HOMOLOGICAL DOMINO EFFECTS AND THE

FIRST FINITISTIC DIMENSION CONJECTURE

Birge Zimmermann Huisgen

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Dedicated to Maurice Auslander on the occasion of his sixtyfifth birthday

Summary

Counterexamples to the first Finitistic Dimension Conjecture are constructed. In fact, for any field k and any integer m ≥ 2, there exist finite dimensional k-algebras Λ such that the little finitistic dimension of Λ is m, while the big finitistic dimension is m + 1. Our examples are monomial relation algebras; within this class of algebras the big and little finitistic dimensions cannot differ by more than 1. The analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras.

1. Introduction and notation.

Our primary goal is to provide a negative solution to the first of the two finitistic dimension problems which were publicized by Bass more than thirty years ago [6]. In the meantime, the status of these problems, restricted to finite dimensional algebras, has been advanced to that of conjectures; see e.g. [18, p.263]. Finitistic Dimension Conjectures: For a finite dimensional algebra Λ,

(1) fin dim Λ = Fin dim Λ and (2) fin dim Λ < ∞, where

fin dim Λ = sup{p dim M | M is a finitely generated left Λ-module with p dim M < ∞}

is the (left) little finitistic dimension of Λ, and

Fin dim Λ = sup{p dim M | M is an arbitrary left Λ-module with p dim M < ∞}

is the (left) big finitistic dimension of Λ. The ‘little’ invariant was introduced by Auslander and Buchsbaum [3],[4]; the ‘big’ is due to Kaplansky (see the introduction to [6]). Together they afford an accurate measure for the homological complexity of Λ.

A brief historical detour: The 1950’s and early 1960’s saw several milestones in the understanding and exploitation of homological dimensions of commutative noetherian rings through work of Auslander and Buchsbaum, Serre, Krull, Nagata, Bass and many others,

Typeset by AMS-TEX

This research was partially supported by a grant from the National Science Foundation

one of the major achievements being an extremely useful homological characterization of regular local rings, due to Auslander/Buchsbaum [4] and Serre [20]. Its proof incorporates the fact that the little finitistic dimension of a commutative local noetherian ring coincides with its depth − see [4] and [20]; this number may be strictly smaller than the big finitistic dimension, the latter being equal to the Krull dimension for any commutative noetherian ring − see [7] and [13]. During this same period, the homological tools were adapted for use in the non-commutative situation, in particular for finite dimensional algebras (see e.g. [2], [6], [8]), a development which led to the above conjectures. In the sequel, the finitistic dimension conjectures were related to a variety of other homological problems; see e.g. [11] for a bridge to dimensions of classical orders and [22, p.105] for a bridge to the Nakayama Conjecture.

While a substantial amount of existing work confirms the second conjecture for various classes of algebras − see [5], [11], [12], [14], [17], [19], [21] − very little appears to be known on the first. According to Bass, the little finitistic dimension is zero if and only if this is true for the big (this observation is part of a far stronger result [6, Theorem 6.3]). The only other contribution, due to the author [15], guarantees that for a monomial relation algebra Λ (see under Notation) the little and the big finitistic dimensions differ by at most 1, and that in ‘most’ cases both dimensions coincide with the (left) cyclic finitistic dimension,

cyc fin dim Λ = sup{p dim M | M is a cyclic left Λ-module with p dim M < ∞}. Our examples of finite dimensional algebras Λ with (fin dim Λ, Fin dim Λ) = (m, m+1)

are actually monomial relation algebras whose radical is nilpotent of index 4. For lower index of nilpotency the disparity does not occur; in fact, all three of the finitistic dimensions are equal in that case (see Corollary D). The idea underlying the examples is to trigger ‘infinite domino effects’ in submodules of projective modules having projective dimension Fin dim Λ − 1; slightly more precisely, our constructions will force such modules to contain infinite sequences x_{0}, x_{1}, x_{2}, … of elements which are k-linearly independent modulo the radical and which ‘push on one another’ within the radical layering of the module, to the effect that p_{i}x_{i} ∈ J^{n_{i}}x_{i+1} for some paths p_{i} ∈ Λ \ J^{n_{i}}, where J denotes the Jacobson radical of Λ.

The positive and negative results both rest on a tight description of the syzygies of submodules of projective modules over a monomial relation algebra Λ, in other words, on a description of second syzygies of arbitrary Λ-modules. Namely, the syzygy Ω^{1}(M) of a submodule M of a projective left Λ-module is always isomorphic to a direct sum of principal left ideals generated by paths; this was proved by the author in [15]. Here we pin down completely which principal left ideals appear as summands of Ω^{1}(M). It turns out that appearance or non-appearance hinges on certain ‘discontinuities’ within the radical layering J^{l}M of M, as alluded to in the preceding paragraph (see Theorem A for a precise statement).

In a subsequent paper [16], we show that the little finitistic dimension can be strictly larger than the cyclic finitistic dimension and, in fact, need not be attained by an ngenerated module for any pre-assigned positive integer n. Moreover, these phenomena can depend on the choice of the base field k.

Notation. Throughout, Λ = kΓ/I will be a finite dimensional monomial relation algebra; here Γ is a quiver and I an ideal of the corresponding path algebra kΓ which is generated by a set of paths of length ≥ 2 in Γ. (To view our results from a somewhat wider perspective, recall that each finite dimensional algebra over an algebraically closed field k is Morita equivalent to a factor algebra of a suitable path algebra kΓ; in the sequel, we will not impose any restrictions on k however.) Our convention for composing paths in Γ is as follows: ‘βα’ stands for ‘β after α’.

The Jacobson radical of Λ, denoted by J, clearly has the following canonical vectorspace basis ℙ over k. For j ≥ 0 let ℙ_{j} be the set of all nontrivial paths in Λ of length j, i.e., of all nonzero residue classes in Λ of paths of length j in Γ; this terminology is unambiguous since Λ is a monomial relation algebra. If J^{L+1} = 0, define ℙ = ⋃_{1≤j≤L} ℙ_{j}, that is, ℙ is the set of nontrivial paths in Λ of length ≥ 1. Moreover, note that ℙ_{0} is just the set of primitive idempotents of Λ. The primitive idempotent of Λ corresponding to a vertex ∗ of Γ will be denoted by either e(∗) or e_{∗}.

Next we introduce an integer s_{0} ≥ −1, depending on Λ, which will permit us to approximate all the finitistic dimensions of Λ up to an error of 1. Start by considering the set

S_{0} = {q ∈ ℙ | S(q) is a non-source in Γ and p dim Λq < ∞},

where S(q) is the starting point of the path q, and a source in Γ is a vertex which is not the endpoint of any arrow in Γ. If S_{0} = ∅, set s_{0} = −1; otherwise define s_{0} = sup{p dim Λq | q ∈ S_{0}}.

Finally, Ω^{1}(M) will denote the first syzygy of a left Λ-module M, that is, the kernel of a projective cover of M, and Ω^{k+1}(M) = Ω^{1}(Ω^{k}(M)) the higher syzygies.

2. Understanding submodules of projective modules. First applications.

While in [15] we were mainly concerned with projective dimensions of finitely generated modules, Theorem A of this section will permit us to compute projective dimensions also in the non-finitely generated case.

Lemma 1. ([15, Lemma 3]) Given any submodule U of a projective left Λ-module and 0 ≤ j ≤ L, the modules J^{j}U and J^{j}U/J^{j+1}U have k-vectorspace decompositions

J^{j}U = ⨁_{p∈ℙ_{j}} pU and J^{j}U/J^{j+1}U = ⨁_{p∈ℙ_{j}} pπ(U),

where π : U ⟶ U/J^{j+1}U is the canonical epimorphism. □

Lemma 2. ([15, Lemma 4]) Let f : P ⟶ Q be a homomorphism between two projective left Λ-modules P and Q, let T be a subset of ⋃_{q∈ℙ_{j}}(qP) for some j, and set D = ∑_{t∈T} Λt. Moreover, fix q_{0} ∈ ℙ_{j}.

Then any element x ∈ q_{0}P with f(x) ∈ f(D) is congruent, modulo ker f, to an element in D ∩ q_{0}P. □

Given any left Λ-module M, let ℙ_{0}(M) be the set of those primitive idempotents of Λ (= vertices of Γ under our identification) such that e(M/JM) ≠ 0. It will be useful to observe that, in case M embeds into the radical of a projective left Λ-module, ℙ_{0}(M) will contain only non-sources of Γ. Moreover, for 1 ≤ j ≤ L, let ℙ_{j}(M) be the set of those paths in ℙ_{j} which have starting point in ℙ_{0}(M). Write ℙ(M) = ⋃_{1≤j≤L} ℙ_{j}(M).

An element x ∈ M will be called a top element of M if x ∉ JM and if there exists an idempotent e ∈ ℙ_{0}(M) with ex = x (in that case x will also be called a top element of type e).

Part(1) of the following theorem is an upgraded version of [15, Theorem I]. Given a submodule M of a projective module, part(2) determines the structure of Ω^{1}(M) in dependence of that of M.

Theorem A. Let M be a submodule of a projective left Λ-module.

(1) The first syzygy Ω^{1}(M) is isomorphic to a direct sum of principal left ideals of Λ, each of which is generated by some path in ℙ(M).

(2) Suppose that G ⊆ M is a generating set for M which consists of top elements of M and is k-linearly independent modulo JM. Then the following statements are equivalent for a path q′ ∈ ℙ(M):

(i) Λq′ is isomorphic to a direct summand of Ω^{1}(M).

(ii) There exists a path q = αp ∈ ℙ(M), where α is an arrow, together with a nonempty subset G′ of G such that Λq′ ≅ Λq and pG′ is k-linearly independent modulo J^{length(p)+1}M, while qG′ is k-linearly dependent modulo J^{length(q)+1}M.

(iii) There exists a path q = αp ∈ ℙ(M), where α is an arrow, together with a top element x of M such that Λq′ ≅ Λq and px ∉ J^{length(p)+1}M, while qx = 0.

Proof. Given any generating set G for M consisting of top elements of M which are klinearly independent modulo JM, let e(g) ∈ ℙ_{0}(M) be the primitive idempotent with e(g)g = g for g ∈ G. Moreover, consider the projective cover

f : P = ⨁_{g∈G} Λe(g) ⟶ M

with f(e(g)) = g, where e(g) = (…, 0, e(g), 0, …) carries e(g) in the g-th position and zeroes elsewhere. The kernel of f, that is Ω^{1}(M), will be abbreviated by C.

For 0 ≤ j ≤ L, set

B_{j} = {pe(g) | p ∈ ℙ_{j}(M), g ∈ G, and S(p) = e(g)}.

Note that B_{0} = {e(g) | g ∈ G} and that each B_{j} is a k-basis for J^{j}P modulo J^{j+1}P; in particular, B = B_{0} ∪ B_{1} ∪ ⋯ ∪ B_{L} is a k-basis for P. Moreover, observe that J^{j}P = ⨁_{b∈B_{j}} Λb for all j. We will alternately view the B_{j}’s as subsets of P or J^{j}P/J^{j+1}P without using residue class notation. Furthermore, given any two subsets 𝕌, 𝕍 of ℙ, we define

𝕌 ∘ 𝕍 = {uv ∈ ℙ | u ∈ 𝕌, v ∈ 𝕍, uv is a nonzero path in Λ};

if 𝕌 ⊆ ℙ and V ⊆ B, we set

𝕌 ∘ V = {uv ∈ B | u ∈ 𝕌, v ∈ V such that uv ≠ 0}.

By contrast, given a nontrivial path q in Λ and a subset X of P or Q, the set qX = {qx | x ∈ X} may contain zero. Finally, _{k}⟨X⟩ will denote the k-subspace of P, resp. Q, generated by X.

Key induction: For 0 ≤ j ≤ L we will choose subsets S_{j}, T_{j} of B_{j} and pick elements c_{j}(s) ∈ C for s ∈ S_{j} such that

(a) S_{0} = ∅, T_{0} = B_{0}, and for j ≥ 1, the set ℙ_{1} ∘ T_{j−1} is the disjoint union of S_{j} and T_{j}.

(b) The sum C_{j} = ∑_{s∈S_{j}} Λc_{j}(s) is direct for each j, and Λc_{j}(s) ≅ Λp if s = pe(g) ∈ S_{j}.

(c) The map f induces a k-vectorspace isomorphism _{k}⟨T_{j}⟩ ⟶ J^{j}M/J^{j+1}M for j ≥ 0.

(d) If D_{j} = _{Λ}⟨T_{j}⟩ = ∑_{t∈T_{j}} Λt, then D_{0} = P and JD_{j−1} = C_{j} ⊕ D_{j} for j ≥ 1. Supplement: If U_{j} is a subset of B_{j} for 0 ≤ j ≤ L, respectively, such that U_{j+1} ⊆ ℙ_{1} ∘ U_{j} and each set f(U_{j}) is k-linearly independent modulo J^{j+1}M, we may choose T_{j} so as to contain U_{j} for each j.

First we will show how an argument for claim (1) can be completed, once we dispose of sets S_{j}, T_{j} and elements c_{j}(s) as above. Start by observing that for each j ≥ 0 we have a k-vectorspace decomposition

D_{j} = _{k}⟨T_{j}⟩ ⊕ JD_{j}.

In particular D_{L} = _{k}⟨T_{L}⟩ because JD_{L} = 0. Thus repeated use of condition (d) yields successive k-vectorspace decompositions

P = D_{0} = _{k}⟨T_{0}⟩ ⊕ JD_{0} = _{k}⟨T_{0}⟩ ⊕ C_{1} ⊕ D_{1} = ⋯

= ⨁_{j=0}^{L−1}_{k}⟨T_{j}⟩ ⊕ ⨁_{j=1}^{L} C_{j} ⊕ D_{L} = ⨁_{j=0}^{L}_{k}⟨T_{j}⟩ ⊕ ⨁_{j=1}^{L} C_{j}.

By construction, ⨁_{j=1}^{L} C_{j} ⊆ C, and by (c), f induces a k-vectorspace isomorphism

⨁_{j=0}^{L}_{k}⟨T_{j}⟩ ⟶ M;

thus ⨁_{j=1}^{L} C_{j} = C, and (b) guarantees that the C_{j} are direct sums of cyclic modules of the required form. This proves (1).

To launch the induction on j ∈ {0, 1, …, L}, set S_{0} = ∅ and T_{0} = B_{0}, and suppose that for some j ∈ {1, …, L}, sets S_{0}, …, S_{j−1} and T_{0}, …, T_{j−1}, as well as elements c_{l}(s) ∈ C for 1 ≤ l ≤ j − 1 and s ∈ S_{l}, have already been chosen in accordance with our rules (a−d). For any path p ∈ ℙ_{j−1}(M) set T_{j−1}^{(p)} = T_{j−1} ∩ (p ∘ B_{0}); in other words, T_{j−1}^{(p)} consists of all the nonzero elements of the form pe(h), h ∈ G, which belong to T_{j−1}. [If U_{j−1} ⊆ T_{j−1}, then clearly also U_{j−1}^{(p)} ⊆ T_{j−1}^{(p)}; here we canonically extend our terminology to U_{j−1}^{(p)} = U_{j−1} ∩ (p ∘ B_{0}).] Now consider a path q = αp ∈ ℙ_{j}(M), where α ∈ ℙ_{1}. By (c), f induces a k-space isomorphism

k⟨T_{j−1}^{(p)}⟩ ⟶ (pM + J^{j}M)/J^{j}M;

in particular, f(α ∘ T_{j−1}^{(p)}) generates the k-space qM modulo J^{j+1}M. Let us pick a subset T_{j}^{(q)} ⊆ α ∘ T_{j−1}^{(p)} such that f induces a k-space isomorphism

k⟨T_{j}^{(q)}⟩ ⟶ (qM + J^{j+1}M)/J^{j+1}M,

and let us denote the disjoint union of the sets T_{j}^{(q)} for q ∈ ℙ_{j}(M) by T_{j}. [If U_{j−1} ⊆ T_{j−1}, then U_{j}^{(q)} ⊆ α∘U_{j−1}^{(p)} ⊆ α∘T_{j−1}^{(p)}, and, due to the fact that f(U_{j}^{(q)}) is k-linearly independent modulo J^{j+1}M, we may clearly choose T_{j}^{(q)} so as to contain U_{j}^{(q)}.] Moreover, we define S_{j} = (ℙ_{1} ∘ T_{j−1}) \ T_{j} and D_{j} = ∑_{t∈T_{j}} Λt. Then condition (a) holds by construction, and Lemma 1 guarantees that this choice of T_{j} again satisfies (c).

To find elements c_{j}(s) ∈ C for s ∈ S_{j} which meet our requirements, we start by observing that condition (c) applied to T_{j} entails the equality f(D_{j}) = J^{j}M. Combining it with Lemma 2, we find for each s = qe(g) ∈ S_{j} an element t(s) ∈ D_{j} ∩ qP such that the difference c_{j}(s) = s − t(s) belongs to C. To check the legitimacy of these choices with respect to conditions (b) and (c), it will be useful to record the equalities

( ∑_{s∈S_{j}} Λs) ∩ D_{j} = ( ⨁_{s∈S_{j}} Λs) ∩ D_{j} = 0  (^{∗})

both obvious consequences of Lemma 1.

To verify that for any s = qe(g) ∈ S_{j} we have Λc_{j}(s) ≅ Λq, note that the left annihilator l ann_{Λ} c_{j}(s) of c_{j}(s) in Λ contains l ann_{Λ} q because c_{j}(s) ∈ qP. On the other hand, since Λs ∩ D_{j} = 0, we obtain l ann_{Λ} c_{j}(s) = l ann_{Λ} s ∩ l ann_{Λ} t(s) ⊆ l ann_{Λ} s = l ann_{Λ} q, which proves the desired equality l ann_{Λ} c_{j}(s) = l ann_{Λ} q.

Combining (^{∗}) with the fact that c_{j}(s) ≡ s modulo D_{j} for each s ∈ S_{j} by construction, we next observe that the sum C_{j} = ∑_{s∈S_{j}} Λc_{j}(s) is direct, which completes the verification of (b). But the same argument yields directness of the sum C_{j} + D_{j}, and in light of the straightforward equalities JD_{j−1} = _{Λ}⟨ℙ_{1} ∘ T_{j−1}⟩ = _{Λ}⟨T_{j}⟩ + _{Λ}⟨S_{j}⟩ = D_{j} + C_{j}, condition (d) is reestablished as well. This completes the induction.

We will now derive (2) from the result and argument of the key induction. As above, let T_{j}^{(q)} = T_{j} ∩ (q ∘ B_{0}) and S_{j}^{(q)} = S_{j} ∩ (q ∘ B_{0}) for any path q ∈ ℙ_{j}(M).

‘(ii) ⇒ (i)’. Suppose that G′ ⊆ G and q = αp ∈ ℙ(M), say q ∈ ℙ_{j}(M), are as under (ii). Writing p = α_{j−1}α_{j−2} ⋯ α_{1} with arrows α_{i} ∈ ℙ_{1}, define U_{l} = {α_{l} ⋯ α_{1}e(g) | g ∈ G′} ⊆ B_{l} for l ≤ j − 1, and set U_{l} = ∅ for j ≤ l ≤ L. By hypothesis, the family (U_{l})_{0≤l≤L} then satisfies the conditions of the supplement to our induction, which yields sets T_{l} such that U_{l} ⊆ T_{l} for each l. In particular, the nonempty set U_{j−1} = {pe(g) | g ∈ G′} is contained in T_{j−1}^{(p)}. By hypothesis, the set f(α ∘ U_{j−1}) = qG′ is k-linearly dependent modulo J^{j+1}M. Thus T_{j}^{(q)} ⫋ α ∘ T_{j−1}^{(p)}, which in turn implies that S_{j}^{(q)} = (α ∘ T_{j−1}^{(p)}) \ T_{j}^{(q)} ≠ ∅. But this means that S_{j} contains an element of the form s = qe(g) for some g ∈ G, and thus Λq′ ≅ Λq ≅ Λc_{j}(s) occurs as a direct summand of C.

‘(i) ⇒ (ii)’. Now suppose that Λq′ is isomorphic to a direct summand of C. Due to the indecomposability of Λq′ and the cyclic summands Λc_{j}(−), this is the same as to say that Λq′ ≅ Λc_{j}(s) for some j ≥ 1 and s ∈ S_{j}. The last isomorphism, in turn, translates into Λq′ ≅ Λq if s = qe(g) for some g. In particular, we glean that S_{j}^{(q)} ≠ ∅. But, as we saw in the preceding paragraph, if q = αp for some arrow α, this means that T_{j}^{(q)} ⫋ α ∘ T_{j−1}^{(p)}. From this nonequality we conclude that the set G′ = {g ∈ G | pe(g) ∈ T_{j−1}^{(p)}} meets the requirements of (ii) as follows. On one hand, the set pG′ = f(T_{j−1}^{(p)}) is k-linearly independent modulo J^{j}M, since by condition (c), f induces a k-space isomorphism

k⟨T_{j−1}^{(p)}⟩ ⟶ (pM + J^{j}M)/J^{j}M.

On the other hand, by the above strict inclusion, qG′ = f(α∘T_{j−1}^{(p)}) is k-linearly dependent modulo J^{j+1}M; here we again invoke condition (c) for the fact that f induces a k-space isomorphism

k⟨T_{j}^{(q)}⟩ ⟶ (qM + J^{j+1}M)/J^{j+1}M.

‘(iii) ⇒ (i)’ is a special case of the implication ‘(ii) ⇒ (i)’, because x can be supplemented to a generating set for M having the properties of G.

‘(i) ⇒ (iii)’. Again we suppose that C contains a direct summand isomorphic to Λq′ and, as under ‘(i) ⇒ (ii)’, we find an index j, an element g ∈ G, and a path q = αp ∈ ℙ(M) such that Λq′ ≅ Λc_{j}(s) ≅ Λq, where s = qe(g) ∈ S_{j}. Recall the construction pattern for the elements c_{j}(s), namely c_{j}(s) = s − t(s) with t(s) ∈ (∑_{t∈T_{j}} Λt) ∩ qP. Accordingly, write t(s) as a Λ-linear combination of elements q_{i}e(h_{i}) ∈ T_{j}, say t(s) = ∑_{i} λ_{i}q_{i}e(h_{i}) with λ_{i}q_{i} ∈ qΛ for each i; this is possible in view of Lemma 1. Since s = qe(g) ∉ T_{j}, we have h_{i} ≠ g whenever q_{i} = q. As a consequence, t(s) is congruent, modulo qJP, to an element of the form q ∑_{h_{i}≠g} k_{i}e(h_{i}), where the k_{i} are suitable scalars in k. With these preparatory remarks in mind, we can pick an element u ∈ P with qu = t(s) and u congruent to ∑_{h_{i}≠g} k_{i}e(h_{i}) modulo JP. Since q = qe(g), we may further assume that u = e(g)u, that is, e(h_{i}) = e(g) for all i. Setting y = e(g) − u, we thus obtain:

• y is a top element of P (of type e(g)), whence f(y) is a top element of M (of the same type).

• qy = c_{j}(s) ∈ C, whence qf(y) = 0.

• pe(h) ∈ T_{j−1}^{(p)} whenever qe(h) ∈ S_{j}^{(q)} ∪ T_{j}^{(q)} = α ∘ T_{j−1}^{(p)}, since αp = q; in particular, pe(g) ∈ T_{j−1}^{(p)}. Consequently, f(pe(g) − ∑_{h_{i}≠g} k_{i}pe(h_{i})) ∉ J^{j}M by the choice of T_{j−1}^{(p)}; this in turn implies pf(y) ∉ J^{j}M, since py ≡ (pe(g) − ∑_{h_{i}≠g} k_{i}pe(h_{i})) modulo J^{j}P.

Thus the element x = f(y) meets all the requirements of (iii). □

Corollary B. cyc fin dim Λ, fin dim Λ, Fin dim Λ ∈ {s_{0} + 1, s_{0} + 2}

Proof. It clearly suffices to check that cyc fin dim Λ ≥ s_{0} + 1, and Fin dim Λ ≤ s_{0} + 2.

The first inequality is clear if s_{0} = −1. Otherwise pick q_{0} ∈ S_{0} with p dim Λq_{0} = s_{0} ≥ 0. In view of Λq_{0} ⊆ J, we have p dim Λ/Λq_{0} = s_{0} + 1, showing that indeed cyc fin dim Λ ≥ s_{0} + 1.

For the second inequality, let X be any left Λ-module of finite projective dimension, and apply Theorem A(1) to the module M = Ω^{1}(X). Since M is contained in the radical of a projective left Λ-module, ℙ_{0}(M) consists of non-sources of Γ, and consequently each summand in the pertinent cyclic decomposition of Ω^{1}(M) is of the form Λq with q ∈ S_{0}. Therefore, p dim Ω^{1}(M) ≤ s_{0} and p dim X ≤ s_{0} + 2. (We adopt the convention p dim 0 = −1 to make the former inequality true also in case s_{0} = −1.) □

Remark. The invariant s_{0} of Λ considered here may be strictly smaller than the invariant s introduced in [15]. The new choice has a clear edge over the old, since it involves only paths starting in non-sources of Γ. Combining Corollary B above with [15, Corollary II], we see that all the finitistic dimensions of Λ actually belong to the intersection {s_{0} + 1, s_{0} + 2} ∩ {s + 1, s + 2}, a proper strengthening of both corollaries.

The next corollary is really a consequence of the proof of Theorem A. We will subsequently use it to see that monomial relation algebras with J^{3} = 0 are homologically harmless. Call a submodule U of a left Λ-module M saturated (with respect to M) if J^{j}U = U ∩ J^{j}M for 0 ≤ j ≤ L.

Corollary C. Let M be a submodule of a projective left Λ-module, and U a saturated submodule of M. Then Ω^{1}(U) is isomorphic to a direct summand of Ω^{1}(M).

Proof. Start by choosing a generating set G(U) of U which consists of top elements of U (and thus of top elements of M) and which is k-linearly independent modulo JU (and thus also modulo JM). Supplement G(U) to a generating set G for M, in turn consisting of top elements of M and k-linearly independent modulo JM. Moreover, let

f : P = ⨁_{g∈G} Λe(g) ⟶ M

be a projective cover as in the proof of Theorem A. Then the restriction of f to P (U) = ⨁_{g∈G(U)} Λe(g) is a projective cover of U, and the canonical k-basis 𝔹 of P contains the canonical k-basis 𝔹(U) = ⋃_{j=0}^{L} 𝔹_{j}(U) of P (U).

Next pick sets S_{j}(U) and T_{j}(U) in 𝔹_{j}(U) as in the key induction backing Theorem A, and note that the family (U_{j})_{0≤j≤L} = (T_{j}(U))_{0≤j≤L} satisfies the hypotheses of the supplement. Thus we may choose sets S_{j}, T_{j} ⊆ 𝔹 for the module M such that T_{j}(U) ⊆ T_{j} for each j. In light of property (a) of the sets we just picked, these containments force S_{j}(U) to be contained in S_{j} for each j. Thus the direct sum Ω^{1}(M) ≅ ⨁_{j=1}^{L}(⨁_{qe(g)∈S_{j}} Λq)

contains Ω^{1}(U) ≅ ⨁_{j=1}^{L}(⨁_{qe(g)∈S_{j}(U)} Λq) as a subsum, which proves our claim □

Corollary D. If J^{3} = 0, then cyc fin dim Λ = fin dim Λ = Fin dim Λ.

Proof. By Corollary B there is nothing to prove if Fin dim Λ = s_{0} + 1, and hence we may assume that Fin dim Λ = s_{0} + 2. Let M ⊆ JΛ^{(I)} be a submodule of the radical of a free left Λ-module Λ^{(I)} such that p dim M = s_{0} + 1.

If s_{0} = −1, we have Fin dim Λ = 1 and may quote [6, Theorem 6.3] for the equality of the big and little finitistic dimensions. Alternately, we observe that the nonzero projective module M, and a fortiori the module J^{(I)}, contains a cyclic submodule U = Λx isomorphic to an indecomposable left ideal Λe. For the conclusion that indeed cyc fin dim Λ = 1 in that case, we refer to the last paragraph of the proof.

Now suppose that s_{0} ≥ 0. By Theorem A(2) there exists a path q = αp ∈ S_{0}, where α is an arrow, together with a top element x of M satisfying p dim Λq = s_{0} and px ∉ J^{length(p)+1}M, while qx = 0. Using the fact that J^{2}M = 0 by hypothesis, we see that the cyclic submodule U = Λx of M is saturated in the sense of Corollary C: Indeed, U/JU being simple, the assumption that JU is strictly contained in U ∩JM would imply U ∩JM = U, which is incompatible with the choice of x ∈ M \ JM. Thus Corollary C guarantees that Ω^{1}(U) is isomorphic to a direct summand of Ω^{1}(M), which shows in particular that p dim U is finite. Due to the choice of the element x, we moreover have px ∉ J^{length(p)+1}U, while qx = 0. Therefore Theorem A provides us with a summand of Ω^{1}(U) isomorphic to Λq. This shows p dim Ω^{1}(U) ≥ s_{0}.

Next we observe that, in either of the two cases considered, Ω^{1}(U) ≅ l ann_{Je} x = l ann_{Je} A, where e is the primitive idempotent with ex = x and A is the set of all those arrows in Λ which occur nontrivially in one of the components of x ∈ J^{(I)}. The isomorphism is clear; to check the equality, we observe that Jr = 0 for each path r of length ≥ 2 in Λ by hypothesis and that the annihilator of any homogeneous element z ∈ Λ^{(I)} annihilates all the individual paths occurring nontrivially in one of the components of z. Setting y = ∑_{a∈A} a ∈ J, we can for the same reason supplement the above annihilator equality by ‘l ann_{Je} A = l ann_{Je} y’, and conclude p dim Λy = p dim U ≥ s_{0} + 1. This shows p dim Λ/Λy ≥ s_{0} + 2, and the inequality cyc fin dim Λ ≥ Fin dim Λ follows. □

3. The inequality ‘fin dim Λ < Fin dim Λ’.

In [15, Section 5] the author developed an easy algorithm to compute the invariant s_{0} of Λ based on the quiver and the relations; it involves a description of the left ideals Λq, q ∈ ℙ, and their syzygies via labeled undirected graphs. Similar graphical means of communicating modules appear to be part of mathematical folklore. Recent treatments were given by Alperin [1] and Fuller [9]; in particular, Fuller pointed out that monomial relation algebras can be represented by, and in fact constructed from, such diagrams. The labeled graphs we use here store structural information in a slightly different fashion which is considerably more convenient for us. In our example, existence and uniqueness of a monomial relation algebra corresponding to the specified set of graphs for the indecomposable projective modules will be obvious.

Given a submodule M of a projective left Λ-module, top elements x, y of M and a path p ∈ ℙ, we will call y a p-domino for x in case px ∈ J^{length(p)+1}y; in view of Lemma 1, this is the same as to say px ∈ pJy. Due to Theorem A, p-dominoes will turn out to be directly responsible for the occurrence of direct summands isomorphic to Λp in the syzygy Ω^{1}(M).

Let n ≥ 1. To construct a monomial relation algebra Λ with fin dim Λ < Fin dim Λ = n + 2, we will use the control over second syzygies provided by Theorem A in the following manner. On one hand, we will make sure that, given any element x_{0} which lies in the radical of a projective left Λ-module Q, the cyclic module Λx_{0} will have infinite projective dimension whenever Ω^{1}(Λx_{0}) contains a cyclic direct summand of projective dimension n. Say Ω^{1}(Λx_{0}) contains a direct summand Λp, where p is a path such that p dim Λp = ∞. Then, if x_{0} is a top element of a submodule M ⊆ JQ with p dim M < ∞, it follows that p_{1}x_{0} ∈ J^{length(p_{1})+1}M for a suitable right subpath p_{1} of p. This containment will translate into the existence of a p_{1}-domino x_{1} ∈ M for x_{0}. Next, we will make sure that Ω^{1}(Λx_{0} + Λx_{1}) will again contain a direct summand Λp, and, in view of the finiteness of the projective dimension of M, the element x_{1} will therefore in turn call for a domino x_{2} ∈ M, and so on. In this way our construction will guarantee that dim_{k} M = ∞ whenever M ⊆ JQ has projective dimension n + 1. On the other hand, we will arrange for the existence of an infinite sequence y_{0}, y_{1}, y_{2}, … in the radical of some projective left Λmodule such that each y_{i} is a p_{1}-domino for y_{i−1} with respect to the module N = ∑_{i≥0} Λy_{i}, and such that p dim N = n + 1.

Theorem E. For each integer n ≥ 1 there exists a finite dimensional monomial relation algebra Λ with J^{4} = 0 such that

fin dim Λ = n + 1  and  Fin dim Λ = n + 2

Remark. Theorem E shows in particular that all pairs of natural numbers (m, m + 1) for m ≥ 2 are attained by the pair (fin dim Λ, Fin dim Λ), where Λ is a finite dimensional monomial relation algebra; that the pairs (m, m) for m ≥ 0 are attained as well, is obvious. On the other hand, (0, 1) does not occur − see [6, Theorem 6.3] − and the only remaining possibility, namely (1, 2), we believe to be excluded as well. While the construction scheme described below will also yield a monomial relation algebra Λ_{0} in case n = 0, it is easily checked that fin dim Λ_{0} = Fin dim Λ_{0} = 2. (Indeed, since s_{0} = 0 in that case, Corollary B yields Fin dim Λ_{0} ≤ 2; on the other hand, the left ideal Λ_{0}x with x = γ_{1} + γ_{2} + τγ_{2} has projective dimension 1, since the left annihilator of x in Λ_{0}e_{1} is Λ_{0}χ_{1} ⊕ Λ_{0}(χ_{2} − χ_{2}τ) ≅ Λ_{0}χ_{1} ⊕ Λ_{0}χ_{2}, and thus p dim Λ_{0}/Λ_{0}x = 2.)

Proof of Theorem E. Let k be any field and fix n ≥ 1.

(I) The quiver Γ:

Define Λ = kΓ/I, where I is the ideal of kΓ generated by those paths in Γ which do not make an appearance in any of the trees shown in Diagram II. For instance, the path α_{1}α_{0}γ_{1} in Γ belongs to I, that is, it vanishes in Λ.

(II) The graphs of the indecomposable projective left Λ-modules:

For a sample interpretation of one of these trees, let S(∗) be the simple module corresponding to a given vertex ∗. That the tree topped by the vertex a_{0} depicts the indecomposable projective module X = Λe(a_{0}) means: X/JX ≅ S(a_{0}), and if x is a generator for X, then α_{1}x and β_{1}x generate copies of S(a_{1}) and S(b) modulo J^{2}X, while α_{2}α_{1}x and β_{2}β_{1}x generate copies of S(a_{2}) and S(b) in the socle of X.

(III) Claim: S_{0} = {α_{1}α_{0}, α_{2}α_{1}, α_{i}, β_{1}, χ_{1}, χ_{2} | i = 1, …, n + 1}; moreover, s_{0} = n, and α_{1}α_{0} is the only path q ∈ S_{0} with p dim Λq = n.

Proof. Since the vertex ‘2’ is a source of Γ, we need only consider paths of length ≥ 1 which do not originate in ‘2’. We illustrate the algorithm of [15, Section 5] with two sample computations:

p dim Λα_{0} = p dim( \begin{matrix}a_{0} \\ ^{α_{1}} / \^{β_{1}} \\ a_{1} b \end{matrix} ) = 1 + max\{p dim \binom{a_{2}}{•}, p dim \binom{b}{•}\} = ∞

and

p dim Λα_{1}α_{0} = p dim \binom{a_{1}}{•} = 1 + p dim \binom{a_{2}}{•} = … = n + p dim \binom{a_{n+1}}{•} = n.

The remaining verifications are left to the reader.

(IV) Claim: fin dim Λ ≤ n + 1.

Proof. Let Q = ⨁_{i∈I} Λe(i) be a projective left Λ-module, where the e(i) are primitive idempotents of Λ, and suppose that M is a submodule of JQ having projective dimension n + 1.

In showing that dim_{k}M = ∞, the following notion of support will be useful. For p ∈ ℙ_{1} and x ∈ Q, say x = (∑_{q∈ℙ} k_{iq}q)_{i∈I} with ∑_{q∈ℙ} k_{iq}q ∈ Λe(i) and k_{iq} ∈ k, we define the p-support of x to be 𝔰_{p}(x) = (k_{ip})_{i∈I} ∈ k^{(I)}. Note that this definition depends only on the chosen decomposition of Q. Thus 𝔰_{p}(x) ≠ 0 if and only if the path p occurs nontrivially in some component of x. If x ∈ M, then this event is only possible in case the length of p is nonzero, since M ⊆ JQ.

Our domino game will be as follows. We will open it by securing a suitable top element x_{0} of type e_{1} in M. Our choice will be based on the fact that Ω^{1}(M) necessarily contains a direct summand isomorphic to Λα_{1}α_{0}. In view of the finiteness of the projective dimension of M, the element x_{0} will entail the existence of a χ_{2}-domino x_{11} ∈ M such that x_{0} and x_{11} are k-linearly independent modulo JM. Finiteness of p dim M and our choice of x_{11} will next necessitate the existence of χ_{2}-dominoes, x_{21} for x_{0} and x_{22} for x_{11}, such that the three elements x_{0}, x_{21}, x_{22} are in turn k-linearly independent modulo JM; in due course, these will call for χ_{2}-dominoes x_{31}, x_{32}, x_{33}, and so on. In other words, in each move we increase the preceding row of dominoes by one and attach x_{0} to the new row, before we proceed to the next move. This yields an array of the form shown below: where the columns consist of dominoes, each of which is pushing on the one located above it, and where each row consists of elements which are k-linearly independent modulo JM. In particular, the game of course entails dim_{k}M = ∞. Clearly, this state of affairs then allows us to conclude that p dim N ≤ n for each finitely generated left Λ-module N contained in the radical of a projective module, which in turn yields fin dim Λ ≤ n + 1 as desired. Before we launch an induction describing a typical move, we repeatedly use Theorem A to establish the rules of the game.

Observation 1: There exists a top element x_{0} ∈ M of type e_{1} such that 𝔰_{γ_{2}}(x_{0}) = 0. Indeed, Ω^{1}(M) has projective dimension n and therefore contains a direct summand isomorphic to Λα_{1}α_{0}, the path α_{1}α_{0} being the only one in ℙ(M) that generates a left ideal of projective dimension n. Theorem A thus yields a top element x_{0} ∈ M of type e_{1} such that α_{0}x_{0} does not belong to J^{2}M, whereas α_{1}α_{0}x_{0} = 0. Since α_{1}α_{0}γ_{2} ≠ 0, the element x_{0} has the property singled out above.

Observation 2: If x ∈ M is any top element of type e_{1}, then 𝔰_{τ} (x) = 0, but either 𝔰_{τγ_{1}}(x) ≠ 0 or 𝔰_{τγ_{2}}(x) ≠ 0. To show vanishing of the τ-support of x, observe that for i = 1, 2 we have χ′_{i}χ_{i}x = 0 because x ∈ JQ and χ′_{i}χ_{i}J = 0. The left ideal Λχ′_{i}χ_{i}, having infinite projective dimension, is not isomorphic to a direct summand of Ω^{1}(M), and thus Theorem A(2) tells us that χ_{i}x ∈ J^{2}M ⊆ J^{3}Q; in particular, 𝔰_{τ} (x) = 𝔰_{χ_{i}τ} (χ_{i}x) = 0. Moreover, if we had 𝔰_{τγ_{1}}(x) = 𝔰_{τγ_{2}}(x) = 0, the definition of Λ would imply σx = 0 and hence, by Theorem A(2), force a summand Λσ into the first syzygy of M. But this is impossible because p dim Λσ = ∞, and thus the second part of the observation is backed up as well.

As an immediate consequence of Observation 2 we obtain the following: If x_{1}, …, x_{m} are top elements of type e_{1} of M which are k-linearly independent modulo JM and such that 𝔰_{τγ_{1}}(x_{i}) = 0 for 1 ≤ i ≤ m, then the vectors 𝔰_{τγ_{2}}(x_{1}), …, 𝔰_{τγ_{2}}(x_{m}) in k^{(I)} are klinearly independent. Indeed, let k_{1}, …, k_{m} be scalars such that 0 = ∑_{i=1}^{m} k_{i}𝔰_{τγ_{2}}(x_{i}) = 𝔰_{τγ_{2}}(∑_{i=1}^{m} k_{i}x_{i}). From Observation 2 we know that ∑_{i=1}^{m} k_{i}x_{i} cannot be a top element of type e_{1} of M, i.e., ∑_{i=1}^{m} k_{i}x_{i} ∈ JM. By the independence hypothesis on the x_{i}, we thus conclude that k_{1} = k_{2} = ⋯ = k_{m} = 0 as desired.

Observation 3: If x ∈ M is a top element of type e_{1} and if i ∈ {1, 2} is such that 𝔰_{τγ_{i}}(x) ≠ 0, then there exists a χ_{i}-domino y ∈ M for x such that 𝔰_{τγ_{i}}(x) = 𝔰_{γ_{i}}(y). As we observed above, χ′_{i}χ_{i}x = 0, and consequently χ_{i}x ∈ J^{2}M ⊆ J^{3}Q. Because 𝔰_{τ} (x) = 0 by Observation 2, we compute χ_{i}x = χ_{i}τγ_{i}𝔰_{τγ_{i}}(x) + χ_{i}τγ_{3}𝔰_{τγ_{3}}(x) ∈ χ_{i}τQ, and, using Lemma 1, we infer that χ_{i}x ∈ χ_{i}τM \ J^{3}M. Thus there exists a top element y ∈ M of type e_{1} such that χ_{i}x = χ_{i}τy; in other words, y is a χ_{i}-domino for x. Comparing χ_{i}x with χ_{i}τy = χ_{i}τγ_{i}𝔰_{γ_{i}}(y) + χ_{i}τγ_{3}𝔰_{γ_{3}}(y) and using again Lemma 1, we finally conclude that 𝔰_{τγ_{i}}(x) = 𝔰_{γ_{i}}(y).

Observation 4, a normalization step: We may choose an element x_{0} as in Observation 1 with the additional property that 𝔰_{τγ_{1}}(x_{0}) = 0. For if this last vector is nonzero, Observation 3 supplies us with a χ_{1}-domino y_{0} ∈ M for x_{0} such that 𝔰_{τγ_{1}}(x_{0}) = 𝔰_{γ_{1}}(y_{0}); the modified top element x_{0} − τy_{0} then combines both of the desired properties.

Keeping these observations in mind, we will now show by induction on m ≥ 1 that M contains top elements x_{m1}, ..., x_{mm} of type e_{1} such that the vectors 𝔰_{γ_{2}}(x_{m1}), ..., 𝔰_{γ_{2}}(x_{mm}) in k^{(I)} are k-linearly independent. Since x_{0} is a top element of M with 𝔰_{γ_{2}}(x_{0}) = 0, this will clearly imply that the elements x_{0}, x_{m1}, …, x_{mm} of M are k-linearly independent modulo J^{2}Q and a fortiori modulo JM, as required for the game sketched at the outset.

‘m = 1’. In view of Observations 2 and 4 above, we may assume that 𝔰_{τ} (x_{0}) = 𝔰_{τγ_{1}}(x_{0}) = 0, whence Observation 2 yields 𝔰_{τγ_{2}}(x_{0}) ≠ 0. Next Observation 3 provides us with a χ_{2}-domino x_{11} ∈ M for x_{0}, with the property that 𝔰_{τγ_{2}}(x_{0}) = 𝔰_{γ_{2}}(x_{11}). In particular, 𝔰_{γ_{2}}(x_{11}) ≠ 0.

Now suppose that for some m ≥ 1 we have already found elements x_{m1}, …, x_{mm} ∈ M meeting our conditions. Write x_{i} = x_{mi} for brevity. We may assume w.l.o.g. that 𝔰_{τγ_{1}}(x_{i}) = 0 for 1 ≤ i ≤ m. Indeed, if 𝔰_{τγ_{1}}(x_{i}) ≠ 0, then, as in the procedure for ‘normalizing’ x_{0}, Observation 3 supplies us with a top element y_{i} ∈ M of type e_{1} such that 𝔰_{τγ_{1}}(x_{i}) = 𝔰_{γ_{1}}(y_{i}). Replacement of x_{i} by x_{i} − τy_{i} therefore yields the situation we want to produce, without interfering with the induction hypothesis; namely, 𝔰_{τγ_{1}}(x_{i} − τy_{i}) = 0 and 𝔰_{γ_{2}}(x_{i} − τy_{i}) = 𝔰_{γ_{2}}(x_{i}).

Moreover, as we have already observed above, the induction hypothesis guarantees that the top elements x_{0}, x_{1}, …, x_{m} are k-linearly independent modulo JM. Invoking again Observation 2, we further see that 𝔰_{τ} (x_{i}) = 0 for all i. Thus we start out with a situation where 𝔰_{τγ_{1}}(x_{i}) = 𝔰_{τ} (x_{i}) = 0 for 0 ≤ i ≤ m, and therefore Observation 2 forces the vectors 𝔰_{τγ_{2}}(x_{0}), …, 𝔰_{τγ_{2}}(x_{m}) in k^{(I)} to be k-linearly independent. On the other hand, Observation 3 yields χ_{2}-dominoes y_{0}, y_{1}, …, y_{m} ∈ M for the elements x_{0}, x_{1}, …, x_{m} respectively, with the property that 𝔰_{τγ_{2}}(x_{i}) = 𝔰_{γ_{2}}(y_{i}) for 0 ≤ i ≤ m. In particular, the vectors 𝔰_{γ_{2}}(y_{0}), 𝔰_{γ_{2}}(y_{1}), …, 𝔰_{γ_{2}}(y_{m}) are in turn k-linearly independent, and thus the choice x_{m+1,i} = y_{i−1} for 1 ≤ i ≤ m + 1 completes our induction.

(V) Claim: Fin dim Λ ≥ n + 2.

Proof. Consider the projective left Λ-module Q = ⨁_{i≥0} Λu_{i} where u_{i} = e_{2} for i ≥ 0. It will clearly suffice to construct a submodule M ⊂ JQ with p dim M = n + 1.

Define M = ∑_{i≥0} Λx_{i}, where

x_{0} = (γ_{1} + γ_{3} + τγ_{2})u_{0} and
x_{i} = γ_{2}u_{i−1} + (γ_{1} + τγ_{2})u_{i} for i ≥ 1.

The module M can be visualized as follows:

Clearly, the set G = {x_{i} | i ≥ 0} consists entirely of top elements of M of type e_{1}, and consequently ℙ(M) is the set of all nontrivial paths of length ≥ 1 in Λ having starting point ‘1’. Thus ℙ(M) = {χ_{i}, χ′_{i}χ_{i}, χ_{i}τ, σ, τ, στ, α_{0}, α_{1}α_{0}, β_{1}α_{0}}. We will now use Theorem A(2) to determine which of the principal left ideals Λq with q ∈ ℙ(M) are isomorphic to direct summands of Ω^{1}(M). We begin by noting that Λχ′_{i}χ_{i} ≅ Λχ_{i}τ for i = 1, 2 and Λστ ≅ Λσ are distinct isomorphism classes, and that the left ideals Λα_{0}, Λα_{1}α_{0}, Λβ_{1}α_{0}, Λχ_{1}, Λχ_{2}, and Λτ are pairwise nonisomorphic. We retain the notion of support of an element of M, relative to the given decomposition of Q, as introduced in the proof of Claim IV.

Observe that χ_{1}G = 0, whence Λχ_{1} is isomorphic to a direct summand of Ω^{1}(M). To see that Λχ′_{1}χ_{1} is not isomorphic to a direct summand of Ω^{1}(M), we still need to check the status of the path q = χ_{1}τ: Clearly 𝔰_{γ_{1}}(x) ≠ 0 for every top element x ∈ M, whence χ_{1}τx ≠ 0, and the conditions of Theorem A(2)(iii) are not satisfied for χ_{1}τ. Thus Λχ′_{1}χ_{1} ≅ Λχ_{1}τ is ruled out as a direct summand of Ω^{1}(M).

In a second step, we compute χ_{2}x_{i} = χ_{2}τx_{i+1} ∈ J^{2}M for all i ≥ 0, whence χ_{2}M ⊆ J^{2}M, and Λχ_{2} is isomorphic to a direct summand of Ω^{1}(M). To rule out the appearance of Λχ′_{2}χ_{2} ≅ Λχ_{2}τ in Ω^{1}(M), note that, by construction of M, we have either 𝔰_{γ_{3}}(x) ≠ 0 or 𝔰_{γ_{2}}(x) ≠ 0 for any top element x of M; in particular, χ_{2}τx ≠ 0 for any top element x, which proves that no copy of Λχ′_{2}χ_{2} ≅ Λχ_{2}τ occurs as a direct summand of Ω^{1}(M).

Next we will check that σx ≠ 0 and στx ≠ 0 for any top element x of M; together, these verifications will guarantee that Ω^{1}(M) has no direct summands isomorphic to Λσ ≅ Λστ. For the second of these two assertions, it suffices to recall that 𝔰_{γ_{1}}(x) ≠ 0 for any top element x of M; since στγ_{1} is a nontrivial path in Λ, this implies that στx ≠ 0. For the first assertion, assume that σx = 0 for some top element x of M. This clearly implies the existence of scalars k_{1}, …, k_{m} ∈ k, not all zero, such that σ(k_{1}x_{1} + ⋯ + k_{m}x_{m}) ∈ J^{2}M. Set y = ∑_{i=1}^{m} k_{i}x_{i}. Now σy = στγ_{2}(∑_{i=1}^{m} k_{i}u_{i}) does not belong to J^{4}Q and a fortiori not to J^{3}M. Since moreover σy ∈ e_{1}J^{2}M = στM, we infer that σy ∈ στM \ J^{3}M. In other words, there exists a top element z ∈ M of type e_{1} such that σy = στz. In particular, we deduce that 𝔰_{γ_{1}}(z) = 0, for the definition of y implies 𝔰_{τγ_{1}}(y) = 0. On the other hand, modulo JM, the top element z is congruent to a nontrivial k-linear combination of the x_{i} which, again by construction, entails 𝔰_{γ_{1}}(z) ≠ 0. This contradiction backs our assertion concerning σx.

Using once more the fact that 𝔰_{γ_{1}}(x) ≠ 0 for any top element x of M, and checking that qγ_{1} ≠ 0 for any path q ∈ {τ, α_{0}, β_{1}α_{0}}, we obtain further that qx ≠ 0 for any top element x of M. Again by Theorem A, this proves that none of the corresponding left ideals Λq is isomorphic to a direct summand of Ω^{1}(M).

Finally Λα_{1}α_{0} does occur as a direct summand of Ω^{1}(M), because α_{0}x_{0} ∉ J^{2}M whereas α_{1}α_{0}x_{0} = 0.

In summary, we have shown that among the cyclic left ideals Λq with q ∈ ℙ(M), the only ones isomorphic to direct summands of Ω^{1}(M) are Λα_{1}α_{0}, Λχ_{1} and Λχ_{2}. In view of part III of our argument, this proves that p dim Ω^{1}(M) = n, and hence p dim M = n + 1 as required. □

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16 BIRGE ZIMMERMANN HUISGEN

NOTE TO THE PRINTER Fonts used:

Greek: α, β, γ, Γ, λ, Λ, π, σ, τ, χ, Ω Special Roman: 𝔹, ℙ, 𝕌, 𝕍 Boldface: b, B, e, s, S, t, T, u, U, v, V Script: S Fraktur: 𝔰

Additional Request: If at all possible, please insert Diagrams I and II in exactly the places marked. The placement of Diagrams III and IV is less sensitive.