FIELD–DEPENDENT HOMOLOGICAL BEHAVIOR

OF FINITE DIMENSIONAL ALGEBRAS

Birge Zimmermann Huisgen

1. Introduction and notation.

From a spurt of recent work it has become apparent that, also in non-commutative situations, the finitistic dimensions introduced by Auslander-Buchsbaum [2] and Bass [4] are highly informative invariants which are, moreover, accessible in many interesting contexts, including non-artinian situations such as enveloping algebras of Lie algebras and rings of generic matrices (see, e.g., [3], [8−16]). In particular, given a finite dimensional algebra Λ over a field k, the combination of the following three dimensions tells much of the algebra’s homological tale:

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

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

where p dim M denotes the projective dimension of M. We refer to these invariants as the cyclic, little and big finitistic dimensions of Λ, respectively.

In [13] the author showed that the big finitistic dimension may exceed the little; however, so far no examples had been known where the little finitistic dimension is not attained on a cyclic module. While it may not come as a surprise that fin dim Λ, in turn, can be strictly larger than cyc fin dim Λ, it was completely unexpected that this phenomenon may depend on the cardinality of the base field. More precisely, starting with any natural number d ≥ 3, we exhibit a quiver Γ_{d} and a set ρ_{d} of paths in Γ_{d}, resulting in a monomial relation algebra Λ_{d} = kΓ_{d}/⟨ρ_{d}⟩ of finite dimension over any base field k, such that cyc fin dim Λ_{d} < fin dimΛ_{d} precisely when |k| < d. Going still further in this direction, we construct for each finite field k and each natural number m ≥ 2 a finite dimensional monomial relation algebra Λ over k such that fin dim Λ is not attained on any module with fewer than m generators, whereas fin dim K ⊗_{k} Λ = cyc fin dim K ⊗_{k} Λ for every extension field K of k of sufficiently high cardinality. Amusingly, the extent to which the base field can influence the various finitistic dimensions of a monomial relation algebra is limited to adding a term ±1, although we expect this to be no longer true in non-monomial situations. These phenomena are all the more surprising as the global dimension of monomial relation algebras is not affected by the choice of field (see [11, p. 178] or Observation A below).

Typeset by AMS-TEX

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

The inequality ‘cyc fin dim Λ < fin dim Λ’ can also be realized in a field-independent fashion; namely, there exist a quiver Γ and a set ρ of paths in Γ such that, irrespective of the choice of k, the cyclic finitistic dimension of kΓ/⟨ρ⟩ is strictly smaller than the little finitistic dimension. On the other hand, we know of no finite dimensional monomial relation algebra Λ over an infinite field for which fin dim Λ is not attained on a module with at most three generators.

To understand the bearing of the base field on the homology of a finite dimensional algebra appears as a rather pressing necessity at this point. Indeed, it is required to assess the theoretical potential of the increasingly sophisticated algorithms which are being developed for the computation of minimal projective resolutions in the case of a finite field, on the basis of, e.g., [6].

All of our examples are assembled from a small collection of building blocks which we present in a preliminary section as a ‘construction kit’. In particular, we invite the reader to modify our constructions on the basis of this kit.

Notation and Prerequisites. Throughout, Λ = kΓ/I will denote a finite dimensional monomial relation algebra over a field k, that is, 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. 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 corresponding to a vertex ∗ of Γ will be written as either e_{∗} or e(∗).

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

Finally, we recall some results from [13] which will be repeatedly applied in the sequel. Given a left Λ-module X, call x ∈ X a top element of X if x ∉ JX and ex = x for some primitive idempotent e ∈ ℙ_{0}; in that case x will also be called a top element of type e. Moreover, denote by ℙ(X) the set of all those paths of length at least 1 in ℙ which have a starting point e with the property that e(X/JX) ≠ 0.

Theorem 1. [13, 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. □

An easy consequence of Theorem 1 permits 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; 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}}.

Corollary 2. [13, Corollary B] cyc fin dim Λ, fin dim Λ, Fin dim Λ ∈ {s_{0} + 1, s_{0} + 2}. □

2. Recyclable building blocks for examples.

In [11] the author developed an easy graphical algorithm for the computation of the key invariant s_{0} of Λ = kΓ/I, based on the quiver and the relations. It involves labeled undirected graphs to describe the principal left ideals Λq, q ∈ ℙ, and their syzygies up to isomorphism, a form of communication that appears to be a piece of time-honored mathematical folklore. Systematic treatments were given by Alperin [1], Fuller [7], and the author [11]; Fuller, in particular, already constructed monomial relation algebras from diagrams. For the reader’s convenience, we include a small example illustrating how one computes syzygies, and consequently projective dimensions, using our algorithm.

Given, e.g., a quiver Γ

we can define relations for a finite dimensional monomial relation algebra Λ = kΓ/I by ‘drawing’ the indecomposable projective left Λ-modules; we can thereby avoid unwieldy lists of generators for I. This graphical information turns out to be very manageable, even in complicated examples. For instance, that the indecomposable projective modules Λe_{i} have graphs

signifies that I is generated by the paths α_{1}βα_{1}, α_{2}βα_{1}, γα_{1}, βα_{2}, δ^{2}γα_{2}, βα_{1}β, βα_{2}β, γα_{1}β, γα_{2}β, δ^{2}γ, δ^{3}. If, moreover, S_{i} is the simple module Λe_{i}/Je_{i}, the graph of the left ideal Λβ, for instance, is the part of the graph of Λe_{2} that lies below the edge labeled β, namely \begin{matrix}1 \\ ^{α_{1}} / \^{α_{2}} \\ 2 2 \end{matrix} , indicating that Λβ/Jβ ≅ S_{1} and that, given a generator x ∈ Λβ,

the elements α_{1}x and α_{2}x generate two copies of S_{2} in Jβ/J^{2}β = Jβ. Moreover, the graph of Ω^{1}(Λβ) is the ‘leftover’ part of the graph of Λe_{1} as the latter covers the graph

of Λβ, namely {1 \atop •} |_{3}^{3} δ, the left hand vertex {1 \atop •} standing for a direct summand of Ω^{1}(Λβ)

isomorphic to S_{1}. See [11, Section 5] for further detail.

While first syzygies over monomial relation algebras, i.e., arbitrary submodules of the radicals of projectives, have an extremely complicated structure in general, second syzygies are quite accessible in view of Theorem 1. In all of our examples we will therefore aim at the construction of submodules of projective modules with suitable homological properties. To provide a ‘grid’ for such constructions, we fix a decomposition Q = ⨁_{i∈I} A_{i} of a projective left Λ-module Q, where the A_{i} are direct summands of the free left Λ-module Λ (the A_{i} are not necessarily indecomposable; e.g., A_{i} = Λ is permitted). For p ∈ ℙ and x ∈ Q, say x = (∑_{q∈ℙ} k_{iq}q)_{i∈I} with ∑_{q∈ℙ} k_{iq}q ∈ A_{i} and k_{iq} ∈ k, we define 𝔰_{p}(x) = (k_{ip})_{i∈I} ∈ k^{(I)} to be the p-support of x. Note that this definition depends only on the chosen decomposition of Q, a dependence which we will not emphasize unless there is danger of confusion. Clearly, we have 𝔰_{p}(x) ≠ 0 if and only if the path p occurs nontrivially in some component of x. Finally, given a submodule M of Q, together with top elements x, y of M and a path p ∈ ℙ, we will call y a p-domino for x if px ∈ J^{length(p)+1}y; in view of [11, Lemma 1], this is the same as to say px ∈ pJy.

In the following Lemmas 3,4,5, we assume that Q = ⨁_{i∈I} A_{i} is a projective left Λmodule as above and that M ⊆ JQ has finite projective dimension.

We start by exhibiting a building block that will allow us to control the invariant s_{0}. The purpose of the arrows β_{1} and β_{2} − not instantly obvious − is to keep the number of paths q with p dim Λq = s_{0} as low as possible in our later constructions.

Lemma 3. Let n ≥ 1, and suppose that Γ contains a subquiver of the form

such that α_{1}α_{0} ≠ 0 and β_{1}α_{0} ≠ 0 in Λ, while α_{2}α_{1}α_{0} = β_{2}β_{1}α_{0} = 0; moreover, suppose that the graphs of the indecomposable projective left Λ-modules Λe(a_{i}), 0 ≤ i ≤ n + 1, and Λe(b) are as follows

(That the graph of Λe(a_{n+1}) consists only of the vertex a_{n+1} just means that Je(a_{n+1}) = 0.) Then p dim Λα_{1}α_{0} = n, p dim Λα_{2}α_{1} = n − 1, p dim Λα_{i} = n + 1 − i for 2 ≤ i ≤ n + 1, and p dim Λα_{1} = p dim Λβ_{1} = 0. On the other hand, p dim Λα_{0} = p dim Λβ_{2} = p dim Λβ_{1}α_{0} = p dim Λβ_{2}β_{1} = ∞.

Proof. It is essential to note that the shapes of the graphs of Λe(b) and Λe(a_{i}) preclude the existence of additional arrows (other than those shown) leaving the vertices b, a_{0}, ..., a_{n+1} in Γ; for instance,

Λα_{0} = \begin{matrix}a_{0} \\ ^{α_{1}} / \^{β_{1}} \\ a_{1} b \end{matrix} .

Using the algorithm sketched above, we compute:

p dim Λα_{1}α_{0} = p dim({a_{1} \atop •} ) = 1 + p dim({a_{2} \atop •} ) = ⋯ = n + p dim({a_{n+1} \atop •} ) = n.

The remaining verifications are left to the reader. □

Next we construct a building block for Λ which forces certain paths to occur nontrivially in top elements x of M, whenever M is as specified above. Our primary interest will be in arranging for alternatives of the form ‘either 𝔰_{p}(x) ≠ 0 or else 𝔰_{q}(x) ≠ 0’ for prescribed paths p = ep and q = eq and an arbitrary top element x of type e; see part (b) of the lemma below.

In the following, vertices denoted by distinct symbols need not be distinct, e.g., a quiver drawn as 1 ← ∗ may coincide with the quiver .

Lemma 4. Suppose that Γ contains a subquiver of the form

such that σp = 0 in Λ for all paths p ∈ ℙ \ {e_{1}, τ, τγ_{1}, …, τγ_{s}} ending in ‘1’. Then a) p dim Λσ = ∞.

b) If x_{1}, …, x_{m} are top elements of type e_{1} of M which are k-linearly independent modulo JM such that 𝔰_{q}(x_{i}) = 0 for 1 ≤ i ≤ m and all q ∈ {τ, τγ_{1}, …, τγ_{s−1}}, then the vectors 𝔰_{τγ_{s}}(x_{1}), …, 𝔰_{τγ_{s}}(x_{m}) in k^{(I)} are k-linearly independent.

In particular, if s = 2 and x is a top element of type e_{1} of M such that 𝔰_{τ} (x) = 0, then either 𝔰_{τγ_{1}}(x) ≠ 0 or 𝔰_{τγ_{2}}(x) ≠ 0.

c) Given any top element x ∈ M of type e_{1}, the module M does not contain a σ-domino for x.

Proof. Part (a). From ‘σ^{2} = 0’ we derive that Λσ is isomorphic to a direct summand of Ω^{1}(Λσ).

As for part (b), we will start by showing that, given any top element x of type e_{1} of M with 𝔰_{τ} (x) = 𝔰_{τγ_{1}}(x) = ⋯ = 𝔰_{τγ_{s−1}}(x) = 0, we have 𝔰_{τγ_{s}}(x) ≠ 0. Note first that 𝔰_{e_{1}}(x) = 0, because x ∈ M ⊆ JQ. If we had 𝔰_{τγ_{s}}(x) = 0, we would thus infer that σx = σe_{1}x = 0 by hypothesis, while e_{1}x = x ∉ JM. By Theorem 1(2), this would force Λσ to be isomorphic to a direct summand of Ω^{1}(M), which is impossible in view of our blanket hypothesis ‘p dim M < ∞’. Hence indeed, 𝔰_{τγ_{s}}(x) ≠ 0.

Now suppose that x_{1}, …, x_{m} are as under (b) above and let k_{1}, …, k_{m} be scalars such that 0 = ∑_{i=1}^{m} k_{i}𝔰_{τγ_{s}}(x_{i}) = 𝔰_{τγ_{s}}(∑_{i=1}^{m} k_{i}x_{i}). From the previous paragraph we know that ∑_{i=1}^{m} k_{i}x_{i} is not a top element of type e_{1} of M, i.e., ∑_{i=1}^{m} k_{i}x_{i} ∈ JM. By hypothesis we thus conclude that k_{1} = k_{2} = ⋯ = k_{m} = 0.

Part (c) is clear, since the existence of a σ-domino for x would imply that σx ∈ J^{2}M, which by Theorem 1(2) would again show Λσ to be isomorphic to a direct summand of Ω^{1}(M). But this we have already seen to be impossible. □

In our last lemma we present a device to force the following property on every Λmodule M as specified at the outset: Whenever x is a top element of type e_{1} of M such that 𝔰_{q}(x) ≠ 0 for some prescribed path q in Λ, then M contains a domino for x.

Lemma 5. Suppose that Γ has a subquiver of the form

such that χτγ_{i} ≠ 0 in Λ for 1 ≤ i ≤ t, while χp = 0 in Λ for all paths p of length ≥ 1 not

in {τ, τγ_{1}, …, τγ_{t}}; moreover we assume that Λe(c) has graph |_{c}^{c} χ′ and χ′χ ≠ 0, while

χ′χJ = 0. Then:

a) p dim Λχ = 0, while p dim Λχ′χ = p dim Λχ′ = ∞.

b) If x ∈ M is a top element of type e_{1}, then 𝔰_{τ} (x) = 0. If, moreover, 𝔰_{τγ_{i}}(x) ≠ 0 for some i, then there exists a χ-domino y ∈ M for x such that 𝔰_{τγ_{j}}(x) = 𝔰_{γ_{j}}(y) for 1 ≤ j ≤ t.

Proof. (a) By the last set of hypotheses, Λχ has graph |_{c}^{c} χ′ and thus p dim Λχ = 0.

Clearly, p dim Λχ′χ = p dim Λe(c)/Je(c) = ∞, and likewise p dim Λχ′ = ∞.

(b) Letting x ∈ M be a top element of type e_{1}, observe that χ′χx = 0 because x ∈ JQ and χ′χJ = 0. Since the left ideal Λχ′χ, having infinite projective dimension, is not isomorphic to a direct summand of Ω^{1}(M), Theorem 1(2) tells us that χx ∈ J^{2}M ⊆ J^{3}Q; in particular, 𝔰_{τ} (x) = 𝔰_{χτ} (χx) = 0. Now suppose that 𝔰_{τγ_{i}}(x) ≠ 0. This clearly implies 𝔰_{χτγ_{i}}(χx) ≠ 0 and thus χx ∈ J^{2}M \ J^{3}M. Consequently, in view of [11, Lemma 1], the fact that χx = ∑_{i=1}^{t} χτγ_{i}𝔰_{τγ_{i}}(x) ∈ χτQ tells us that χx ∈ χτM \ J^{3}M, and therefore provides us with a top element y ∈ M of type e(∗) such that χx = χτy ∈ J^{2}y. Comparing χx with χτy = ∑_{i=1}^{t} χτγ_{i}𝔰_{γ_{i}}(y) and using again [11, Lemma 1], we finally conclude that 𝔰_{τγ_{i}}(x) = 𝔰_{γ_{i}}(y) for 1 ≤ i ≤ t. □

Roughly speaking, our lemmas will combine as follows. If p dim M = s_{0} + 1, then a subquiver of Γ as presented in Lemma 3 will provide us with a first top element, say x, of a previously specified type. A subquiver as in Lemma 4 will entail that 𝔰_{q}(x) ≠ 0 for some q within a specified set of paths. As a consequence, subquivers as in Lemma 5 will guarantee the existence of a domino for x. We can in turn force such a domino to involve a path from our specified set, so that this domino in turn calls for another domino, and so forth. Thus suitable amalgamation of subquivers of the ilk discussed will give us tight control over the minimal number of generators for submodules M of projectives having high finite projective dimension.

3. Where is fin dim Λ attained?

In light of the results in [11, Theorem VI], the author had suspected that, at least for every finite dimensional monomial relation algebra Λ, the equality ‘cyc fin dim Λ = fin dim Λ’ should be true. It is not. From the building blocks of Section 2 we will rig up classes of counterexamples. In contrast, both the global dimension of Λ and the critical number s_{0} = s_{0}(Λ) are independent of the base field [11, p. 178, second paragraph of Section 6]. Since the invariant s which we attached to Λ in [11] is not always equal to the invariant s_{0} considered here, we sketch the short argument for the convenience of the reader. Note that this argument actually provides an easy combinatorial algorithm for the computation of gl dim Λ. A similar algorithm has been developed independently by Cibils in [5].

Observation A. Let Λ = kΓ/I be a finite dimensional monomial relation algebra.

(1) The key invariant s_{0} depends only on the quiver and the relations, i.e., only on Γ and the set I ∩ {paths in Γ}. In particular,

| cyc fin dim K ⊗_{k} Λ − cyc fin dim Λ| ≤ 1 and | fin dim K ⊗_{k} Λ − fin dim Λ| ≤ 1

for any extension field K of k.

(2) The global dimension of Λ depends only on the quiver and the relations.

Proof. For (1) we refer to the algorithm discussed at the beginning of the previous section. Using this algorithm, the projective dimensions of the left ideals Λq − q any path in Λ − can be computed solely from the graphs of the indecomposable projective left Λ-modules. Our first claim thus follows immediately from the definition of s_{0}, and, in view of Corollary 2, this in turn proves the supplementary statement under (1).

(2) Keeping the preceding paragraph in mind and noting that

gl dim Λ < ∞ ⟺ p dim_{Λ} J < ∞ ⟺ p dim_{Λ} Λq < ∞ for all paths q in Γ,

we see that the field k does not enter into the decision whether the global dimension of Λ is finite or infinite. Now suppose that gl dim Λ < ∞. Define t to be −1 if Γ has no arrows, and set t = sup{p dim Λα | α is an arrow} otherwise. Then gl dim Λ = t + 1; indeed, if t = −1, this is clear, and if t ≥ 0, then gl dim Λ = 1 + p dim J, while J is the direct sum of all left ideals Λα, where α runs through the set of arrows of Γ. Since by the proof of part (1) the choice of k does not impinge on the invariant t, this equality completes the argument. □

Due to this observation, the extent to which a change of field can potentially affect the finitistic dimensions of a monomial relation algebra is to add a term of absolute value ≤ 1.

Theorem B. Let n ≥ 1. For each integer d ≥ 3, there exist a quiver Γ_{d} and a set ρ_{d} of paths in Γ_{d} such that, for any field k, the factor algebra Λ_{d} = kΓ_{d}/⟨ρ_{d}⟩ is a finite dimensional monomial relation algebra with the following finitistic dimensions:

(1) If |k| < d, then (cyc fin dim Λ_{d}, fin dim Λ_{d}) = (n + 1, n + 2).

(2) If |k| ≥ d, then cyc fin dim Λ_{d} = fin dim Λ_{d} = n + 2.

Proof. Fix d, and consider the following quiver Γ_{d} = Γ for Λ_{d} = Λ = kΓ/⟨ρ⟩:

where 1 ≤ i < j ≤ d and 1 ≤ l ≤ d. Moreover, let ρ_{d} = ρ be a set of paths in Γ such that the hypotheses of Lemma 3 are satisfied − a requirement that determines the graphs of the indecomposable projective left ideals Λe(a_{i}) and Λe(b) − and such that the graphs of the left ideals Λe_{1}, …, Λe_{4}, Λe(c_{l}) are as follows:

where 1 ≤ i < j ≤ d and 1 ≤ l ≤ d, and

where 1 ≤ i < l ≤ d and 1 ≤ l < j ≤ d.

In particular, the subquivers then satisfy the hypotheses of Lemma 4, and Lemma 5, respectively.

We start by computing

S_{0} = {α_{1}α_{0}, α_{2}α_{1}, α_{i}, β_{1}, χ_{l} | 1 ≤ i ≤ n + 1, 1 ≤ l ≤ d}

and s_{0} = n, again with the aid of the algorithm sketched at the beginning of Section 2. This computation shows in particular that α_{1}α_{0} is the only path q in S_{0} with the property that p dim Λq = n. To give a little more detail: Lemmas 3,4,5 give the projective dimensions of the left ideals Λq, where q is any path of length ≥ 1 starting in one of the vertices a_{0}, …, a_{n+1}, b, 1, c_{l}; thus the only paths that remain to be checked are q = τ, ψ, σ_{ij}τ, χ_{l}τ.

From Corollary 2 we therefore obtain that

n + 1 ≤ cyc fin dim Λ ≤ fin dim Λ ≤ n + 2.

Claim I. fin dim Λ = n + 2 for any field k.

Proof. It clearly suffices to exhibit a submodule M ⊆ JQ, where Q = ⨁_{i=1}^{d} Λ1_{i} and Λ1_{i} ≅ Λ, such that p dim M = n + 1. Set x_{0} = (γ + ∑_{i=1}^{d−1} τγ_{i})1_{1},

x_{i} = γ_{i}1_{1} + (δ + ∑_{{j=1 \atop j≠i}}^{d} γ_{j})1_{i+1} for 1 ≤ i ≤ d − 1,

and define M = ∑_{i=0}^{d−1} Λx_{i}. Then the generating set G = {x_{i} | 0 ≤ i ≤ d − 1} of M is k-linearly independent modulo JM, x_{0} is a top element of type e_{1}, and the x_{i}, i ≥ 1, are top elements of type e_{3}. We will repeatedly apply Theorem 1(2) to check that Ω^{1}(M) is a direct sum of copies of Λα_{1}α_{0} and Λχ_{l} for 1 ≤ l ≤ d. Since M has only top elements of types e_{1}, e_{3}, we need only consider left ideals of the form Λq, where q is a path of length ≥ 1 starting in ‘1’ or ‘3’, namely

q ∈ {α_{0}, α_{1}α_{0}, β_{1}α_{0}, χ_{l}, χ′_{l}χ_{l}, σ_{ij}, τ, χ_{l}τ, σ_{ij}τ, ψ | 1 ≤ l ≤ d}.

Observe moreover that the only isomorphisms occurring among the principal left ideals generated by these paths are Λχ′_{l}χ_{l} ≅ Λχ_{l}τ for 1 ≤ l ≤ d and Λσ_{ij} ≅ Λσ_{ij}τ for 1 ≤ i < j ≤ d.

By construction α_{1}α_{0}x_{0} = 0, while α_{0}x_{0} ∉ J^{2}M; indeed, 𝔰_{α_{0}γ}(α_{0}x) ≠ 0 and hence α_{0}x ∉ J^{3}Q. Using Theorem 1(2), we deduce that Λα_{1}α_{0} is isomorphic to a direct summand of Ω^{1}(M), while Λα_{0} is not. Similarly, β_{1}α_{0}x_{0} ∉ J^{3}M ⊆ J^{4}Q, because 𝔰_{β_{1}α_{0}γ}(β_{1}α_{0}x_{0}) ≠ 0, and hence no copy of Λβ_{1}α_{0} occurs as a direct summand of Ω^{1}(M).

We further observe χ_{l}M ⊆ J^{2}M for all l. Indeed, χ_{l}Jx_{0} = 0 and χ_{l}x_{0} = 0 if l = d, while χ_{l}x_{0} = χ_{l}τx_{l} ∈ J^{2}M for l < d; this proves χ_{l}M = χ_{l}e_{1}M ⊆ χ_{l}τM as required. Therefore the left ideals Λχ_{l} all make an appearance in Ω^{1}(M). On the other hand, occurrence of any summands Λχ′_{l}χ_{l} ≅ Λχ_{l}τ in Ω^{1}(M) can be ruled out in view of the preceding remark on χ_{l}M and the k-linear independence of the elements χ_{l}τx_{1}, …, χ_{l}τx_{d−1} modulo J^{3}M which we check by looking at their χ_{l}τγ_{l}-supports. Namely, 𝔰_{χ_{l}τγ_{l}}(χ_{l}τx_{i}) = 1_{i+1} whenever i ≥ 1 and i ≠ l, while 𝔰_{χ_{l}τγ_{l}}(χ_{l}τx_{l}) = 1_{1} if l < d, and thus 𝔰_{χ_{l}τγ_{l}}(χ_{l}τx_{1}), …, 𝔰_{χ_{l}τγ_{l}}(χ_{l}τx_{d−1}) are linearly independent vectors in the space k^{d}. Hence, the elements χ_{l}τx_{1}, …, χ_{l}τx_{d−1} are k-linearly independent modulo J^{4}Q and, a fortiori, modulo J^{3}M.

Next we verify that for each pair (i , j) with i < j the element σ_{ij}x_{0} does not belong to J^{2}M and that the elements σ_{ij}τx_{1}, …, σ_{ij}τx_{d−1} are k-linearly independent modulo J^{3}M. By Theorem 1(2) this will rule out appearance of summands of the form Λσ_{ij} ≅ Λσ_{ij}τ in Ω^{1}M. If we had σ_{ij}x_{0} ∈ J^{2}M, the equality

σ_{ij}x_{0} = σ_{ij}τγ_{i}𝔰_{σ_{ij}τγ_{i}}(σ_{ij}x_{0}) + σ_{ij}τγ_{j}𝔰_{σ_{ij}τγ_{j}}(σ_{ij}x_{0}) = σ_{ij}τ(γ_{i} + ∗γ_{j})1_{1}

with ∗ ∈ {0, 1} – in fact ∗ = 1 if j < d and ∗ = 0 otherwise – would yield σ_{ij}x_{0} ∈ σ_{ij}τM ⊆ ⨁_{u=2}^{d} Λ1_{u}; see [11, Lemma 1]. On the other hand, each nonzero element of σ_{ij}τM = ∑_{u=1}^{d−1} kσ_{ij}τx_{u} has a nonzero component in ⨁_{u=2}^{d} Λ1_{u}, whereas the nonzero element σ_{ij}x_{0} of M does not. Thus σ_{ij}x_{0} does not belong to J^{2}M. For the independence statement it suffices to observe that the vectors 𝔰_{σ_{ij}τγ_{i}}(σ_{ij}τx_{1}), …, 𝔰_{σ_{ij}τγ_{i}}(σ_{ij}τx_{d−1}) are k-linearly independent, which makes σ_{ij}τx_{1}, …, σ_{ij}τx_{d−1} linearly independent modulo J^{4}Q and, a fortiori modulo J^{3}M.

That the elements qx_{1}, …, qx_{d−1} are k-linearly independent modulo J^{2}M for q ∈ {τ, ψ}, finally, we see by checking that their τγ_{1}-supports are linearly independent if q = τ, and their ψδ-supports are linearly independent if q = ψ. This excludes the occurrence of summands Λτ and Λψ in Ω^{1}(M). We conclude that p dim Ω^{1}(M) = n, which gives us p dim M = n + 1 as required.

Claim II. If |k| ≥ d, then cyc fin dim Λ = n + 2. Indeed, choose nonzero elements k_{1}, …, k_{d−1} in k. Then the left ideal M = Λ(γ + ∑_{i=1}^{d−1} τγ_{i}) + Λ(δ + γ_{d} + ∑_{i=1}^{d−1} k_{i}γ_{i}) ⊆ J has projective dimension n + 1, since its syzygy Ω^{1}(M) is again a direct sum of copies of Λα_{1}α_{0} and the left ideals Λχ_{l}. We infer that p dim Λ/M = n + 2, and our claim follows. Thus the algebra Λ = Λ_{d} satisfies condition (2) above.

Claim III. If |k| < d, then cyc fin dim Λ = n + 1.

Proof. In view of the inequality we derived from Corollary 2, it clearly suffices to show that, whenever a submodule M of the radical of a finitely generated free left Λ-module Q = ⨁_{i=1}^{t} Λ1_{i} has projective dimension n + 1, then t ≥ 2. Let M be as described.

In a first step we will show that M must contain two top elements of type e_{3} which are k-linearly independent modulo JM. Since α_{1}α_{0} is the only path q ∈ S_{0} with p dim Λq = s_{0} = n, Theorem 1(2) guarantees the existence of a top element x_{0} of type e_{1} in M such that α_{1}α_{0}x_{0} = 0. Moreover, Lemma 5 tells us that 𝔰_{τ} (x_{0}) = 0, and from Lemma 4 we therefore deduce 𝔰_{τγ_{l}}(x_{0}) ≠ 0 for all but possibly one index l in {1, ..., d}. By renumbering if necessary, we can make sure that 𝔰_{τγ_{l}}(x_{0}) ≠ 0 for 1 ≤ l ≤ d − 1. For each index l in this range, Lemma 5 now provides us with a χ_{l}-domino y_{l} for x_{0} which has the property that 𝔰_{τγ_{l}}(x_{0}) = 𝔰_{γ_{l}}(y_{l}). If the k-span of the elements y_{1}, ..., y_{d−1} had dimension 1 modulo JM, we could, in view of the inclusion JM ⊆ J^{2}Q, find scalars b_{l} ∈ k \ {0} such that 𝔰_{γ_{l}}(y_{l}) = b_{l}𝔰_{γ_{l}}(y_{1}) for 1 ≤ l ≤ d − 1. These scalars b_{l} would have to be distinct. Indeed, from b_{u} = b_{v} with u < v, we would infer 𝔰_{τγ_{u}}(x_{0}) = 𝔰_{γ_{u}}(b_{u}y_{1}) and 𝔰_{τγ_{v}}(x_{0}) = 𝔰_{γ_{v}}(b_{u}y_{1}); using these equalities and keeping in mind that 𝔰_{τ} (x_{0}) = 0, we could further compute

\begin{aligned}σ_{uv}x_{0} & = σ_{uv}τγ_{u}𝔰_{τγ_{u}}(x_{0}) + σ_{uv}τγ_{v}𝔰_{τγ_{v}}(x_{0}) \\ & = σ_{uv}τγ_{u}𝔰_{γ_{u}}(b_{u}y_{1}) + σ_{uv}τγ_{v}𝔰_{γ_{v}}(b_{u}y_{1}) = σ_{uv}τb_{u}y_{1}, \end{aligned}

which would make b_{u}y_{1} a σ_{uv}-domino for x_{0}. On the other hand, the existence of such a domino was excluded in Lemma 4, a contradiction showing that b_{1}, ..., b_{d−1} are indeed distinct. But this is incompatible with our hypothesis that |k \ {0}| < d − 1, and thus the k-dimension of _{k}⟨y_{1}, ..., y_{d−1}⟩ modulo JM is at least 2. In particular, we can find top elements y_{1}, y_{2} of type e_{3} with the desired independence property.

Since p dim Λψ = ∞, the left ideal Λψ is not isomorphic to a direct summand of Ω^{1}(M), which forces the vectors 𝔰_{δ}(y_{1}), 𝔰_{δ}(y_{2}) ∈ k^{t} to be k-linearly independent. Indeed, if we had scalars k_{1}, k_{2} ∈ k, not both zero, with 0 = k_{1}𝔰_{δ}(y_{1}) + k_{2}𝔰_{δ}(y_{2}) = 𝔰_{δ}(k_{1}y_{1} + k_{2}y_{2}), then k_{1}y_{1} + k_{2}y_{2} would again be a top element of type e_{3} by the choice of the y_{i}, and the equality ψ(k_{1}y_{1} + k_{2}y_{2}) = ψδ𝔰_{δ}(k_{1}y_{1} + k_{2}y_{2}) = 0 would entail occurrence of Λψ as a direct summand of Ω^{1}(M) – this being again a consequence of Theorem 1(2). Hence 𝔰_{δ}(y_{1}) and 𝔰_{δ}(y_{2}) are linearly independent vectors in k^{t}, proving that t ≥ 2. This completes the proof of Claim III, and thus Λ = Λ_{d} also satisfies condition (1). □

The next class of examples demonstrates that, in general, we have to consider finitely generated modules requiring arbitrarily high numbers of generators in order to get hold of the little finitistic dimension. A further abbreviation will be convenient: For m ≥ 2 we define fin dim_{m−1} Λ to be

sup{p dim M | M is an (m − 1)-generated left Λ-module with p dim M < ∞}

Theorem C. Let n ≥ 1. For each finite field k and each m ≥ 2 there exists a finite dimensional monomial relation algebra Λ over k such that:

(1) fin dim Λ = n + 2 > fin dim_{m−1} Λ = n + 1; in particular, fin dim Λ is not attained on any left Λ-module with fewer than m generators.

(2) For every extension field K of k of cardinality greater than |k|^{m−1},

fin dim K ⊗_{k} Λ = cyc fin dim K ⊗_{k} Λ = n + 2.

Proof. Fix k and m, and set d = |k|^{m−1} + 1. Moreover, specify some integer n ≥ 1 and define Λ = kΓ_{d}/⟨ρ_{d}⟩, where Γ_{d} and ρ_{d} are the quiver and relations constructed in the proof of Theorem B. Given any field K of cardinality at least d, we thus know that fin dim KΓ_{d}/⟨ρ_{d}⟩ = cyc fin dim KΓ_{d}/⟨ρ_{d}⟩ = n + 2, which shows that our algebra Λ satisfies The proof of Theorem B further tells us that

n + 1 ≤ cyc fin dim Λ ≤ fin dim_{m−1} Λ ≤ fin dim Λ ≤ n + 2

and that α_{1}α_{0} is the only path q in S_{0} with the property that p dim Λq = s_{0} = n. We will show that, whenever M is a submodule of some JΛ^{t}, where Λ^{t} is the free left Λ-module on t generators, then ‘p dim M = n + 1’ forces the exponent t to be at least m. Part (1) will then follow.

Let M be a module as described. Then M contains a top element x_{0} of type e_{1} such that α_{1}α_{0}x_{0} = 0, while α_{0}x_{0} ∉ J^{2}M. On one hand, we see that 𝔰_{τ} (x_{0}) = 0 by Lemma 5. On the other hand, as in the proof of Theorem B, Claim III, we may assume that 𝔰_{τγ_{i}}(x_{0}) ≠ 0 for 1 ≤ i ≤ d − 1. For each index i between 1 and d − 1, Lemma 5 then provides us with a χ_{i}-domino y_{i} ∈ M for x_{0} with the property that 𝔰_{τγ_{i}}(x_{0}) = 𝔰_{γ_{i}}(y_{i}). Note that, in particular, y_{i} ∉ JM for 1 ≤ i ≤ d − 1.

Claim I. If z ∈ M is a top element of type e_{3}, then the residue class z + e_{3}JM contains y_{i} for at most one index i.

Indeed, suppose to the contrary that y_{u}, y_{v} ∈ z + e_{3}JM for distinct indices u and v, say u < v. We infer that 𝔰_{γ_{u}}(z) = 𝔰_{γ_{u}}(y_{u}) = 𝔰_{τγ_{u}}(x_{0}), as well as 𝔰_{γ_{v}}(z) = 𝔰_{γ_{v}}(y_{v}) = 𝔰_{τγ_{v}}(x_{0}), and conclude that

σ_{uv}x_{0} = σ_{uv}τz ∈ J^{2}z

as in the proof of Theorem B, Claim III. But this makes z a σ_{uv}-domino for x_{0}, which is incompatible with Lemma 4 and thus proves our first claim.

Claim II. dim_{k}e_{3}M/e_{3}JM ≥ m. On one hand, we have y_{1}, …, y_{d−1} ∈ e_{3}M \ e_{3}JM. On the other hand, if dim_{k}e_{3}M/e_{3}JM = r, then e_{3}M/e_{3}JM contains precisely |k|^{r} − 1 nonzero elements; denote them by z_{i}, where 1 ≤ i ≤ |k|^{r} − 1. In other words, e_{3}M \ e_{3}JM is the disjoint union of the residue classes z_{i} + e_{3}JM for 1 ≤ i ≤ |k|^{r} − 1. By Claim I we deduce that |k|^{r} − 1 ≥ d − 1, and, keeping in mind that d − 1 = |k|^{m−1}, we obtain r ≥ m as desired.

Claim III. t ≥ m.

Choose any sequence z_{1}, …, z_{m} ∈ M of top elements of type e_{3} which are k-linearly independent modulo e_{3}JM or, equivalently, modulo JM. Since p dim Λψ = ∞, the left ideal Λψ is not isomorphic to a direct summand of Ω^{1}(M). Consequently, the elements ψz_{1}, …, ψz_{m} are k-linearly independent modulo J^{2}M by Theorem 1(2) and, a fortiori, k-linearly independent. In view of the equalities ψz_{i} = ψδ𝔰_{δ}(z_{i}) which are due to the construction of Λ, we conclude that the vectors 𝔰_{δ}(z_{1}), …, 𝔰_{δ}(z_{m}) ∈ k^{t} are linearly independent. But this clearly implies that t ≥ m. □

The final class of examples assembled from our recyclable building blocks realizes the phenomenon ‘cyc fin dim Λ < fin dim Λ’ in a field-independent fashion.

Theorem D. For each positive integer n, there exist a quiver Γ and a set ρ of paths in Γ such that for any field k, the factor Λ = kΓ/⟨ρ⟩ is a finite dimensional monomial relation algebra over k with (cyc fin dim Λ, fin dim Λ) = (n + 1, n + 2).

Proof. Let Γ be the quiver Moreover, choose a set ρ of paths in Γ in such a way that the indecomposable projective modules Λe(a_{i}), 0 < i ≤ n + 1, and Λe(b) are as in Proposition 3, the graph of Λe(c) is

|_{c}^{c} χ′ , and the graphs of Λe_{i}, i = 1, 2, are as follows:

In particular, the subquivers

satisfy the hypotheses of Lemma 4 and Lemma 5, respectively.

Once more one computes s_{0} = n, but this time the set S_{0} contains two paths which generate principal left ideals of projective dimension n, namely α_{1}α_{0} and α_{1}α_{0}τ.

To show that fin dim Λ = n + 2, we construct a submodule M ⊆ J(Λ ⊕ Λ ⊕ Λ) with p dim M = n + 1 as follows: Let M = Λx_{0} + Λx_{1}, where x_{0} = (γ_{1}, τγ_{2} + δ, τδ) and x_{1} = (0, γ_{2} + τγ_{1}, γ_{1} + δ). Then Ω^{1}(M) is a direct sum of copies of Λα_{1}α_{0} ≅ Λα_{1}α_{0}τ and Λχ. To prove this with the aid of Theorem 1, we need only consider left ideals Λq, where q is a path of length ≥ 1 starting in the vertex ‘1’, since e_{1}(M/JM) = M/JM. We single out one step of the discussion and leave the rest to the reader: To see that Λσ ≅ Λστ ≅ Λψ is not isomorphic to a direct summand of Ω^{1}(M), we first check that the elements σx_{0}, σx_{1} are k-linearly independent modulo J^{2}M; indeed, dependence would mean existence of scalars k_{0}, k_{1} ∈ k, not both zero, and scalars l_{0}, l_{1} ∈ k, such that

k_{0}σx_{0} + k_{1}σx_{1} = στ(0, k_{0}γ_{2} + k_{1}γ_{1}, 0) = στ(l_{0}x_{0} + l_{1}x_{1}) = στ(l_{0}γ_{1}, l_{1}γ_{2}, l_{1}γ_{1}),

which is clearly impossible. Moreover, it is easily seen that the elements στx_{0}, στx_{1} are k-linearly independent modulo J^{3}M (even modulo J^{4}Q), and ψx_{0}, ψx_{1} are k-linearly independent modulo J^{2}M (even modulo J^{3}Q). In view of Theorem 1, this completes the step we singled out. We obtain fin dim Λ ≥ n + 2, and hence, due to Corollary 2, fin dim Λ = n + 2.

To see that cyc fin dim Λ = n + 1, suppose that M is any submodule of the radical of a finitely generated projective left Λ-module Q, say Q = Λ^{t}, such that p dim M = n + 1. We show that t ≥ 2. By Theorem 1, the equality ‘p dim M = n + 1’ forces M to contain a top element x_{0} of type e_{1} which is either annihilated by α_{1}α_{0}, but not by α_{0}, or else annihilated by α_{1}α_{0}τ but not by α_{0}τ. Since α_{0}τ(r ann_{e_{1}J} α_{1}α_{0}τ) = 0, the last possibilty is ruled out, and annihilation of x_{0} by α_{1}α_{0} forces the vectors 𝔰_{τγ_{1}}(x_{0}), 𝔰_{τ} (x_{0}) and 𝔰_{γ_{2}}(x_{0}) to be zero. Using Lemma 4, we infer 𝔰_{τγ_{2}}(x_{0}) ≠ 0 and deduce further that M contains a χ-domino x_{1} for x_{0} with the property that 𝔰_{γ_{2}}(x_{1}) = 𝔰_{τγ_{2}}(x_{0}) ≠ 0. We have thus found top elements x_{0}, x_{1} of M which are k-linearly independent modulo JM. Moreover, the left ideal Λψ, having infinite projective dimension, is not isomorphic to a direct summand of Ω^{1}(M). Consequently, ψx_{0}, ψx_{1} are k-linearly independent modulo JM, from which we derive that the vectors 𝔰_{δ}(x_{0}) and 𝔰_{δ}(x_{1}) are k-linearly independent. This yields t ≥ 2 and completes the proof. □

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