SYZYGIES AND HOMOLOGICAL DIMENSIONS OVER LEFT SERIAL RINGS

BIRGE ZIMMERMANN HUISGEN* Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

1 INTRODUCTION

Our primary goal is to explore the structure of syzygies over left serial rings and, in particular, to understand and compute the various finitistic dimensions of such rings. A secondary goal is to enlarge the collection of tools presently available for tackling submodules of projective modules over artinian rings.

Let R be a unitary left serial ring with Jacobson radical J. Seriality means that R is a finite direct sum of uniserial left ideals, in other words, of left ideals each having a unique finite composition series. Observe that the unique composition series of a uniserial module M is necessarily of the form M ⊇ JM ⊇ J^{2}M ⊇ ⋯ ⊇ J^{l}M = 0 for some l. Recall moreover that the rings which are left and right serial are of finite representation type (combine [N, Theorem 17] and [EG, Theorem 1.2] with [FR]), while, even in the case of a left serial finite dimensional algebra R, the category of left R-modules may be highly complicated. Finally, note that there are left serial rings which fail to be right artinian [RZ].

Recently, some of the homological initiatives of the late fifties and early sixties (e.g., [AB], [B]) have been rekindled, this being in particular true for the study of finitistic dimensions; see, e.g., [H1] for an outline and further references. Here we will consider the following (left) big, little, and cyclic finitistic dimensions of R:

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

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16E10, 16P20.

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

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

cyc fin dim R = sup{p dim M | M is a cyclic left R-module with p dim M < ∞},

where p dim M denotes the projective dimension of M. In view of the failure of the first finitistic dimension conjecture [H1], which asserted that the big finitistic dimension of a finite dimensional algebra is always equal to the little, and in view of the fact that the little finitistic dimension may in turn exceed the cyclic dimension [H2], it has become apparent that an understanding of more than one of these dimensions is required for a sharp homological picture of a ring.

Since we believe these three dimensions to coincide for ‘well-behaved’ classes of artinian rings, we expected them to be equal for left serial rings from the outset. This turned out to be true. However, we found the equality ‘Fin dim R = fin dim R’ far less obvious than we had expected. In particular, its proof requires a thorough understanding of submodules of projectives. One of our principal results along this line addresses a somewhat more general situation. Namely, given a left serial ring R and a submodule M of a direct sum of uniserial left R-modules (note that the submodules of projectives are of this type), the syzygies of M are direct sums of copies of uniserial left ideals J^{l}e, where l ≥ 1 and e is a primitive idempotent of R; moreover − this being the crucial point − M contains a uniserial submodule having the same projective dimension as M. As a consequence we deduce the (in)equalities

Fin dim R = fin dim R = cyc fin dim R = m + 1 ≤ 2n − 2,

where m is the maximum of the projective dimensions of those uniserial left ideals J^{l}e with 1 ≤ l ≤ L − 1 and e a primitive idempotent of R, which have finite projective dimension; here L is minimal with the property that J^{L} = 0 and n is the number of inequivalent primitive idempotents of R; moreover, we set m = −1 if all the pertinent left ideals J^{l}e have infinite projective dimension. The upper bound of 2n−2 on the finitistic dimensions was also obtained by Y. Wang [W], simultaneously with our work. In case R has finite global dimension, the corresponding inequality gl dim R ≤ 2n − 2 was established much earlier by Gustafson [G] and subsequently extended to a larger class of rings of finite global dimension by Zacharia [Za2].

The equalities Fin dim R = fin dim R = cyc fin dim R = m + 1 make it a cinch to compute the finitistic dimensions of left serial rings. In fact, as is easily seen, the computation of the above maximum m can be based solely on the sequences of composition factors Re/Je, Je/J^{2}e, … of the indecomposable left ideals Re of R; in particular, this implies that the finitistic dimensions of a finite dimensional left serial path algebra modulo relations depend only on the quiver and the relations, not on the base field, a simplicity of behavior which again singles out the left serial case as being particularly tractable (for contrast, see [H2]). We will illustrate the easy combinatorial procedure with an example (Section 2).

The bulk of our work (Sections 3,4) is concerned with a comparison of non-finitely generated objects with finitely generated ones. Since, with simpler tools, we arrive at sharper results when R is left serial and right artinian, we precede the study of Fin dim R in the general case by a discussion of the twosided artinian situation (Section 3). In particular, we obtain that all submodules of projective left modules are direct sums of copies of uniserial left ideals under this restriction.

Additional notation: We will throughout assume that R is indecomposable and basic; neither of these assumptions will impinge on the generality of our results. By e_{1}, e_{2}, …, e_{n} we denote a basic set of primitive idempotents of R; the corresponding simple left modules Re_{i}/Je_{i} we abbreviate by S_{i}.

For a left R-module M, we denote by Ω^{1}(M) the first syzygy, that is, the kernel of a projective cover of M. Moreover, we will call an element x ∈ M normed if x = e_{i}x for some i; in that case, we will also say that x is normed of type e_{i}.

2 SUBMODULES OF FINITELY GENERATED PROJECTIVE MODULES AND THE LITTLE FINITISTIC DIMENSION

PROPOSITION 1. If R is left serial and M a finitely generated submodule of a direct sum of uniserial left R-modules Q_{i}, then M is in turn a direct sum of uniserial components, each of which is isomorphic to a submodule of some Q_{i}. In particular, each finitely generated submodule of a projective left R-module is isomorphic to a direct sum of uniserial left ideals.

Proof. The module M, being finitely generated, is contained in a finite direct sum of the uniserial modules Q_{i}, say M ⊆ ⨁_{i=1}^{t} Q_{i}. We prove our claim by an induction on the composition length of M. Suppose that M ≠ 0, and choose l ≥ 1 minimal with the property that J^{l}M = 0. Moreover, pick a normed element x in M, say x = e_{1}x, such that J^{l−1}x ≠ 0. Writing x in terms of its Q_{i}-components, x = ∑_{i=1}^{t} q_{i} with q_{i} ∈ Q_{i}, we may assume that J^{l−1}q_{1} ≠ 0. It is then easily verified that l.ann_{R}(x) = l.ann_{R}(q_{1}) = J^{l}e_{1} ⊕ ⨁_{j≥2} Re_{j}. In particular, Rx ≅ Rq_{1} ⊆ Q_{1}. We claim that M = Rx ⊕ M′, where M′ = M ∩ ⨁_{i≥2} Q_{i}. Indeed, the sum Rx + M′ is direct, since from rx ∈ M′ we deduce r ∈ l.ann_{R}q_{1} = l.ann_{R}x, that is, rx = 0. On the other hand, let y = ∑_{i=1}^{t} p_{i} ∈ M with p_{i} ∈ Q_{i} for each i. Clearly, the equality J^{l}M = 0 implies J^{l}p_{1} = 0. But, Q_{1} being uniserial, we then have Rq_{1} = J^{l_{1}}Q_{1} and Rp_{1} = J^{m_{1}}Q_{1} for suitable integers l_{1} ≤ m_{1}. In other words p_{1} = rq_{1} for some r ∈ R, which implies that y − rx ∈ M′. This verifies our claim. Since the composition length of M′ is strictly smaller than that of M, we need only invoke the induction hypothesis to complete the proof. □

Our first corollary is a mild extension of an observation in [BFVH]; it will smooth the road toward the theorem to follow.

COROLLARY 2. Let R be left serial, suppose that p dim S_{1} ≤ 1, and let e = 1 − e_{1}. If M is a finitely generated left R-module of finite projective dimension, then eM has finite projective dimension over eRe, and

p dim_{R} M ≤ p dim_{eRe} eM + 2.

Proof. Suppose that p dim M = s < ∞. If M is a direct sum of uniserial modules, then a proof for our conclusion can be carried over verbatim from [BFVH, Lemma 4], where the case M = S_{i} is treated. In particular, this takes care of the case s = 0, and we may assume s ≥ 1.

Note that multiplying an R-projective resolution of M from the left by e turns it into an eRe-projective resolution of eM; this is based on Zacharia’s observation [Za1, p. 355] that eP is projective over eRe, provided that P is a projective left R-module. In particular, p dim_{eRe} eM ≤ p dim_{R} M, and eΩ_{R}^{1}(M) differs from Ω_{eRe}^{1}(eM) only by an eRe-projective direct summand. By Proposition 1, the syzygy Ω_{R}^{1}(M) is a direct sum of uniserial modules, whence, by the first paragraph

s − 1 = p dim Ω_{R}^{1}(M) ≤ p dim_{eRe} eΩ^{1}(M) + 2 = p dim_{eRe} eM − 1 + 2.

Thus s ≤ p dim_{eRe} eM + 2 as required. □

The following theorem will be supplemented by the equality ‘Fin dim R = fin dim R’ in Corollary 11 of Section 4.

THEOREM 3. If R is left serial, then fin dim R = cyc fin dim R = m + 1 ≤ 2n − 2, where

m = max{p dim J^{l}e_{i} | 1 ≤ l ≤ L − 1, 1 ≤ i ≤ n, p dim J^{l}e_{i} < ∞},

or m = −1 if the above set is empty.

Proof. We first prove the two equalities. If m = −1, that is, if the above set is empty, then all nonzero finitely generated submodules of projective left modules have infinite projective dimension by Proposition 1, which shows that fin dim R = cyc fin dim R = 0 in that case.

Now suppose that m ≥ 0 and pick integers l ≥ 1 and i ∈ {1, …, n} such that p dim J^{l}e_{i} = m. Then p dim Re_{i}/J^{l}e_{i} = m+1, which proves that cyc fin dim R is bounded from below by m + 1. In order to verify the two equalities, it thus suffices to show that fin dim R ≤ m + 1. Start with any finitely generated left R-module M of finite projective dimension. In view of Proposition 1, Ω^{1}(M) is isomorphic to a direct sum of uniserial left ideals contained in J, and consequently, p dim Ω^{1}(M) ≤ m as desired.

Next we prove by induction on n that fin dim R ≤ 2n − 2. If n = 1, then fin dim R = 0. Now consider the case ‘n ≥ 2’, and assume that fin dim R ≥ 2, this assumption being clearly harmless.

We start by guaranteeing the existence of a simple left module of projective dimension 1. Let X be a finitely generated left R-module with 2 ≤ p dim X < ∞, and let X_{0} be a non-projective uniserial direct summand of Ω^{1}(X); for the existence of such a summand X_{0} we again invoke Proposition 1. Consider the ‘tail end’ of a minimal projective resolution of X_{0}; it clearly provides us with an embedding f : Re_{i} → Je_{j} for suitable indices i, j ∈ {1, …, n}. Since Re_{j} is uniserial, we have f(Re_{i}) = J^{m}e_{j} for some m ≥ 1. If m = 1, then Je_{j} ≅ Re_{i} and p dim S_{j} = 1. If m > 1, look at a projective cover Re_{k} → J^{m−1}e_{j} and note that the induced epimorphism Je_{k} → J^{m}e_{j} ≅ Re_{i} splits. But Je_{k} being indecomposable, we infer Je_{k} ≅ Re_{i} and conclude p dim S_{k} = 1.

Once more, let X be a finitely generated left R-module of finite projective dimension. By the induction hypothesis, fin dim eRe ≤ 2(n − 1) − 2, which, in view of Corollary 2, yields

p dim_{R} X ≤ p dim_{eRe} eX + 2 ≤ 2(n − 1) − 2 + 2 = 2n − 2. □

As already mentioned above, the upper bound of 2n − 2 on the finitistic dimensions of R was independently found by Wang [W]; he actually established this in a slightly more general context, namely for almost left serial rings in the sense of Burgess and Fuller [BF]. For a treatment of the (twosided) serial case see also [FS]. The bound ‘2n − 2’ is sharp. In fact, Gustafson [G] was the first to notice that it is even attained by the global dimension of serial rings.

Next we demonstrate how easy it is to compute finitistic dimensions on the basis of Theorem 3.

EXAMPLE 4. Let K be a field, Γ the quiver

and KΓ the corresponding path algebra. Define R = KΓ/I, where I is the ideal of KΓ generated by α_{6}α_{1}, α_{8}α_{7}α_{4}α_{2}, α_{7}α_{4}α_{3}, α_{8}α_{7}α_{4}, α_{6}α_{8}α_{5}, α_{6}α_{8}α_{7}α_{6}, α_{7}α_{6}α_{8}α_{7}, α_{6}α_{8}α_{7}α_{6}α_{8}; here ‘α_{i}α_{j}’ stands for ‘α_{i} after α_{j}’. Since there is at most one arrow departing from each vertex of Γ, the finite dimensional algebra R is left serial. In an obvious fashion, the indecomposable left R-modules Re_{i} can be represented by the following single-branched trees pinning down the succession of simple composition factors:

1  2  3  4  5  6  7  8
|  |  |  |  |  |  |  |
6  4  4  7  8  7  8  6
|  |  |  |  |  |  |
7  7  8  6  8  6  7
|  |  |  |
8  6  7  8

| 6

We compute, for instance, that

p dim Je_{1} = p dim S_{6} = 1 + p dim \begin{matrix}7 \\ | \\ 8 \\ | \\ 6 \end{matrix} = 2 + p dim S_{7} = 3 + p dim \begin{matrix}8 \\ | \\ 6 \\ | \\ 7 \end{matrix}

= 4 + p dim \begin{matrix}8 \\ | \\ 6 \end{matrix} = 5 + p dim \begin{matrix}7 \\ | \\ 8 \\ | \\ 6 \end{matrix} = ∞

and p dim Je_{3} = p dim \begin{matrix}4 \\ | \\ 7 \end{matrix} = 1 + p dim S_{8} = 2 + p dim Je_{8} = 2. In this manner, we

readily obtain m = 2, where m is the maximum addressed in Theorem 3. This shows that fin dim R = cyc fin dim R = 3. In view of the results to follow, we may add Fin dim R = 3. □

3 SYZYGIES OF NON-FINITELY GENERATED MODULES – THE TWOSIDED ARTINIAN CASE

In this section we adopt an additional hypothesis on R which is actually considerably weaker than the requirement that R be right artinian. Namely, all we need is that R is left Σ-algebraically compact, meaning that arbitrary direct sums of copies of the left Rmodule R are algebraically compact = pure-injective (for basic properties and examples, see e.g. [HZ]). It is well-known that the descending chain condition for right ideals of R forces R to be left Σ-algebraically compact [Zi, Satz 2.5]. To better assess this compactness condition, note that the left serial – but not right artinian – ring exhibited by Rosenberg and Zelinsky [RZ] is still Σ-algebraically compact on the left.

THEOREM 5. Suppose that R is left serial and left Σ-algebraically compact. Then every submodule of a (not necessarily finite) direct sum of uniserial left R-modules Q_{i} is in turn a direct sum of uniserial modules, each of which is isomorphic to a submodule of some Q_{i}. In particular, every submodule of a projective left R-module is a direct sum of copies of uniserial left ideals of R.

Proof. We begin by observing that any uniserial left R-module which has the same Loewy length as R is isomorphic to one of the indecomposable projective left ideals Re_{i} and is thus in turn Σ-algebraically compact. (Recall that the Loewy length of a nonzero R-module M is the smallest natural number l with J^{l}M = 0.)

Now let M be a nonzero submodule of a direct sum ⨁_{i∈I} Q_{i} of uniserial left R-modules Q_{i}. We will prove the theorem by an induction on the Loewy length l of M. Since M is an R/J^{l}-submodule of the direct sum of uniserial left modules ann_{Q_{i}}(J^{l}) over the left serial ring R/J^{l}, we may assume without loss of generality that l = L. In case l = L = 1, the ring R is semisimple artinian, and there is nothing to prove.

Hence, suppose that L ≥ 2 and pick a uniserial submodule Rm_{1} of M, where m_{1} is a normed element of M, with the property that J^{L−1}m_{1} ≠ 0, say m_{1} = (q_{i})_{i∈I} with q_{i} ∈ Q_{i} and J^{L−1}q_{i(1)} ≠ 0. As in the proof of Proposition 1, we then check that M = Rm_{1} ⊕ M_{1}, where Rm_{1} ≅ Rq_{i(1)} ⊆ Q_{i(1)} and M_{1} = (⨁_{i∈I\{i(1)}} Q_{i}) ∩ M. If the Loewy length of M_{1} is smaller than L, we are done by the induction hypothesis; otherwise we repeat the described procedure with M_{1} in place of M. We then either arrive at a situation

M = Rm_{1} ⊕ ⋯ ⊕ Rm_{k} ⊕ M_{k}

with Rm_{i} uniserial and J^{L−1}M_{k} = 0 after finitely many steps, or else we obtain a sequence (Rm_{i})_{1≤i<ω} of uniserial R-submodules Rm_{i} of M, each of Loewy length L, such that ⨁_{i≤k} Rm_{i} is a direct summand of M for each k < ω. In the first case we are done by the induction hypothesis. In the second case we observe that ⨁_{i<ω} Rm_{i} is a pure submodule of M. Moreover, we note that the family (Rm_{i})_{i<ω} contains only finitely many isomorphism types, since, by the opening paragraph, each Rm_{i} is isomorphic to Re for a suitable primitive idempotent e. Thus the sum ⨁_{i<ω} Rm_{i} is (Σ)-algebraically compact, and consequently a direct summand of M, say

M = ⨁_{i<ω} Rm_{i} ⊕ M_{ω}.

If J^{L−1}M_{ω} is still nonzero, an obvious (subsidiary) transfinite induction takes us to a situation

M = ⨁_{i<α} Rm_{i} ⊕ M_{α},

where α is an ordinal number, Rm_{i} is uniserial for each i < α and J^{L−1}M_{α} = 0. Now apply the hypothesis of our finite induction to decompose M_{α} into a direct sum of uniserial components. □

The next example shows that, even when R is a finite dimensional algebra, the hypothesis that R be left serial is essential in the previous theorem, as well as in Proposition 1.

EXAMPLE 6. Let K be a field, Γ the quiver

and R = KΓ/I, where I ⊆ KΓ is the ideal generated by α^{2}. Then R is clearly not left serial; in particular Je_{1} ≅ S_{1} ⊕ S_{2}. On the other hand, R is right serial, since each vertex in Γ is the endpoint of only one arrow. Thus each indecomposable projective right R-module is uniserial, whence, by duality, each indecomposable injective left R-module is uniserial. In particular, the injective hull E(_{R}R) is a direct sum of uniserial modules, while _{R}R is not. □

The following immediate consequence of Theorem 5 is merely a precursor of the more general Corollary 11 of Section 4. On dropping the compactness hypothesis, we will, however, have to work considerably harder to arrive at the same conclusion.

REMARK 7. If R is left serial and left Σ-algebraically compact, then Fin dim R = fin dim R. □

4 SYZYGIES OF NON-FINITELY GENERATED MODULES – THE GENERAL CASE

First we need to supplement our notation.

In the sequel, we will assume, without exception, that R is a left serial ring. Consequently, there exists a function σ : {1, 2, …, n} ⟶ {1, 2, …, n} such that Je_{i}/J^{2}e_{i} ≅ S_{σ(i)}, whenever Je_{i} ≠ 0.

For each i ∈ {1, …, n} with Je_{i} ≠ 0 pick α_{i} ∈ e_{σ(i)}Re_{i} such that Je_{i} = Rα_{i}. Observe that, given any uniserial module M with J^{l}M ≠ 0 and M/JM ≅ S_{i}, we have

J^{l}M = Rα_{σ^{l−1}(i)} ⋯ α_{σ^{2}(i)}α_{σ(i)}α_{i}m,

where m is any element in M \ JM. Indeed, let f : Re_{i} ⟶ M be a projective cover with f(e_{i}) = e_{i}m; then J^{l}M = f(J^{l}e_{i}) = Rα_{σ^{l−1}(i)} ⋯ α_{σ(i)}α_{i}m as claimed. Any nonzero product of the form a = α_{σ^{l−1}(i)} ⋯ α_{σ^{2}(i)}α_{σ(i)}α_{i} will be called a string of order l; we will write ord(a) = l. The primitive idempotents e_{1}, …, e_{n} will be the strings of order 0. Note that this definition of ‘order’ is unambiguous since ord(a) = l precisely if a ∈ J^{l} \ J^{l+1}. If e, e′ are primitive idempotents with a = e′ae we will, moreover, say that a is a string from e to e′. Finally, we set

𝔸 = ⋃_{0≤l≤L−1} 𝔸_{l},

where 𝔸_{l} is the set of all strings of order l. The following straightforward comments will be useful in the sequel:

REMARK 8. (1) For each i ∈ {1, …, n} and l ≥ 0, the set 𝔸_{l}e_{i} is either empty or a singleton {a}, depending on whether J^{l}e_{i} is zero or not. In the latter case, every normed element x ∈ J^{l}e_{i} \ J^{l+1}e_{i}, say x = e′x, can be written in the form x = rae_{i}, where r is a unit in the local ring e′Re′.

(2) Suppose that a is a string from e_{i} to e_{j}, that b is a string from e_{j} to e_{k}, and that r, s are units in e_{j}Re_{j} and e_{k}Re_{k}, respectively. Then ba is either zero, or else ba is a string of order ord(a) + ord(b) from e_{i} to e_{k}, and sbra = tba for some unit t ∈ e_{k}Re_{k} – the last statement following from (1).

(3) If a is a string from e_{i} to e_{j} and r an element in e_{j}Re_{j}, then l. ann_{Re_{j}}(ra) ⊇ l. ann_{Re_{j}}(a) and consequently l. ann_{R}(ra) ⊇ l. ann_{R}(a). This is due to the fact that Rra = J^{m}e_{i} and Ra = J^{l}e_{i} for some m ≥ l, which shows that l. ann_{Re_{j}}(ra) = J^{L(j)−m}e_{j} and l. ann_{Re_{j}}(a) = J^{L(j)−l}e_{j}, where L(j) ≥ 1 is minimal with the property that J^{L(j)}e_{j} = 0. □

At first sight, it may appear as though the study of syzygies over left serial rings should be analogous to that of syzygies over monomial relation algebras (i.e., factor algebras of path algebras modulo ideals generated by paths), with the set of strings in the left serial case replacing the set of nontrivial paths (= nonzero residue classes of paths) in the monomial case. However, there is a major caveat. Namely, sums of strings do not satisfy the uniqueness properties that we have for sums of nontrivial paths in monomial relation algebras. For example, if a_{1} = e_{1}a_{1}e, a_{2} = e_{2}a_{2}e, …, a_{s} = e_{s}a_{s}e are strings in 𝔸 starting in e with ord(a_{1}) < ord(a_{2}) < ⋯ < ord(a_{s}) and if the r_{i} are units in e_{i}Re_{i}, respectively, then

r_{1}a_{1} + ⋯ + r_{s}a_{s} = ta_{1}

for a suitable unit t ∈ e_{1}Re_{1}. (Indeed, note that ord(e_{i}a_{i}) = ord(a_{i}) for all i, and hence ord(r_{1}a_{1} + ⋯ + r_{s}a_{s}) = ord(a_{1}).) In fact, we have to make do with a comparatively weak replacement for uniqueness as spelled out in the following lemma; unfortunately, this key prerequisite has a rather technical proof.

LEMMA 9. Let R be left serial, M a submodule of a direct sum of uniserial left Rmodules, and N any left R-module. Moreover, let f : N ⟶ M be a homomorphism such that f(J^{l}N) = J^{l}M for some positive integer l. Then f(RpN) = RpM for each p ∈ 𝔸_{l}.

Proof. Step (1). We precede the main argument with a rather harmless observation. Namely, if X = Rx is uniserial, A a subset of ⋃_{i≥l} 𝔸_{i} and

∑_{a∈A} r(a)ax ∈ J^{l+1}X

for some r(a) ∈ R, then r(a)ax ∈ J^{l+1}X for each a ∈ A.

Indeed, r(a)ax ∈ J^{l+1}X, whenever a ∈ ⋃_{i>l} 𝔸_{i}, and hence

∑_{ord(a)=l} r(a)ax ∈ J^{l+1}X.

But there exists at most one string a_{0} of order l such that a_{0}x ≠ 0 – see Remark 8. If there is none, our claim is trivially satisfied; otherwise, we obtain r(a_{0})a_{0}x = ∑_{ord(a)=l} r(a)ax ∈ J^{l+1}X as required. Step (2). Without loss of generality, we may assume that N is a direct sum of uniserial left modules. If not, let g : P → N be an epimorphism from a projective module P onto N, and replace f by fg; note that fg inherits the hypothesis on f, and g(RpP) = RpN. We now prove the lemma for the special case where both M and N are finitely generated. By Proposition 1, M and N are then finite direct sums of uniserial modules, say

M = ⨁_{i=1}^{s} M_{i}  and  N = ⨁_{j=1}^{t} N_{j}

with M_{i} = Rm_{i} and N_{j} = Rn_{j} uniserial. Let p ∈ 𝔸_{l}, and let e be the primitive idempotent with p = ep. We may clearly assume that RpM_{i} ≠ 0 for all i. If RpM_{i} = J^{l_{i}}m_{i}, there exists a unique string b_{i} of order l_{i} − l such that pb_{i} is a string of order l_{i} and RpM_{i} = Rpb_{i}m_{i}.

To verify the lemma in this situation, it clearly suffices to show that f induces an epimorphism

RpN ⟶ RpM/JpM,

in other words, it suffices to prove that, for each i ∈ {1, …, s} and each x ∈ RpM_{i}, there exists an element y ∈ RpN such that

x − f(y) ∈ JpM.

Assume this last statement to be false, and let k ∈ {1, …, s} be an index with maximal value l_{k} such that the statement fails for k. For simplicity, we assume k = 1 and abbreviate the string pb_{1} of order l_{1} to p_{1}. Then, clearly, p_{1}m_{1} ∉ f(RpN) + JpM. Since l_{1} ≥ l, we have f(J^{l_{1}}N) = J^{l_{1}}M by hypothesis, whence we can choose an element z ∈ eJ^{l_{1}}N = ∑_{j=1}^{t} eJ^{l_{1}}n_{j} so that

f(z) = p_{1}m_{1} = ep_{1}m_{1}.

Clearly, z = ∑_{j=1}^{t} r_{j}a_{j}n_{j}, where each r_{j} is a unit in eRe and each a_{j} is either zero or a string of order ≥ l_{1}. Moreover, in view of Remark 8(2), each term r_{j}a_{j}f(n_{j}) can be expressed in the form

r_{j}a_{j}f(n_{j}) = ∑_{i=1}^{s} s_{ij}a_{j}a_{ij}m_{i},

where each s_{ij} is in turn a unit in eRe and each a_{ij} is either zero or a string of order ≥ 0. Assembling this information, we obtain

p_{1}m_{1} = f(z) = ∑_{j=1}^{t} ∑_{i=1}^{s} s_{ij}a_{j}a_{ij}m_{i} = ∑_{i=1}^{s}(∑_{j=1}^{t} s_{ij}a_{j}a_{ij})m_{i},

which yields

(p_{1} − ∑_{j=1}^{t} s_{1j}a_{j}a_{1j})m_{1} = 0  and  (i)

(∑_{j=1}^{t} s_{ij}a_{j}a_{ij})m_{i} = 0  for i ≠ 1.  (ii)

Suppose that 1, …, t_{1} are precisely those indices j among 1, …, t such that a_{j} is a string of the form a_{j} = p_{1}a′_{j} = pb_{1}a′_{j} for some string a′_{j}. Observe that for any choice of u ∈ {1, …, t_{1}} and v ∈ {t_{1} + 1, …, t}, and for all i = 1, …, s, we then have

a_{u}a_{iu} ≠ a_{v}a_{iv}

whenever a_{u}a_{iu} ≠ 0. Using Step (1) of the proof, we therefore deduce from (i) that

(p_{1} − ∑_{j=1}^{t_{1}} s_{1j}a_{j}a_{1j})m_{1} ∈ J^{l_{1}+1}M_{1}

and from (ii) we deduce that

(∑_{j=1}^{t_{1}} s_{ij}a_{j}a_{ij})m_{i} ∈ J^{l_{1}+1}M_{i} ∩ RpM_{i}  for i ≠ 1.  (ii′)

Since J^{l_{i}}M_{i} = RpM_{i} by the choice of l_{i}, we obtain for each i ∈ {1, …, s} with l_{i} ≤ l_{1} that J^{l_{1}+1}M_{i} ⊆ J^{l_{i}+1}M_{i} = JpM_{i}.

On the other hand, for l_{i} > l_{1} we can, due to the maximality property of l_{1}, find an element z_{i} ∈ RpN so that

(∑_{j=1}^{t_{1}} s_{ij}a_{j}a_{ij})m_{i} − f(z_{i}) ∈ JpM.

For

y = ∑_{j=1}^{t_{1}} r_{j}a_{j}n_{j} − ∑_{{i=1 \atop l_{i}>l_{1}}}^{s} z_{i} ∈ RpN

we thus obtain pb_{1}m_{1} − f(y) ∈ RpM, which contradicts our assumption and concludes the proof of Step (2). Step (3). Now let M be an arbitrary submodule of a direct sum of uniserial left Rmodules and p ∈ 𝔸_{l}. Clearly, M is a sum of uniserial modules, and it is therefore enough to show that x = pm belongs to f(RpN) whenever m ∈ M is such that Rm is uniserial with pm ≠ 0. Since RpM ⊆ J^{l}M, our hypothesis yields elements n_{1}, …, n_{t} ∈ N such that

x ∈ f(∑_{j=1}^{t} J^{l}n_{j}).

Define Ñ = ∑_{j=1}^{t} Rn_{j} and M̃ = Rm + ∑_{j=1}^{t} Rf(n_{j}). Then f induces a homomorphism

f̃: Ñ → M̃.

That we in turn have f̃(J^{l}Ñ) = J^{l}M̃, follows by construction and the fact that J^{l}m = Rpm = Rx. Applying the second step of our argument, we thus obtain an element y ∈ RpÑ ⊆ RpN such that f(y) = f̃(y) = x. □

THEOREM 10. Suppose that R is left serial and that M is a submodule of a direct sum of uniserial left R-modules. Then the following statements hold:

(I) The syzygies of M are direct sums of copies of uniserial left ideals of the form J^{l}e_{i} with l ≥ 1 and 1 ≤ i ≤ n.

(II) There exists a uniserial submodule M_{0} of M with the property that p dim M_{0} = p dim M.

Before we prove Theorem 10, we derive our final corollary.

COROLLARY 11. If R is left serial, then

Fin dim R = fin dim R = cyc fin dim R =

1 + max{p dim J^{l}e_{i} | 1 ≤ l ≤ L − 1, 1 ≤ i ≤ n, and p dim J^{l}e_{i} < ∞},

where, again, this maximum is defined to be −1 if the corresponding set is empty.

Proof of Corollary 11. Theorem 3 leaves only the first equality to be shown. Let X be any left R-module with p dim X < ∞. Since M = Ω^{1}(X) ⊆ JP, where P is a projective module, M satisfies the hypothesis of Theorem 10. If Ω^{1}(X) ≠ 0, we can therefore find a uniserial submodule U of Ω^{1}(X) such that p dim U = p dim Ω^{1}(X). Choosing a finitely generated direct summand Q of P with U ⊆ Q, we conclude that p dim Q/U = p dim X. □

Proof of Theorem 10. (I). Let f : P → M be a projective cover, where P = ⨁_{u∈U} Re_{u} and e_{u} = (…, 0, e_{i(u)}, 0, …) for some primitive idempotent e_{i(u)} of R; in particular, Re_{u} ≅ Re_{i(u)}.

Strategy. We will inductively construct a chain of subsets U_{−1} = U_{0} ⊇ U_{1} ⊇ ⋯ ⊇ U_{L−1} and elements c_{v} ∈ ker(f) for each v ∈ V_{i} = U_{i−1} \ U_{i}, i ≥ 0, such that the following two conditions are satisfied:

(1) f induces an R-isomorphism

⨁_{u∈U_{l}} J^{l}e_{u}/J^{l+1}e_{u} → J^{l}M/J^{l+1}M

for 0 ≤ l ≤ L − 1.

(2) The sum ∑_{u∈U_{l}} J^{l}e_{u} + ∑_{v∈V_{l}} Rc_{v} is direct for 0 ≤ l ≤ L − 1, and

⨁_{u∈U_{l−1}} J^{l}e_{u} = ⨁_{u∈U_{l}} J^{l}e_{u} ⊕ ⨁_{v∈V_{l}} Rc_{v}.

Set U_{−1} = U_{0} = U. The validity of condition (1) for l = 0 follows from the fact that f is a projective cover; condition (2) is trivially satisfied for l = 0.

Now suppose that, for some l ≤ L − 1, sets U_{−1} = U_{0} ⊇ ⋯ ⊇ U_{l−1} and elements c_{v} of ker(f), for v ∈ ⋃_{0≤i≤l−1} V_{i} = ⋃_{0≤i≤l−1}(U_{i−1} \ U_{i}), have already been picked as specified above. From condition (1) for U_{l−1} we see that f induces an epimorphism

⨁_{u∈U_{l−1}} J^{l−1}e_{u} → J^{l−1}M,

whence a fortiori, f(⨁_{u∈U_{l−1}} J^{l}e_{u}) = J^{l}M. Each of the factors J^{l}e_{u}/J^{l+1}e_{u} being simple, we can therefore find a subset U_{l} ⊆ U_{l−1} such that f induces an isomorphism

⨁_{u∈U_{l}} J^{l}e_{u}/J^{l+1}e_{u} → J^{l}M/J^{l+1}M.

Note that this, in turn, implies that f induces an epimorphism

⨁_{u∈U_{l}} J^{l}e_{u} → J^{l}M.

Let v ∈ V_{l} = U_{l−1} \ U_{l}. If J^{l}e_{v} = 0, set c_{v} = 0. Otherwise, let p ∈ 𝔸_{l} be the unique string of order l such that J^{l}e_{v} = Rpe_{v}. Applying Lemma 9 to the induced map f : N = ⨁_{u∈U_{l}} Re_{u} → M, we obtain RpM = f(RpN), which shows that we can pick an element d_{v} ∈ RpN with f(d_{v}) = f(pe_{v}) = pf(e_{v}) ∈ RpM. If e is the primitive idempotent with p = ep, then we may, in addition, assume that d_{v} = ed_{v} ∈ eRpN = ⨁_{u∈U_{l}} eRepa_{u}e_{u}, where the a_{u} are suitable strings of order ≥ 0 (note that the product pa_{u} is either zero or in turn a string). By Remark 8(3), we conclude that l. ann_{R}(d_{v}) ⊇ l. ann_{R}(p). Hence, if we define c_{v} = pe_{v} − d_{v}, we obtain l. ann_{R}(c_{v}) = l. ann_{R}(p), since v does not belong to the support supp(d_{v}) ⊆ U of d_{v} ∈ P, relative to the decomposition P = ⨁_{u∈U} Re_{u}. On one hand, this gives us Rc_{v} ≅ Rp ≅ J^{l}e′, where e′ is the primitive idempotent with p = pe′. On the other hand, we see that for each nonzero element x ∈ Rc_{v} the index v belongs to supp(x), whereas v ∉ supp(y) for any

y ∈ ⨁_{u∈U_{l}} J^{l}e_{u} + ∑_{{w∈V_{l} \atop w≠v}} Rc_{w}

by construction. This proves the required directness of the sum under (2). The equality

⨁_{u∈U_{l−1}} J^{l}e_{u} = ⨁_{u∈U_{l}} J^{l}e_{u} ⊕ ⨁_{v∈V_{l}} Rc_{v},

finally, follows from the fact that, for each v ∈ V_{l} = U_{l−1} \ U_{l}, we have

J^{l}e_{v} ⊆ Rd_{v} + Rc_{v} ⊆ ⨁_{u∈U_{l}} J^{l}e_{u} + Rc_{v},

the last inclusion being another immediate consequence of our construction. This completes the induction.

To prove claim (I) on the basis of these preparations, set C_{i} = ⨁_{v∈V_{i}} Rc_{v} for 1 ≤ i ≤ L − 1, and C = ∑_{i} C_{i}. The directness of the sum ∑_{i} C_{i} is again clear, since the sets V_{i} are pairwise disjoint and, for each nonzero element x_{i} ∈ C_{i} we have supp(x_{i}) ∩ V_{i} ≠ ∅, while supp(x) ∩ V_{i} = ∅ for all x ∈ ∑_{j>i} C_{j}.

We want to show that ker(f) = C. That C is contained in ker(f) is a consequence of our construction. For the other inclusion, start by observing that, due to condition (2) above,

JP = ⨁_{u∈U_{0}} Je_{u} = ⨁_{u∈U_{1}} Je_{u} ⊕ C_{1},

J^{2}P = ⨁_{u∈U_{1}} J^{2}e_{u} ⊕ JC_{1} = ⨁_{u∈U_{2}} J^{2}e_{u} ⊕ C_{2} ⊕ JC_{1}.

An obvious induction yields

J^{l}P = ⨁_{u∈U_{l}} J^{l}e_{u} ⊕ C_{l} ⊕ JC_{l−1} ⊕ ⋯ ⊕ J^{l−1}C_{1}

for 1 ≤ l ≤ L−1. Setting \overline{P} = P/C and using condition (1) above, we infer that f induces an isomorphism

J^{l}\overline{P} /J^{l+1}\overline{P} ≅ ⨁_{u∈U_{l}} J^{l}e_{u}/J^{l+1}e_{u} → J^{l}M/J^{l+1}M

for each l, whence f induces an isomorphism \overline{P} → M. This shows that ker(f) = C and completes the proof of part (I).

(II). Let M be nonzero and p dim M = s ∈ ℕ ∪ {∞}. If s = 0, our claim is obvious, so suppose that s ≥ 1. Retaining the notation we built up in part (I) of the proof, we can then find an element c_{v} = pe_{v} − d_{v} ∈ P, where v ∈ V_{l} for some l ≥ 1, and d_{v} ∈ ⨁_{u∈U_{l}} J^{l}e_{u}, such that p dim Rc_{v} = s − 1. Let W^{(l)} be a finite subset of U_{l} such that d_{v} ∈ ⨁_{u∈W^{(l)}} J^{l}e_{u}. Next pick a finite subset W^{(l+1)} ⊆ U_{l+1} with the property that, whenever w ∈ W^{(l)} ∩ V_{l+1}, then d_{w} ∈ ⨁_{u∈W^{(l+1)}} J^{l+1}e_{u}. Continue inductively in order to find a sequence of finite sets W^{(m)} ⊆ U_{m}, l ≤ m ≤ L − 1, with the property that, whenever w ∈ W^{(m−1)} ∩ V_{m}, then d_{w} ∈ ⨁_{u∈W^{(m)}} J^{m}e_{u}.

Define W = {v} ∪ ⋃_{l≤m≤L−1} W^{(m)} and set M_{1} = ∑_{w∈W} Rf(e_{w}). We claim that the kernel of the induced projective cover

f_{1} = f|_{res} : ⨁_{w∈W} Re_{w} → M_{1}

is the direct sum of the cyclic modules Rc_{w}, where w ∈ W ∩ (⋃_{k≥l} V_{k}). Since one verifies this by following along the argument under (I), we will merely outline the proof. In view of the facts that v ∈ V_{l} ⊆ U_{l−1} and that the U_{i} form a descending chain, we first observe that W ⊆ U_{l−1}; in particular, f_{1} induces an isomorphism

⨁_{w∈W} J^{k}e_{w}/J^{k+1}e_{w} → J^{k}M_{1}/J^{k+1}M_{1}

for 0 ≤ k ≤ l − 1. Hence, if we set W_{−1} = W_{0} ⋯ = W_{l−1} = W, these sets play the same role relative to W as is played by the sets U_{−1}, U_{0}, …, U_{l−1} relative to U. We supplement this sequence of subsets of W as follows. We define X_{0} = X_{1} = ⋯ = X_{l−1} = ∅, X_{k} = W ∩ V_{k} for k ≥ l, and W_{k} = W \ (⋃_{0≤i≤k} X_{i}) for k ≥ l. Then X_{l} = {v}, and, for each k ≥ 0, the set W_{k−1} is the disjoint union of W_{k} and X_{k}. By a straightforward induction on k ∈ {0, …, L − 1}, we check moreover that W_{k} = W ∩ U_{k}, that

⨁_{w∈W_{k−1}} J^{k}e_{w} = ⨁_{w∈W_{k}} J^{k}e_{w} ⊕ ⨁_{x∈X_{k}} Rc_{x}

– for the last equality, invest the fact that supp(d_{x}) ⊆ W ∩ U_{k}, whenever x ∈ X_{k}, by construction – and, finally, that f_{1} induces an isomorphism

⨁_{w∈W_{k}} J^{k}e_{w}/J^{k+1}e_{w} → J^{k}M_{1}/J^{k+1}M_{1}

for each k. As under (I), this setup entails

ker(f_{1}) = ⨁ ; x∈⋃_{k≥0} X_{k} Rc_{x}.

In view of the equality ‘⋃_{k≥0} X_{k} = W ∩ (⋃_{k≥l} V_{k})’, this proves the above claim.

In particular, we infer that p dim ker(f_{1}) ≤ p dim ker(f) ≤ s−1, and since v ∈ ⋃_{k≥0} X_{k} and p dim Rc_{v} = s−1, we derive further that p dim ker(f_{1}) = s−1. This proves p dim M_{1} = s. But the finitely generated module M_{1}, being a direct sum of uniserial components by Proposition 1, contains in turn a uniserial summand M_{0} with p dim M_{0} = s. This proves claim (II). □

REFERENCES

[AB] M. Auslander and D. Buchsbaum, “Homological dimension in local rings. II”, Trans. Amer. Math. Soc. 85, 1957, 390-405.

[B] H. Bass, “Finitistic dimensions and a homological generalization of semiprimary rings”, Trans. Amer. Math. Soc. 95, 1960, 466-488.

[BF] W. D. Burgess and K. R. Fuller, “Left almost serial rings and the Cartan determinant conjecture”, This volume.

[BFVH] W. D. Burgess, K. R. Fuller, E. R. Voss, and B. Zimmermann Huisgen, “The Cartan matrix as an indicator of finite global dimension for artinian rings”, Proc. Amer. Math. Soc. 95, 1985, 157-165.

[EG] D. Eisenbud and P. Griffith, “Serial rings”, J. Algebra 17, 1971, 389-400.

[FR] K. R. Fuller and I. Reiten, “Note on rings of finite representation type and decompositions of modules”, Proc. Amer. Math. Soc. 50, 1975, 92-94.

[FS] K. R. Fuller and M. Saorin, “On the finitistic dimension conjecture”, manuscripta math. 74, 1992, 117-132.

[G] W. H. Gustafson, “Global dimension in serial rings”, J. Algebra 96, 1985, 14-16.

[H1] B. Zimmermann Huisgen, “Homological domino effects and the first finitistic dimension conjecture”, Invent. Math., To appear.

[H2] B. Zimmermann Huisgen, “Field dependent homological behavior in finite dimensional algebras”, Preprint.

[HZ] B. Zimmermann Huisgen and W. Zimmermann, “Algebraically compact rings and modules”, Math. Zeitschr. 161, 1978, 81-93.

[N] T. Nakayama, “On Frobeniusean algebras II”, Annals of Math. 42, 1941, 1-21.

[RZ] A. Rosenberg and D. Zelinsky, “Finiteness of the injective hull”, Math. Zeitschr. 70, 1959, 372-380.

[W] Y. Wang, Hand-written notes, University of Iowa, 1991.

[Za1] D. Zacharia, “On the Cartan matrix of an Artin algebra of global dimension two”, J. Algebra 82, 1983, 353-357.

[Za2] D. Zacharia, “Special monomial algebras of finite global dimension”, in Perspectives in Ring Theory (F. Van Oystaeyen and L. Le Bruyn, Eds.), 1988, Kluwer Academic Publishers, pp. 375-378.

[Zi] W. Zimmermann, “Rein injektive direkte Summen von Moduln”, Comm. Algebra 5, 1977, 1083-1117.