the quiver Γ and a generating set for I of the described form can be explicitly (and easily) computed from the valued quiver which was associated to any tiled classical order by Wiedemann and Roggenkamp [23]. (A sketch of this procedure can be found ahead of Example 4.6 below.)

• Each of the simple Λ-modules occurs as a composition factor of multiplicity 1 in each of the indecomposable projective Λ-modules. Consequently, the multiplicities of the simples are equal in all Λ-modules of finite projective dimension, which, in particular, shows that all simple modules have infinite projective dimension for n ≥ 2.

Since O is noetherian and π is a central non-zero-divisor in O which is not a unit, the standard change-of-rings arguments for homological dimensions (see, e.g., [16]) apply to the pair O and Λ = O/πO. They yield a helpful relationship between the little finitistic dimensions of O and Λ, as was observed by Tarsy and by Green, Kirkman, and Kuzmanovich:

Proposition 4.1. l fin dim O = l fin dim Λ + 1 and r fin dim O = r fin dim Λ + 1.

Proof. [22], proof of Corollary 1; [8], Lemma 2.7. □

For the big finitistic dimensions, however, these arguments only yield inequalities

l Fin dim O ≥ l Fin dim Λ + 1  and  r Fin dim O ≥ r Fin dim Λ + 1.

In case the order O has finite global dimension, or more generally, if it has finite injective dimension on both sides, we can combine Proposition 4.1 with Proposition 1.3 to obtain coincidence of the big and little finitistic dimensions of Λ, as follows.

Proposition 4.2. If i dim_{O}O and i dim O_{O} are both finite, then

l Fin dim Λ = r Fin dim Λ = l fin dim Λ = r fin dim Λ

= i dim_{Λ}Λ = i dim Λ_{Λ} = p dim E(_{Λ}(Λ/J)) = p dim E((Λ/J)_{Λ}) = l fin dim O − 1 = r fin dim O − 1.

In particular, if gl dim O = d < ∞, then d ≥ 1 and all the above dimensions equal d − 1.

Proof. Since the order O is not divisible by π on either side, it is not injective as a right or left O-module. Therefore, π being a central non-zero-divisor in O, it follows from [16], Theorem 205 that

i dim_{Λ}Λ ≤ i dim_{O}O − 1 < ∞,

and likewise that i dim Λ_{Λ} < ∞. Thus, by Proposition 1.3,

l Fin dim Λ = r Fin dim Λ = l fin dim Λ = r fin dim Λ

= i dim_{Λ}Λ = i dim Λ_{Λ} = p dim E(_{Λ}(Λ/J)) = p dim E((Λ/J)_{Λ}).

If d′ is the common value of these dimensions, Proposition 4.1 shows that l fin dim O = r fin dim O = d′ + 1.

The final statement of the proposition is clear. □

We shall need the following well known variant of Schanuel’s Lemma, which holds for modules over an arbitrary artinian ring: If 0 → L → P → N → 0 is a short exact sequence of Λ-modules with P projective, then L ≅ Ω^{1}(N) ⊕ Q for some projective Λ-module Q. This conclusion can be obtained from Schanuel’s Lemma together with the Krull-Schmidt Theorem, or from [1], Lemma 17.17.

Theorem 4.3. If M is a nonzero left Λ-module with p dim_{O}M = m < ∞, then m > 0 and Ω_{Λ}^{m−1}(M) ≅ Ω_{Λ}^{m+1}(M) ⊕ Q for some projective module Q. In particular, rep(_{Λ}M) ≤ m − 1, and if M is finitely generated, then _{Λ}M has finite syzygy type.

Proof. Since M is torsion as an O-module, m ≥ 1.

Suppose first that m = 1, and write M ≅ P/K for some projective O-modules K ⊆ P. Since πM = 0, there is an exact sequence

0 → K/πP → P/πP → M → 0

of Λ-modules with P/πP projective, and hence K/πP ≅ Ω_{Λ}^{1}(M) ⊕ Q_{1} for some projective Λ-module Q_{1}. There is also an exact sequence

0 → πP/πK → K/πK → K/πP → 0

of Λ-modules with K/πK projective, and so πP/πK ≅ Ω_{Λ}^{1}(K/πP) ⊕ Q for some projective Λ-module Q. Consequently, πP/πK ≅ Ω_{Λ}^{2}(M) ⊕ Q. Since π is a non-zero-divisor on P, we have πP/πK ≅ P/K ≅ M as well.

Now assume that m > 1, and consider the exact sequence 0 → L → Q_{0} → M → 0 of Λ-modules where Q_{0} is a Λ-projective cover of M and L = Ω_{Λ}^{1}(M). Observe that p dim_{O}Λ = 1 and so p dim_{O}Q_{0} ≤ 1. Since m ≥ 2, it follows from the long exact sequence for Ext_{O} that p dim_{O}L = m − 1. By induction, Ω_{Λ}^{m−2}(L) ≅ Ω_{Λ}^{m}(L)⊕Q for some projective module Q, and therefore Ω_{Λ}^{m−1}(M) ≅ Ω_{Λ}^{m+1}(M) ⊕ Q.

The final conclusions of the theorem now follow easily. □

Corollary 4.4. If gl dim O = d < ∞, then l gl rep Λ = r gl rep Λ = d − 1 and all finitely generated Λ-modules have finite syzygy type.

Proof. Suppose that gl dim O = d < ∞. Theorem 4.3 then shows that l gl rep Λ and r gl rep Λ are bounded above by d−1, and that all finitely generated Λ-modules have finite syzygy type. To obtain the missing inequalities, we use Proposition 4.1 and Observation 2.5 to obtain

d − 1 = l fin dim Λ ≤ rep((Λ/J)_{Λ}) ≤ r gl rep Λ,

and similarly d − 1 ≤ l gl rep Λ. □

The next consequence of Theorem 4.3 addresses the Poincaré-Betti series of a pair (M, N) of Λ-modules, i.e. the formal power series

∑_{i=0}^{∞}(−1)^{i} dim_{K} Ext_{Λ}^{i}(M, N) · t^{i}.

Corollary 4.5. If O has finite global dimension, then the Poincaré-Betti series of any pair of Λ-modules is a rational function.

Proof. Combine Corollary 4.4 with Wilson’s main result in [24]. □

We now briefly sketch the road that leads from a basic tiled classical order O to a presentation of Λ = O/πO in terms of quiver and relations, as it was communicated to us by Kirkman and Kuzmanovich [18]. We would like to warn the reader, however, that the quivers we use are duals of those considered in [17] and [23].

After a conjugation, we may assume that O ⊆ M_{n}(D) and that the diagonal matrix units e_{ii}, 1 ≤ i ≤ n, all lie in O. Now consider the ordinary quiver Γ of the (semiperfect) ring O: It consists of n vertices 1, …, n corresponding to the n non-isomorphic indecomposable projective left O-modules P_{i} = Oe_{ii}, and there is an arrow i → j provided that P_{j} is isomorphic to a direct summand of the projective cover of rad P_{i}. (This is in accordance with the usual convention, since the multiplicity of P_{j} as a direct summand of the projective cover of rad P_{i} is at most 1 in the present situation.) We attach to any arrow i ⟶^{α} j the value v(α) = k where k is the least nonnegative integer such that π^{k}P_{j} ⊆ rad P_{i} (so that Dπ^{k}e_{ji} corresponds, via right multiplication, to Hom_{O}(P_{j}, rad P_{i})). Thus Dπ^{k}e_{ji} = e_{jj}Oe_{ii} if i ≠ j, whereas k = 1 if i = j. This process yields the ‘valued quiver’ Γ̃ of O. (The order O can actually be retrieved from Γ̃ – see [23].) The value function v is extended additively from arrows to paths. Thus the value of a path q = α_{m}α_{m−1} ⋯ α_{1} in Γ̃, where the α_{i} are arrows, is v(q) = ∑_{i=1}^{m} v(α_{i}).

We now specialize to the situation where Γ̃ has no loops. In that case the quiver of Λ = O/πO coincides with the quiver Γ as above, and the following set G of paths and differences of paths in KΓ generates an ideal I ⊆ KΓ such that Λ ≅ KΓ/I. A path q, with starting point i and endpoint j say, belongs to G if and only if there exists a path q′ from i to j in Γ̃ such that v(q′) < v(q). Moreover, given two paths p and q with coinciding starting points and coinciding endpoints, the difference p − q belongs to G if and only if v(p) = v(q).

Since, given a tiled classical order O, the algebra Λ = O/πO is substantially easier to handle, this latter algebra serves as an excellent tool for the investigation of O. In our first illustration of this principle, we exhibit a classical order with differing left and right finitistic dimensions.

Example 4.6. Let D be any discrete valuation ring with uniformizing parameter π and quotient field Q, and consider the following tiled classical order O ⊆ M_{6}(Q):

O = ( \begin{matrix}D & D & D & D & D & D \\ (π) & D & (π) & (π) & D & D \\ (π) & (π) & D & D & D & D \\ (π^{2}) & (π) & (π^{2}) & D & (π) & D \\ (π^{2}) & (π) & (π) & (π) & D & D \\ (π^{2}) & (π^{2}) & (π^{2}) & (π) & (π) & D \end{matrix} )

The valued quiver of O is

[figure: 1; 2; 1; 1; 3; 1; 0; 2; 0; 1; 1; 5; 0; 0; 1; 6; 0; 0; 4; 1; 0]

and Λ = O/πO has the same quiver (non-valued) and relations p and p − q, for suitable paths p and q such that the indecomposable projective right Λ-modules are as in Example 2.8. Combining our computations in the latter example with Proposition 4.1, we thus obtain l fin dim O = 4 and r fin dim O = 1. □

In our final example, we construct a tiled classical order O of infinite global dimension such that Λ = O/πO is Gorenstein, i.e., such that i dim_{Λ}Λ = i dim Λ_{Λ} < ∞. The fact that gl dim O = ∞ will be established by pinpointing a left Λmodule which does not satisfy the conclusion of Theorem 4.3.

Example 4.7. Again, let D be any discrete valuation ring with uniformizing parameter π and quotient field Q, and let O ⊆ M_{6}(Q) be the tiled classical order which has valued quiver

[figure: 1; 2; 1; 1; 3; 1; 0; 2; 0; 1; 5; 0; 1; 6; 0; 0; 4; 1; 0]

Then the quiver Γ of Λ = O/πO is the non-valued version of the one shown and Λ is of the form KΓ/I, where I is generated by paths and differences of paths, such that the indecomposable projective \underline{left} Λ-modules have the following graphs:

[figure: 1; 2; 3; 4; 5; 6; 2; 6; 3; 1; 4; 1; 5; 3; 6; 2; 6; 4; 5; 4; 5; 3; 6; 6; 2; 1; 5; 1; 4; 3; 2; 5; 4; 2; 3; 1]

In particular, the only indecomposable injective object in Λ-mod which fails to be projective is the injective envelope E_{6} of Λe_{6}/Je_{6}, namely the unique left Λ-module with graph

[figure: 2; 3; 4; 1; 5; 6]

One checks that Ω^{1}(E_{6}) ≅ Λe_{1}, whence p dim E_{6} = 1. Thus the minimal injective cogenerator for Λ-mod has projective dimension 1. Analogously, one computes the projective dimension of the minimal injective right cogenerator to be 1, which yields i dim_{Λ}Λ = i dim Λ_{Λ} = 1 by duality. Furthermore, l fin dim Λ = r fin dim Λ = 1 by Proposition 1.3. Consequently, Proposition 4.1 tells us that l fin dim O = r fin dim O = 2.

In order to check that gl dim O = ∞, it thus suffices to procure a left Λ-module M which violates the conclusion of Theorem 4.3 for m = 2. One can readily verify that each of the simple left Λ-modules qualifies for that purpose. We pick one with a ‘slim’ projective resolution, namely M = Λe_{6}/Je_{6}. Then the first three syzygies Ω^{1}(M), Ω^{2}(M), Ω^{3}(M) have the following graphs, respectively:

[figure: 4; 5; 6; 1; 6; 4; 2; 6; 3; 5; 3; 2; 5; 4; 3; 4; 5; 2; 1; 2; 3; 1; 1]

Clearly, these syzygies are indecomposable and Ω^{1}(M) ≅ Ω̸^{3}(M), which shows that indeed p dim_{O} M > 2. (The repetition index of M as a Λ-module is, in fact, infinite.) □

References

1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.

2. H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488.

3. , Injective dimension in noetherian rings, Trans. Amer. Math. Soc. 102 (1962), 18-29.

4. R. Colby and K. R. Fuller, A note on the Nakayama conjectures, Tsukuba J. Math. 14 (1990), 343-352.

5. H. Fujita, Tiled orders of finite global dimension, Trans. Amer. Math. Soc. 322 (1990), 329-341; Erratum, Trans. Amer. Math. Soc. 327 (1991), 919-920.

6. K. R. Fuller and M. Saorín, On the finitistic dimension conjecture for artinian rings, manuscripta math. 74 (1992), 117-132.

7. K. R. Fuller and Y. Wang, Redundancy in resolutions and finitistic dimensions of noetherian rings, Communic. in Algebra 21 (1993), 2983-2994.

8. E. L. Green, E. E. Kirkman, and J. J. Kuzmanovich, Finitistic dimensions of finite dimensional monomial algebras, J. Algebra 136 (1991), 37-51.

9. B. Huisgen-Zimmermann, Homological domino effects and the first finitistic dimension conjecture, Invent. Math. 108 (1992), 369-383.

10. , A convenient source of homological examples over artinian rings, in Ring Theory, Proc. Biennial Ohio State – Denison Conf., 1992 (S. K. Jain and S. T. Rizvi, eds.), World Scientific, Singapore, 1993, pp. 364-372.

11. , Analyzing the structure of representations representations via approximations, in Representation Theory of Algebras and Related Topics, Mexico City 1994 (R. Bautista, R. Martínez-Villa, and J. A. de la Peña, Eds.), Canad. Math. Soc. Conf. Proc. 19 (1996), 373-406.

12. K. Igusa and D. Zacharia, Syzygy pairs in a monomial algebra, Proc. Amer. Math. Soc. 108 (1990), 601-604.

13. J. P. Jans, Some generalizations of finite projective dimension, Illinois J. Math. 5 (1961), 334-344.

14. W. Jansen and C. Odenthal, A tiled order having large global dimension, J. Algebra (to appear).

15. V. A. Jategaonkar, Global dimension of tiled orders over a discrete valuation ring, Trans. Amer. Math. Soc. 196 (1974), 313-330.

16. I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.

17. E. E. Kirkman and J. J. Kuzmanovich, Global dimensions of a class of tiled orders, J. Algebra 127 (1989), 57-72.

18. , Personal communication.

19. E. E. Kirkman, J. J. Kuzmanovich, and L. W. Small, Finitistic dimension of noetherian rings, J. Algebra 147 (1992), 350-364.

20. W. Rump, Discrete posets, cell complexes, and the global dimension of tiled orders, Communic. in Algebra 24 (1996), 55-107.

21. R. B. Tarsy, Global dimension of orders, Trans. Amer. Math. Soc. 151 (1970), 335-340.

22. , Global dimension of triangular orders, Proc. Amer. Math. Soc. 28 (1971), 423-426.

23. A. Wiedemann and K. W. Roggenkamp, Path orders of global dimension two, J. Algebra 80 (1983), 113-133.

24. G. V. Wilson, Ultimately closed projective resolutions and rationality of Poincaré-Betti series, Proc. Amer. Math. Soc. 88 (1983), 221-223.

REPETITIVE RESOLUTIONS 21

25. A. Zaks, Injective dimension of semi-primary rings, J. Algebra 13 (1969), 73-86. Department of Mathematics, University of California, Santa Barbara, CA

93106 E-mail address: goodearl@math.ucsb.edu Department of Mathematics, University of California, Santa Barbara, CA

93106 E-mail address: birge@math.ucsb.edu