Canadian Mathematical Society Conference Proceeding Volume 00, 0000

Homological Assets of Positively Graded Representations of Finite Dimensional Algebras

B. ZIMMERMANN HUISGEN

Abstract. It is shown that positively graded modules have a significant homological edge over modules with a doubly infinite grading. For example: for a large class of finite dimensional algebras, each positively graded module M is a direct limit of finitely generated graded modules, the projective dimensions of which approximate that of M from below. This makes the big and little finitistic dimensions equal when restricted to positively graded representations. The reason for these phenomena lies in the fact that each positively graded module M is a directed union of finitely generated modules which enjoy a strong radical saturation property relative to M. Examples are given to show (a) failure of analogous statements for ℤ-graded representations, and (b) the existence of algebras for which the little finitistic dimension exceeds the supremum of the finite projective dimensions attained on arbitrary positively graded representations.

1. Introduction and Notation

It is well known – folklore, apparently – that to any module M over a positively graded finite dimensional algebra Λ one can associate a ℤ-graded Λ-module which faithfully reflects the good or bad homological behavior of M; we include more detail at the end of the introduction. Our goal here is to show that, by contrast, the positively graded modules over algebras endowed with a positive grading are comparatively well-behaved from a homological point of view. In particular, they are often direct limits of finitely generated modules M_{i} such that p dim M = sup p dim M_{i}, where p dim denotes the projective dimension.

For a more precise outline of our results, call a submodule N of M saturated (with respect to M) if N ∩ J^{i}M = J^{i}N for all i ≥ 1, where J is the Jacobson radical of Λ. The reason for our interest in saturated submodules lies in

1991 Mathematics Subject Classification. 16E10, 16G10, 16W50. This research was partially supported by a grant from the National Science Foundation. This paper is in final form and no version of it will be submitted for publication elsewhere.

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the fact that saturation of N with respect to M entails the existence of a projective cover P ⨁ Q → M which restricts to a projective cover P → N. Our main result states that any finitely generated submodule of a positively graded module X is contained in a finitely generated saturated submodule of X which, moreover, can be chosen to be homogeneous. For different types of algebras this process of saturation without jeopardizing finite generation has consequences of varying strength. A rather obvious follow-up is the fact that a projective cover of X can be successively built up from projective covers of finitely generated saturated and homogeneous submodules of X. The theorem has its strongest impact on algebras subject to the following condition: If M is a submodule of a projective left Λ-module and N a finitely generated saturated submodule of M, then p dim N ≤ p dim M. For these algebras Λ, which include all monomial relation algebras, each positively graded left Λ-module X is a direct limit of finitely generated graded Λ-modules X_{i} such that p dim X = sup p dim X_{i}. In particular, the left little and big finitistic dimensions, restricted to positively graded representations, coincide. More precisely, gr fin dim Λ = pos gr Fin dim Λ, where

gr fin dim Λ =

sup{p dim X | X a finitely generated graded left Λ-module of finite p dim}

and

pos gr Fin dim Λ = sup{p dim X | X a positively graded left Λ-module of finite p dim}.

On one hand these graded dimensions are computationally more accessible than their ungraded counterparts. On the other hand, they may differ from the latter; in fact, the big positively graded finitistic dimension, pos gr Fin dim Λ, may be strictly smaller than the ungraded little finitistic dimension,

fin dim Λ = sup{p dim M | M a finitely generated left Λ-module of finite p dim}.

The ungraded big finitistic dimension, finally, is

Fin dim Λ = sup{p dim M | M an arbitrary left Λ-module of finite p dim}.

We supplement the theory with examples illustrating its scope and limitations.

Throughout, Λ will stand for a strongly positively graded finite dimensional algebra over a field k, that is, Λ has a k-vectorspace decomposition Λ = ⨁_{n≥0} Λ_{n} such that Λ_{n}Λ_{m} ⊆ Λ_{n+m} and the powers J^{l} of the Jacobson radical J of Λ are equal to ⨁_{n≥l} Λ_{n}. For example, all path algebras modulo homogeneous ideals of relations are strongly positively graded via lengths of paths. A Λ-module X is called positively graded if X has a grading X = ⨁_{n≥0} X_{n}; as usual, a submodule Y of X is called homogeneous if Y = ⨁_{n≥0}(Y ∩ X_{n}), or, equivalently, if Y can be generated by homogeneous elements of X.

A top element of a left Λ-module M is an element m ∈ M \ JM such that m = em for some primitive idempotent e of Λ; in this case, we will also say that m is a top element of type e of M.

A piece of folklore. There is an exact functor M ↦ M′ from the category of all left Λ-modules to the category of all ℤ-graded left Λ-modules which preserves and reflects (minimal) projective resolutions. In particular, p dim M = p dim M′.

Set M′ = ⨁_{r∈ℤ} M_{r}, where M_{r} = M for r ∈ ℤ, and define

\begin{aligned}& & r & & & & & & r + i \\ λ(⋯, & 0, & x, & 0, & ⋯) & = & (⋯, & 0, & λx, & 0, & ⋯) \end{aligned}

for λ ∈ Λ_{i} and x ∈ M; here the left-hand sequence carries x in the r-th position, while the right-hand sequence carries λx in the (r + i)-th position.

Thus ℤ-graded modules cannot be expected to have a homological edge over ungraded modules.

2. Saturated submodules of positively graded modules. Applications

Theorem 1. If X = ⨁_{n≥0} X_{n} is a positively graded left Λ-module and Y a finitely generated submodule of X, then Y is contained in a finitely generated homogeneous submodule of X which is saturated with respect to X.

Proof. We start by observing that saturation is transitive in the following sense: If W ⊆ Z ⊆ X, where W is saturated with respect to Z and Z saturated with respect to X, then W is saturated with respect to X. Choosing N′ ∈ ℕ such that Y ⊆ ∑_{n=0}^{N′} ΛX_{n} and noting that this last sum is saturated with respect to X, we may thus assume that X = ∑_{n=0}^{N′} ΛX_{n}. If J^{L+1} = 0, this implies in particular that X_{n} = 0 whenever n ≥ N′ +L+1; in other words, we may assume X to be finitely graded. Let X = ⨁_{n=0}^{N} X_{n} be such a finite grading. Since clearly Y embeds into a finitely generated homogeneous submodule of X, it is moreover harmless to assume that Y is homogeneous in X to begin with. Finally, we extend the concept of saturation as follows: Given any submodule Z of X, not necessarily containing Y, call Y saturated with respect to Z if J^{l}Z ∩ Y ⊆ J^{l}Y for all l ≥ 1.

Setting Z(j) = X_{j} + X_{j+1} + ... + X_{N} for 0 ≤ j ≤ N, we will construct a chain

Y = Y (N) ⊆ Y (N − 1) ⊆ ... ⊆ Y (0)

of finitely generated homogeneous Λ-submodules Y (j) of X such that each Y (j) is saturated with respect to Z(j). Since Z(0) = X, this will yield in particular that Y (0) is saturated with respect to X.

The construction will be carried out via a downward induction on j ≤ N. The beginning is trivial since, in view of JZ(N) = 0, the module Y (N) is saturated with respect to Z(N).

Now suppose that for some index j with 0 ≤ j < N we already have sets Y = Y (N) ⊆ ... ⊆ Y (j + 1) as described. To construct Y (j), start by observing that

J^{l}Z(j) = _{k}⟨Λ_{l}X_{j}⟩ + J^{l}Z(j + 1)

for any l ∈ {1, ..., L}; here _{k}⟨Λ_{l}X_{j}⟩ denotes the k-subspace of X spanned by Λ_{l}X_{j}. Since clearly J^{l}Z(j) is homogeneous in X and J^{l}Z(j) ∩ X_{j+l} = _{k}⟨Λ_{l}X_{j}⟩, homogeneity of Y (j + 1) entails the equality

J^{l}Z(j) ∩ Y (j + 1) = (_{k}⟨Λ_{l}X_{j}⟩ ∩ Y (j + 1)) + (J^{l}Z(j + 1) ∩ Y (j + 1)).

Invoking saturation of Y (j + 1) with respect to Z(j + 1), we deduce that

J^{l}Z(j) ∩ Y (j + 1) ⊆ (_{k}⟨Λ_{l}X_{j}⟩ ∩ Y (j + 1)) + J^{l}Y (j + 1) (containment l)

for each l. Moreover, Y (j + 1) being finitely generated over Λ, the intersection k⟨Λ_{l}X_{j}⟩∩Y (j +1) is a finite dimensional k-space. Pick finitely many elements in Λ_{l}X_{j}, say λ_{li}x_{li} with i ∈ I_{l}, where the λ_{li} are homogeneous elements in Λ_{l}, the x_{li} are elements of X_{j} and I_{l} is a finite index set, such that _{k}⟨Λ_{l}X_{j}⟩∩Y (j +1) ⊆ k⟨λ_{li}x_{li} | i ∈ I_{l}⟩.

Now set

Y (j) = Y (j + 1) + ∑_{l=1}^{L} ∑_{i∈I_{l}} Λx_{li}.

By definition Y (j) is finitely generated and homogeneous in X. It remains to be checked that, given any integer m ≥ 1, the intersection Y (j) ∩ J^{m}Z(j) is contained in J^{m}Y (j). Let y ∈ Y (j) ∩ J^{m}Z(j) be a homogeneous element. We start by decomposing y in this intersection according to the definition of Y (j), namely

y = y_{j+1} + ∑_{l=1}^{L} ∑_{i∈I_{l}} ∑_{h} μ_{lih}x_{li},

where y_{j+1} ∈ Y (j + 1) and the μ_{lih} are homogeneous elements in Λ. Since Y (j + 1) is a homogeneous submodule of X by the induction hypothesis and since the elements x_{li} ∈ X are homogeneous of degree j, we may clearly assume that y_{j+1} lies in Y (j+1)∩Z(j+m) and that all the elements μ_{lih} are homogeneous of degree ≥ m. In particular, this implies that the sum ∑_{l=1}^{L} ∑_{i∈I_{l}} ∑_{h} μ_{lih}x_{li} belongs to both J^{m}Y (j) and J^{m}Z(j). From the last fact we infer that y_{j+1} ∈ J^{m}Z(j) ∩ Y (j + 1). Hence ‘containment m’, together with the choice of the elements x_{mi}, forces y_{j+1} into J^{m}Y (j). Putting our observations together, we see that y ∈ J^{m}Y (j), which completes the proof of the theorem. □

Corollary 2. Let X be a positively graded ℵ-generated left Λ-module, where ℵ is an infinite cardinal number, viewed as the least ordinal number of the pertinent cardinality.

Then X is the union of a chain (X(α))_{α<ℵ} of homogeneous saturated submodules of X, where X(α + 1)/X(α) is finitely generated for each α < ℵ. Moreover, there exists a family of homomorphisms f_{α} : P (α) → X(α), where P (α) is finitely generated projective such that the maps

⨁_{β≤α} f_{β} : ⨁_{β≤α} P (β) → X(α)

are projective covers which in the limit α → ℵ converge to a projective cover

⨁_{α<ℵ} f_{α} : ⨁_{α<ℵ} P (α) → X.

Proof. Let (x_{α})_{α<ℵ} be a generating set for X. We will find X(α) ⊆ X with the required properties such that ∑_{β≤α} Λx_{β} ⊆ X(α), as well as maps f_{α} : P (α) → X(α) such that ⨁_{β≤α} f_{β} : ⨁_{β≤α} P (β) → X(α) is a projective cover, by a transfinite induction on α < ℵ. Since this procedure will, in particular, yield ⋃_{α<ℵ} X(α) = X, the direct sum ⨁_{α<ℵ} f_{α} will then automatically be a projective cover for X.

Theorem 1 permits us to pick a finitely generated homogeneous and saturated submodule X(0) of X with x_{0} ∈ X(0). Let f_{0} : P (0) → X(0) be its projective cover.

Now suppose that for some γ < ℵ a chain of modules (X(α))_{α<γ} and maps f_{α} : P (α) → X(α) for α < γ have already been constructed as desired. Since Y = ⋃_{α<γ} X(α) is a homogeneous submodule of X, the factor X/Y is in turn positively graded, and Theorem 1 supplies us with a finitely generated homogeneous and saturated submodule Z of X/Y = \overline{X} that contains \overline{x}_{γ} = x_{γ} + Y. Let X(γ) be the preimage of Z under the canonical epimorphism X → \overline{X}. Then clearly x_{γ} ∈ X(γ), the quotient X(γ)/ ⋃_{α<γ} X(α) is finitely generated, and X(γ) is homogeneous. To check that X(γ) is saturated, let u ∈ X(γ) ∩ J^{l}X. Then \overline{u} ∈ J^{l}Z by construction, i.e., there exist u_{1}, …, u_{m} ∈ X(γ) and λ_{1}, …, λ_{m} ∈ J^{l} such that u − ∑_{i=1}^{m} λ_{i}u_{i} ∈ Y ∩ J^{l}X. Since Y = ⋃_{α<γ} X(α), being the union of a chain of saturated submodules, is in turn saturated with respect to X, we obtain that u − ∑_{i=1}^{m} λ_{i}u_{i} ∈ J^{l}Y ⊆ J^{l}X(γ), and hence u ∈ J^{l}X(γ). Let ⨁_{α<γ} f_{α} : ⨁_{α<γ} P (α) → Y be the projective cover existing by the induction hypothesis, and let z_{1}, …, z_{n} ∈ X(γ) \ Y be a minimal number of top elements of X(γ) such that Y + ∑_{1≤i≤n} Λz_{i} = X(γ). If z_{i} is a top element of type e(i), where e(i) is a primitive idempotent, set P (γ) = ⨁_{1≤i≤n} Λe(i) and define f_{γ} : P (γ) → X(γ) via e(i) ↦ z_{i}. Then the map ⨁_{α≤γ} f_{α} : ⨁_{α≤γ} P (α) → X(γ) is clearly surjective. It is a projective cover because, due to saturation of Y with respect to X and hence with respect to X(γ), top elements of Y continue to be top elements in X(γ). This completes the induction. □

To prepare for the next corollary, suppose that Λ is a monomial relation algebra, i.e., Λ = kΓ/I, where Γ is a quiver and I ⊆ kΓ an ideal generated by paths of length at least 2. Note that Λ carries a natural strong positive grading Λ = ⨁_{n≥0} Λ_{n}, where each Λ_{n} is the k-subspace of Λ generated by the residue classes in Λ of the paths of length n in kΓ.

Lemma 3. [6, Corollary C] Suppose that Λ is a finite dimensional monomial relation algebra and M a submodule of a projective left Λ-module. Moreover, let N ⊆ M be a saturated submodule of M. Then the first syzygy Ω^{1}(N) is isomorphic to a direct summand of Ω^{1}(M). In particular, p dim N ≤ p dim M. □

Recall that, according to [5, Corollary II], s + 1 ≤ fin dim Λ ≤ Fin dim Λ ≤ s + 2, where

s = sup{p dim Λq | q is a path of length ≥ 1 in kΓ\I such that p dim Λq < ∞}

if the above set is nonempty and s = −1 otherwise.

As we will presently see, the positively graded finitistic dimensions of a finite dimensional monomial relation algebra fall into that same interval. But, while the ungraded big and little finitistic dimensions differ in general (see [6, Theorem E]), the positively graded variants of these dimensions are always the same. This is an immediate consequence of Theorem 1 and Lemma 3 as follows.

Corollary 4. Let Λ be a finite dimensional monomial relation algebra and X a positively graded left Λ-module. If X is a submodule of a projective left Λmodule (not necessarily homogeneous), then X is the directed union of a family of finitely generated homogeneous submodules X(i), i ∈ I, such that p dim X = sup_{i∈I} p dim X(i).

Moreover, the graded finitistic dimensions of Λ in turn belong to the interval [s + 1, s + 2], namely,

s + 1 ≤ gr fin dim Λ = pos gr Fin dim Λ ≤ fin dim Λ ≤ s + 2.

Proof. By Theorem 1, the family (X(i))_{i∈I} of all finitely generated, homogeneous and saturated submodules of X is upward directed and has union X. Now Lemma 3 yields p dim X(i) ≤ p dim X for all i and therefore sup_{i∈I} p dim X_{i} ≤ p dim X. Since each of the functors Tor_{n}^{Λ}(Z, −), n ≥ 0, commutes with direct limits [2, Theorems 9.2* and 9.4*], the reverse inequality is also true. This proves the first claim, as well as the statement

gr fin dim Λ = pos gr Fin dim Λ ≤ fin dim Λ.

To see that gr fin dim Λ ≥ s + 1, finally, it suffices to observe that for each path q ∈ kΓ \ I of positive length, the module Λ/Λq is (positively) graded and has projective dimension 1 + p dim Λq. □

In Section 4, we will construct a finite dimensional monomial relation algebra Λ and a ℤ-graded submodule M of a projective left Λ-module such that M has finite projective dimension, but contains no nonzero finitely generated submodule of finite projective dimension. In particular, M thus fails to contain any nontrivial finitely generated saturated submodule. Perhaps more surprisingly, the big positively graded finitistic dimension, pos gr Fin dim Λ, may be strictly smaller than the ungraded little finitistic dimension, fin dim Λ. An example of this phenomenon can also be found in Section 4.

3. Which modules are direct limits of finitely generated modules of finite projective dimension?

Whenever fin dim Λ < ∞, then clearly the Λ-modules of the title have finite projective dimension, since the functor Tor_{n}^{Λ}(Z, −) commutes with direct limits for all n ≥ 0 and each right Λ-module Z [2, Theorems 9.2* and 9.4*]. On the other hand, not all modules of finite projective dimension can be expressed as direct limits of directed systems of finitely generated modules of finite projective dimension. In particular, whenever fin dim Λ ≠ Fin dim Λ, there are modules of finite projective dimension illustrating this fact.

The purpose of this section is to exhibit a class of positively graded algebras, including the finite dimensional monomial relation algebras, for which each positively graded module has the property of the title. Our main result is preceded by three easy observations which hold for any finite dimensional algebra Δ, not necessarily positively graded.

Observation 5. Let M be a left Δ-module of projective dimension 1. Then M is a direct limit of finitely generated modules of projective dimension ≤ 1.

Proof. Let 0 ⟶ ⨁_{v∈V} Δe(v) ⟶^{g} ⨁_{u∈U} Δe(u) ⟶^{f} M ⟶ 0 be an exact sequence, where the e(u), e(v) are primitive idempotents of Δ and f is a projective cover of M. Consider the set I of pairs (I_{1}, I_{2}) where I_{1} is a finite subset of U and I_{2} the largest subset of V with the property that g(⨁_{v∈I_{2}} Δe(v)) ⊆ ⨁_{u∈I_{1}} Δe(u), and equip I with the obvious partial order. Moreover, for i = (I_{1}, I_{2}) define M_{i} = (⨁_{u∈I_{1}} Δe(u))/g(⨁_{v∈I_{2}} Δe(v)); the maps f_{ij} : M_{i} → M_{j} for i ≤ j in I are the canonical ones. Then, clearly, p dim M_{i} ≤ 1 for all i ∈ I and M ≅ \underrightarrow{lim}_{i∈I} M_{i}. □

Note that Observation 5 contains the well known fact that Fin dim Δ = fin dim Δ provided that fin dim Δ = 0 [1, Theorem 6.3]. While for all d ≥ 2 there exist finite dimensional algebras Δ with d = fin dim Δ < Fin dim Δ, it is still open whether fin dim Δ = Fin dim Δ when fin dim Δ = 1. We conjecture that, at least, gr fin dim Δ = pos gr Fin dim Δ in case Δ is positively graded with fin dim Δ = 1.

Observation 6. Each finitely generated saturated submodule of a projective left Δ-module is projective.

Proof. Assume this statement to be false, and let U ⊆ P = ⨁_{1≤i≤m} Δe(i) be a counterexample such that dim_{k} U is minimal, where the e(i) are primitive idempotents of Δ. Clearly, U is nonzero. It follows that U ⊈ JP, where J denotes the Jacobson radical of Δ for the moment; for otherwise, letting l be the maximal integer with U ⊆ J^{l}P, we would obtain U = U ∩ J^{l}P = J^{l}U ⊆ J^{l+1}P due to saturation of U in P. But this is clearly absurd.

Thus we may assume, without loss of generality, that the canonical projection π_{1} : P → Δe(1) maps U onto Δe(1). Letting π be the restriction of π_{1} to U, we obtain U = Ker(π) ⨁ U_{1}, where U_{1} ≅ Δe(1). Now Ker(π) is again saturated in P and has strictly smaller k-dimension than U, yielding a counterexample of strictly smaller k-dimension to our claim. □

Now consider the following condition (SAT) for a finite dimensional algebra Δ:

(SAT) Suppose that M is a submodule of the radical of a projective left Δ-module. If N ⊆ M is a finitely generated saturated submodule of M, then p dim N ≤ p dim M.

The class of algebras satisfying (SAT) includes all finite dimensional monomial relation algebras (Lemma 3), as well as all finite dimensional algebras Δ with Fin dim Δ ≤ 1 (Observation 6). A much weaker connection between first syzygies than that given in Lemma 3 actually suffices to guarantee that Δ satisfies (SAT). In fact,

Observation 7. Suppose that Δ satisfies the following condition: Whenever M is a submodule of the radical of a projective left Δ-module and N ⊆ M is finitely generated and saturated with respect to M, then Ω^{1}(N) is isomorphic to a saturated submodule of Ω^{1}(M). In that case, (SAT) holds for Δ.

Proof. Again let N ⊆ M be as in the hypothesis of (SAT). We may clearly assume that p dim M < ∞. An obvious induction on p dim M, starting out with Observation 6, then yields p dim N ≤ p dim M. □

We now return to the positively graded algebra Λ.

Theorem 8. If Λ satisfies (SAT), then each positively graded left Λ-module X is the direct limit of a directed family (X(i))_{i∈I} of finitely generated graded modules such that

p dim X = sup_{i∈I} p dim X(i).

In particular, pos gr Fin dim Λ = gr fin dim Λ ≤ fin dim Λ.

Remark. One can actually choose the family X(i) such that X is the direct limit in the category of graded left Λ-modules, but since we do not need this in the sequel, we leave the verification to the reader.

Proof. Without loss of generality, we may assume that p dim X ≥ 1 and that X has no nonzero projective direct summands. Consider a projective cover

0 ⟶ Ω^{1}(X) ⟶ P = ⨁_{a∈A} Λe(a) ⟶^{f} X ⟶ 0,

where the e(a) are primitive idempotents of Λ, each Λe(a) is endowed with a grading that results from the canonical grading of Λe(a) by means of a shift, and f is homogeneous of degree 0 (for the existence of f adapt the argument given in [4, Proposition 1.3] to non-finitely generated modules). The module X being positively graded, so is M = Ω^{1}(X). From Theorem 1 we therefore know that M is the directed union of the family (M(i))_{i∈I} of its finitely generated homogeneous and saturated submodules. By (SAT) we moreover have p dim M(i) ≤ p dim M for i ∈ I. Equip I with the upward directed partial order ‘i ≤ j ⟺ M(i) ⊆ M(j)’.

For i ∈ I, let A(i) be the smallest subset of A with M(i) ⊆ ⨁_{a∈A(i)} Λe(a), and set

X(i) = ( ⨁_{a∈A(i)} Λe(a))/M(i).

Note that ⋃ A(i) = A, since X has no nonzero projective direct summands. Letting u_{ij} : X(i) → X(j) be the canonical map for i ≤ j, we obtain a direct system (X(i))_{i∈I} with \underrightarrow{lim}_{i∈I} X(i) = X. Since the sets A(i) are finite, the X(i) are clearly finitely generated. Moreover, since M is homogeneous in P and each M(i) homogeneous in M and thus also in ⨁_{a∈A(i)} Λe(a), all the X(i) are graded. Finally, p dim X(i) ≤ p dim X for i ∈ I, because p dim X(i) ≤ 1 + p dim M(i) ≤ 1 + p dim M = p dim X in view of p dim X ≥ 1. Therefore

sup_{i∈I} p dim X(i) ≤ p dim X.

The reverse inequality follows from [2, Theorems 9.2* and 9.4*], which proves the first statement of the theorem. The supplement about finitistic dimensions is an immediate consequence of this statement. □

4. Examples

For both of our examples, we need to recall a bit of notation from [6, Section 3]. Let Λ = kΓ/I be a monomial relation algebra based on a quiver Γ, and identify the vertices of Γ with the primitive idempotents e_{1}, …, e_{n} of Λ.

Given a projective left Λ-module P = ⨁_{i∈I} P (i), where each of the P (i) is of the form Λe(i) for some primitive idempotent e(i), every element x_{i} ∈ P (i) ⊆ Λ can be uniquely written as a linear combination x_{i} = ∑_{q∈ℙ} k_{iq}q, where ℙ is the k-basis of Λ consisting of the nonzero residue classes in Λ of the paths of length ≥ 0 in kΓ, and the k_{iq} are scalars in k. As long as we leave the decomposition of P fixed it thus makes sense to define, for x = (x_{i})_{i∈I} ∈ P and p ∈ ℙ, the p-support 𝔰_{p}(x) = (k_{ip})_{i∈I} ∈ k^{(I)} of x. Clearly, we have 𝔰_{p}(x) ≠ 0 if and only if the path p occurs nontrivially in some component x_{i} of x. Moreover, given a submodule M of P, together with top elements x, y ∈ M and p ∈ ℙ, we call y a p-domino for x if px ∈ J^{length(p)+1}y; in other words, y is directly responsible for px lying in ‘too deep’ a radical layer of M, when compared with the radical stratification of P. By [6, Lemma 1], y is a p-domino for x if and only if px ∈ pJy.

For an interpretation of the graphs we use, we refer to [5, Section 5] and [7, Section 2].

We start with an example showing that the conclusion of Corollary 4 may fail for ℤ-graded submodules of projective modules over a finite dimensional monomial relation algebra. In fact, we will construct a finite dimensional monomial relation algebra Λ and a ℤ-graded submodule M of a projective left Λ-module such that p dim M = 1, while M has no nontrivial finitely generated submodules of finite projective dimension. The structure of this module is rather amusing; it consists of a string of identically shaped links which are hooked into each other as in a doubly infinite zipper.

Example 9. Let k be any field and Λ the monomial relation algebra kΓ/I, where Γ is the quiver

and I is an ideal generated by paths in kΓ such that the indecomposable projective left Λ-modules have the following graphs:

These graphs determine I uniquely (compare [5] and [3]); namely, I is the ideal of kΓ generated by all the paths of kΓ that do not make an appearance in any of the above graphs. It is readily checked that s = 0 and p dim Λ/Λ(γ_{1} + γ_{2}) = 2, where s is as defined after Lemma 3, Section 2. We conclude that fin dim Λ = Fin dim Λ = 2.

Now let P be the projective left Λ-module ⨁_{i≥0} Λu_{i}, where u_{i} = e_{2} for all i, and define a submodule M of JP as follows: M = ∑_{i≥0} Λx_{i}, where

x_{i} = γ_{1}(u_{i}) + (τγ_{1} + γ_{2})(u_{i+1})

for i ≥ 0. Using [6, Theorem A(2)], we check that Ω^{1}(M) is isomorphic to a direct sum of copies of Λχ_{1} and Λχ_{2}, whence Ω^{1}(M) is nonzero projective, and p dim M = 1. Observe that M has a ℤ-grading, e.g. as follows: M = ⨁_{n∈ℤ} M_{n}, where

M_{−i} = _{k}⟨x_{i}, τx_{i+1}, χ_{1}x_{i+1}, χ_{2}τx_{i+2}⟩ for i ≥ 0,

while M_{1} = _{k}⟨τx_{0}, χ_{1}x_{0}, χ_{2}τx_{1}⟩ and M_{2} = _{k}⟨χ_{1}τx_{0}, χ_{2}τx_{0}⟩; finally M_{n} = 0 for n ≥ 3.

To see that any nonzero finitely generated submodule U of M has infinite projective dimension, observe first that JM = ⨁_{i≥0} Λτx_{i}. The cyclic left modules

Λτx_{i} = Λ(τγ_{1}u_{i} + τγ_{2}u_{i+1})

are clearly all isomorphic to Λτ and have graph \begin{matrix}1 \\ ^{χ_{1}} / \^{χ_{2}} \\ c_{1} c_{2} \end{matrix} . It is now straight-

forward to verify that, whenever U ⊆ JM, then U is a direct sum of copies of the cyclic module Λτ and the simple modules Λe(c_{i})/Je(c_{i}), i = 1, 2 (this fact does not even require finite generation of U). Since all three of these potential summands have infinite projective dimension, so does U.

Now suppose that U is not contained in JM, and assume p dim U < ∞. Since U is finitely generated, there exists a largest integer m such that U contains a top element of the form x = ∑_{i} λ_{i}x_{i} with λ_{m} ∈ Λe_{1} \Je_{1}, say λ_{m} = (k_{m} +l_{m})e_{1}, where k_{m} ∈ k \ {0} and l_{m} ∈ J. Letting π_{n} : ⨁_{i≥0} k ⟶ k be the canonical kvectorspace projection onto the n-th component, we obtain that π_{m+1}(𝔰_{τγ_{1}}(x)) = k_{m} + l_{m+1}, where l_{m+1} ∈ k is such that

λ_{m+1} = l_{m+1}τ + a linear combination of other paths of length ≥ 1.

First assume that k_{m} +l_{m+1} ≠ 0. Then χ_{1}x ≠ 0 and χ′_{1}χ_{1}x = 0 by the definition of Λ. Moreover, since Λχ′_{1} ≅ Λe(1)/Je(1) has infinite projective dimension, while p dim U < ∞, Λχ′_{1} is not isomorphic to a direct summand of Ω^{1}(U). Therefore [6, Theorem A(2)] tells us that χ_{1}x ∈ J^{2}M, which, in view of [6, Lemma 1], means that χ_{1}x ∈ χ_{1}JM = χ_{1}τM (the last equality follows from the fact that χ_{1}p = 0 for p ∈ ℙ ∩ J unless p = τp′). On the other hand, χ_{1}x ∉ J^{3}M = 0, whence there exists a χ_{1}-domino y ∈ U with the property that 𝔰_{τγ_{1}}(x) = 𝔰_{γ_{1}}(y). We derive that π_{m+1}(𝔰_{γ_{1}}(y)) = π_{m+1}(𝔰_{τγ_{1}}(x)) = k_{m} + l_{m+1} ≠ 0, which in view of the definition of the x_{i} shows that y = ∑ μ_{i}x_{i} with μ_{m+1} ∈ Λe_{1} \ Je_{1}. But y being a top element of U, this contradicts the maximal choice of m.

Thus we must have k_{m}+l_{m+1} = 0, and hence π_{m+2}(𝔰_{τγ_{2}}(x)) = l_{m+1} = −k_{m} ≠ 0. In particular, χ_{2}x ≠ 0 while χ′_{2}χ_{2}x = 0. By an argument symmetric to the one carried out for χ_{1} in the preceding paragraph, we obtain a χ_{2}-domino z ∈ U for x such that 𝔰_{τγ_{2}}(x) = 𝔰_{γ_{2}}(z). This time, we deduce π_{m+2}(𝔰_{γ_{2}}(z)) = −k_{m} ≠ 0, which shows that z = ∑ ν_{i}x_{i} with ν_{m+1} ∈ Λe_{1} \ Je_{1}. This is again impossible by the maximal choice of m.

Thus p dim U = ∞ as claimed. □

Our final example shows that, even for a finite dimensional monomial relation algebra Λ, the big positively graded finitistic dimension, pos gr Fin dim Λ, may be strictly smaller than the ungraded little finitistic dimension, fin dim Λ.

Example 10. Our new quiver Γ results from that in Example 9 via attachment of an additional loop to vertex ‘1’. Namely, Γ is Let Λ = kΓ/I be the monomial relation algebra whose indecomposable projective left ideals have the following graphs:

On the basis of the algorithm described in [5, Section 5], it is easy to check that the only paths in ℙ of positive length which generate left ideals of finite projective dimension are χ_{1} and χ_{2}, and thus s = p dim Λχ_{1} = p dim Λχ_{2} = 0. By Corollary 4, we thus know that pos gr Fin dim Λ = gr fin dim Λ and fin dim Λ belong to [1, 2].

We first check that fin dim Λ = 2. Setting U = Λu ⊆ Λ, where u = γ_{1} + γ_{2} + τγ_{1}, we can use [6, Theorem A(2)] to check that Ω^{1}(U) ≅ Λχ_{1} ⨁ Λχ_{2}; indeed, χ_{1}u = χ_{1}τu ∈ J^{2}U, χ_{2}u = 0, σu = στγ_{1} ∉ J^{2}U, because στu = στγ_{1} + στγ_{2}, and none of the following elements belongs to J^{3}U:

στu = στγ_{1} + στγ_{2},  χ_{1}τu = χ_{1}τγ_{1},  χ_{2}τu = χ_{2}τγ_{2}.

Thus Ω^{1}(U) is a nontrivial projective module and p dim Λ/U = 2.

Next we prove that pos gr Fin dim Λ = 1 by showing that there is no positively graded submodule of a projective left Λ-module which has projective dimension 1. Since χ_{1} and χ_{2} are the only paths in ℙ of length ≥ 1 which generate left ideals of finite projective dimension, [6, Theorem A(1)] shows that any submodule U of a projective module that has projective dimension 1 must have syzygy Ω^{1}(U) ≅ (Λχ_{1})^{(I_{1})} ⨁(Λχ_{2})^{(I_{2})} with either I_{1} ≠ ∅ or I_{2} ≠ ∅. In particular, U must contain a top element of type e_{1}. For our purpose it thus suffices to show that there is no positively graded submodule U of the radical of a projective left Λ-module such that U contains a top element of type e_{1} and p dim U = 1.

Assume to the contrary that U, contained in the radical of a projective module P = ⨁_{i∈I} P (i), has all of the above properties, and let u ∈ U be a top element of type e_{1}. We may clearly choose u homogeneous, of degree d say. Since the cyclic left ideal Λσ has infinite projective dimension, it is not isomorphic to a direct summand of Ω^{1}(U). Therefore σu ≠ 0, which shows that either 𝔰_{τ} (u) ≠ 0 or 𝔰_{τγ_{1}}(u) ≠ 0 or 𝔰_{τγ_{2}}(u) ≠ 0; here the support of u, as introduced prior to Example 9, is relative to a fixed decomposition P = ⨁_{i∈I} P (i) with indecomposable summands P (i). To rule out the possibility that 𝔰_{τ} (u) ≠ 0, note that this nonequality would imply χ_{1}u ∈ J^{2}P \ J^{3}P and consequently χ_{1}u ∉ J^{2}U in view of U ⊆ JP. On the other hand, the equality χ′_{1}χ_{1}JP = 0 (definition of Λ) yields χ′_{1}χ_{1}u = 0, and by [6, Theorem A(2)] we would therefore deduce that Λχ′_{1}χ_{1} is isomorphic to a direct summand of Ω^{1}(U). But p dim Λχ′_{1}χ_{1} = ∞, and this conclusion is incompatible with finiteness of the projective dimension of U. Thus 𝔰_{τγ_{1}}(u) ≠ 0 or 𝔰_{τγ_{2}}(u) ≠ 0. Suppose that 𝔰_{τγ_{1}}(u) ≠ 0 – the other case can be treated symmetrically. Again invoking [6, Theorem A(1)] and the fact that χ′_{1}χ_{1}u = 0 while p dim Λχ′_{1}χ_{1} = ∞, we see that χ_{1}u ∈ J^{2}U. Due to [6, Lemma 1], we even obtain χ_{1}u ∈ χ_{1}τU. On the other hand, χ_{1}u ∈ J^{3}P \ J^{4}P, and thus χ_{1}u ∉ J^{3}U. Thus there exists a χ_{1}-domino u_{1} of u such that χ_{1}u = χ_{1}τu_{1}. In particular, u_{1} ∈ U is a top element of type e_{1} which may clearly again be chosen homogeneous since χ_{1}u is homogeneous. Since χ_{1}τ ∈ Λ_{2}, while χ_{1} ∈ Λ_{1}, we conclude that u_{1} is homogeneous of degree d − 1.

Repeating the above argument for u_{1} in place of u yields a top element u_{2} of type e_{1} in U which is homogeneous of degree d−2, and so forth. In this manner, we obtain a sequence u, u_{1}, u_{2}, u_{3}, … of homogeneous elements of U, the degrees of which are strictly decreasing. But this contradicts our hypothesis that U is positively graded. □

References

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Department of Mathematics, University of California, Santa Barbara, CA 93106

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