ALMOST SPLIT SEQUENCES FOR DEDEKIND–LIKE RINGS II

David M. Arnold and Reinhard C. Laubenbacher

Abstract. Let R be a two–pullback ring, that is, the pullback of two Dedekind domains over a common residue field. This paper contains explicit constructions of all the almost split sequences in the category of finitely generated R–modules, under the assumption that the Dedekind domains are local.

1. Introduction

Almost split sequences (Auslander–Reiten sequences) originally arose in the study of modules over finite dimensional Artin algebras by Auslander and Reiten and now play a central role in this theory. Since then they have been studied extensively in various contexts. While it has been shown that almost split sequences exist in many different categories, there are surprisingly few instances in the literature in which these sequences are constructed explicitly.

Let R be a two–pullback ring, that is, a pullback of two Dedekind domains over a common residue field. In [AL1, Theorem 3.4], we showed that an indecomposable finitely generated R–module M is the right–hand term of an almost split sequence if and only if either M is artinian of homological dimension 1, or else the coordinate rings of R are local and M is nonartinian of dimension 1. This paper is devoted to an explicit description of the almost split sequences in the category Mod(R) of finitely generated R–modules.

Two–pullback rings and, more generally, multiple pullback rings occur in a variety of settings, for instance as integral group rings, and subrings of square–free index in the product of two Dedekind domains [L1, Intro.]. There are also connections between finitely generated modules over multiple pullback rings and abelian groups containing a finite direct sum of subgroups of ℚ as a subgroup of finite index [A, AD]. Our construction of almost split sequences is expedited by a complete list of indecomposables in Mod(R) [L1, Theorem 1.9].

Since the endomorphism ring of the right–hand term of an almost split sequence is local, we assume also that the coordinate rings of R are local, in order to allow non–artinian modules as right–hand terms of almost split sequences. Specifically,

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16G70, 16D10.

Key words and phrases. almost split sequence, Dedekind–like ring.

The first author was supported, in part, by NSF grant #DMS-9101000, and the University of Colorado at Colorado Springs. The second author was supported, in part, by Baylor University and Colorado College. He would also like to thank the Cornell University Mathematics Department for its hospitality while part of this work was completed. Both authors would like to thank M. C.

R. Butler and L. Klingler for helpful conversations. They would also like to thank the referee for the many comments leading to an improved exposition.

let R_{1} and R_{2} be discrete valuation domains with maximal ideals ℘_{1} = R_{1}π_{1} and ℘_{2} = R_{2}π_{2} and residue field k. Let v_{1} : R_{1} → k and v_{2} : R_{2} → k be the canonical projections, that is ker(v_{i}) = ℘_{i}. Define the two-pullback ring of (R_{1}, v_{1}) and (R_{2}, v_{2}) as

R = {(r_{1}, r_{2}) ∈ R_{1} × R_{2} | v_{1}(r_{1}) = v_{2}(r_{2})},

with coordinate-wise operations.

Recall that an almost split sequence in Mod(R) is an exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0

of finitely generated R-modules such that (1) it is not split;

(2) if X ∈ Mod(R), and h : X ⟶ C is a homomorphism which is not a split epimorphism, then h can be lifted to B;

(3) if Y ∈ Mod(R) and t : A ⟶ Y is a homomorphism which is not a split monomorphism, then t can be extended to B.

An exact sequence satisfying (1) and (2) is called right almost split. It is well known that an almost split sequence is determined, up to isomorphism, by its right or left hand term [Au, Theorem 1]. Furthermore, both left and right hand terms of an almost split sequence have local endomorphism rings.

Auslander proved a very general existence theorem about almost split sequences for modules. In order to state his theorem, we recall the relevant definitions from [AL1]. Let A be a ring, and M a finitely generated R–module. Choose a projective presentation

P_{1} ⟶^{f} P_{0} ⟶ M ⟶ 0

of M, and define the transpose of M, denoted T r(M), by the exact sequence

Hom_{R}(P_{0}, R) ⟶^{f^{∗}} Hom_{R}(P_{1}, R) ⟶ T r(M) ⟶ 0.

Then T r(M) is well–defined up to projective summands. Define the category \underline{Mod}(R) to be the quotient category of Mod(R), whose objects are finitely generated R–modules, and whose morphism groups are

\underline{Hom}(A, B) = Hom_{R}(A, B)/P_{R}(A, B),

where P_{R}(A, B) is the subgroup of Hom_{R}(A, B) consisting of those homomorphisms which factor through finitely generated projective R–modules. Then the above construction induces a well–defined additive functor

T r : \underline{Mod}(R) ⟶ \underline{Mod}(R^{op}),

which is a duality, that is, T r^{2} is naturally equivalent to the identity functor on \underline{Mod}(R). On morphisms, T r is defined via lifting to projective presentations.

If M is a finitely generated R–module, let T r(M) be defined via a fixed projective presentation. Let Γ = End_{R} (T r(M)) be the endomorphism ring of T r(M). Then an \underline{End}(M)^{op}–module is a Γ–module via the ring surjection given by the composite

Γ ⟶ \underline{End} (T r(M)) ⟶^{T r} \underline{End}(M)^{op}.

Theorem 1.1. [Au, Theorem 4] Let A be a noetherian ring, and let M be a finitely generated, non–projective A–module with local endomorphism ring, and let Γ = End (T r(M)). Let I be the Γ–injective envelope of the unique simple \underline{End}(M)^{op}– module, regarded as a Γ–module. Then there is an almost split sequence

0 ⟶ Hom_{Γ}(T r(M), I) ⟶ B ⟶ M ⟶ 0

in the category of all A–modules.

Observe that injective envelopes of (finitely generated) modules are not finitely generated in general, therefore the left, hence the middle, term of the sequence in the theorem is not finitely generated in general. In [AL1] we determined those finitely generated, indecomposable modules M over a two–pullback ring, for which the short exact sequence of Theorem 1.1 is in the category of finitely generated R–modules.

In this paper we describe the almost split sequences in Mod(R) explicitly, in terms of the right–hand term M. In particular, if M is indecomposable artinian of dimension 1, then the Auslander–Reiten translate DT r(M) = Hom_{Γ}(T r(M), I) ≅ M and the middle term B is the direct sum of at most two indecomposables (Theorem 4.6). If M is indecomposable, nonartinian of dimension 1, then Hom_{Γ}(T r(M), I) is artinian of infinite dimension and B is the direct sum of two indecomposables (Theorem 5.6).

The methods for computing the almost split sequence in Theorem 1.1 depend on whether M is artinian or not. In the non–artinian case, the computation is ad hoc, but DT r(M) turns out to be the maximal artinian submodule of M. The artinian case is more interesting however. First we show that the associated graded ring Λ of a two–pullback ring, with respect to the maximal ideal, is isomorphic to a localization of the k–algebra k[x, y]/⟨xy⟩. Specifically, it is the two–pullback ring defined by the pullback square of local rings

Λ ⟶ k[x]_{(x)}
↓  ↓
k[y]_{(y)} ⟶ k

where the bottom horizontal and right vertical maps send the variable to zero. Furthermore we show (Theorem 4.2) that there is a category equivalence between suitable quotient categories of Mod(R) and Mod(Λ), which induces a one–to–one correspondence between isomorphism classes of objects and preserves indecomposables and almost split sequences. Thus, Λ is the “generic” two–pullback ring for the computation of almost split sequences and the classification of indecomposable modules. Thus, any two local two–pullback rings with common residue field k lead to the same matrix problem as Λ.

If one is only interested in almost split sequences for indecomposable artinian R–modules, then one can reduce further to certain quotients of Λ which are artin algebras over k. These were studied by Butler and Shazamanian [BS], who surveyed the classification of all finitely generated indecomposable modules and computed their almost split sequences. Thus, we not only obtain the almost split sequences for artinian indecomposable R–modules in this way, but also an alternative classification method for these modules. This last observation is a key ingredient in the approach to classifying indecomposable modules over certain kinds of pullback rings, carried out in [AL2].

2. Indecomposable Modules over two-pullback rings

Let R be a two-pullback ring with local coordinate rings, as in the introduction. First we recall a complete list, up to isomorphism, of all finitely generated, indecomposable R-modules from [L1], with notation convenient for our purposes. For 1 ≤ e, f ≤ ∞ define

[e, f] = R/(℘_{1}^{e} ⊕ ℘_{2}^{f}) = {(r_{1} + ℘_{1}^{e}, r_{2} + ℘_{2}^{f}) ∈ (R_{1}/℘_{1}^{e}) × (R_{2}/℘_{2}^{f}) | v_{1}(r_{1}) = v_{2}(r_{2})}

(with the convention that ℘_{i}^{∞} = 0). Abusing notation, we will denote the elements of [e, f] by (r_{1}, r_{2}). There are three types of indecomposables in Mod(R).

(1) Basic building blocks: These are all modules of the form M = [e, f] for integers 1 ≤ e, f ≤ ∞. If f = 1 and e > 1, then M ≅ R_{1}/℘_{1}^{e}, while, if e = 1 and f > 1, then M ≅ R_{2}/℘_{2}^{f}. Furthermore, we have that [1, 1] ≅ k ≅ R/(℘_{1} ⊕ ℘_{2}). The module [∞, ∞] is the only projective indecomposable, hence does not play a role in this paper. Basic building blocks are indecomposable, and the ordered pair of integers (e, f) is a complete isomorphism invariant of [e, f].

(2) Deleted cycles: For integers 1 ≤ e_{1}, f_{n} ≤ ∞, 2 ≤ f_{1}, e_{2}, …, f_{n−1}, e_{n} < ∞

define:

M = [e_{1}, f_{1}; e_{2}, f_{2}; …; e_{n}, f_{n}]

to be the cokernel of the embedding

σ : k^{n−1} ⟶ [e_{1}, f_{1}] ⊕ [e_{2}, f_{2}] ⊕ ⋯ ⊕ [e_{n}, f_{n}]

defined as follows. Let a_{1}, b_{1}, …, a_{n−1}, b_{n−1} be units of k. Then

σ(r_{1}, r_{2}, …, r_{n−1}) =

(0, a_{1}r_{1}π_{2}^{f_{1}−1}; b_{1}r_{1}π_{1}^{e_{2}−1}, a_{2}r_{2}π_{2}^{f_{2}−1}; …; b_{n−1}r_{n−1}π_{1}^{e_{n}−1}, 0) .

In other words, the submodule ℘_{2}^{f_{i}−1}/℘_{2}^{f_{i}} ≅ k of [e_{i}, f_{i}] is identified with the submodule ℘_{1}^{e_{i+1}−1}/℘_{1}^{e_{i+1}} ≅ k of [e_{i+1}, f_{i+1}] for 1 ≤ i ≤ n − 1.

Deleted cycle modules are indecomposable. A complete set of invariants is the ordered sequence of pairs e_{1}, f_{1}; …; e_{n}, f_{n}. Deleted cycles are independent, up to isomorphism, of the choice of the a_{i} and b_{i}. Thus, unless otherwise specified, we will always assume that a_{i} = b_{i} = 1 for all i.

(3) Block cycles: Suppose we are given n ≥ 1, and 2 ≤ e_{i}, f_{i} ≤ ∞ for 1 ≤ i ≤ n, defining a sequence of basic building blocks

[e_{1}, f_{1}], …, [e_{n}, f_{n}],

which consists of r repetitions of the first t terms, with t minimal, that is, n = rt. Suppose furthermore, that g(x) ≠ x is a monic irreducible polynomial in k[x], ϵ is a positive integer, and g(x)^{ϵ} = a_{0} + a_{1}x + ⋯ + a_{r−1}x^{r−1} + x^{r}. Define the block cycle

M = ⟨e_{1}, f_{1}; …; e_{n}, f_{n}⟩

to be the cokernel of an embedding of k into a deleted cycle

σ : k ⟶ [e_{1}, f_{1}; e_{2}, f_{2}; …; e_{n}, f_{n}]

defined as follows:

σ(1_{k}) = (a_{0}π_{1}^{e_{1}−1}, 0; …; 0, 0; a_{1}π_{1}^{e_{1}−1}, 0; …; a_{r−1}π_{1}^{e_{1}−1}, 0; …; 0, π_{2}^{f_{n}−1}).

We call g(x)^{ϵ} a blocking polynomial of M. In other words, σ identifies k ≅ ℘_{2}^{f_{n}−1}/℘_{2}^{f_{n}} in [e_{n}, f_{n}] with a one-dimensional subspace of [e_{1}, f_{1}]^{r}, one copy of [e_{1}, f_{1}] for each of the r repetitions of the sequence [e_{1}, f_{1}], …, [e_{t}, f_{t}].

If r = 1 the block cycle M is indecomposable. If r ≥ 2, then M may or may not be indecomposable, depending on k and the blocking polynomial g(x)^{ϵ} (see [KL, Remark 1.9] or [L1, Theorem 8.2]; the statement of [L1, Theorem 1.8] is incomplete). However, every indecomposable block cycle is of the form described above. Its isomorphism class depends only on its blocking polynomial g(x)^{ϵ} and the ordered sequence of its basic building blocks [KL, Theorem 1.7].

The nonprojective indecomposable modules in Mod(R) are divided into the following classes [AL, Prop. 3.2]:

(1) The class A_{1} of indecomposable block cycles; these are the artinian indecomposable modules in Mod(R) of homological dimension one.

(2) The class N A_{1} of deleted cycles of the form [∞, f_{1}; …; e_{n}, ∞]; these are the nonartinian indecomposable modules of homological dimension one.

(3) The class A_{∞} of deleted cycles [e_{1}, f_{1}; e_{2}, f_{2}; …; e_{n}, f_{n}] with e_{1}, f_{n} < ∞; these are the artinian indecomposable modules of infinite homological dimension.

(4) The class N A_{∞} of deleted cycles [e_{1}, f_{1}; e_{2}, f_{2}; …; e_{n}, f_{n}] with exactly one of e_{1}, f_{n} equal to ∞; these are the nonartinian indecomposable modules with infinite homological dimension.

The main theorem of [AL] can then be restated as saying that almost split sequences exist exactly for the modules in A_{1} and N A_{1} as right–hand terms.

3. Almost split sequences

We need several well–known facts about almost split sequences, which we collect here.

Lemma 3.1. [AR] Suppose that there exists an almost split sequence

0 ⟶ C ⟶ B′ ⟶ M ⟶ 0

in Mod(R). Let the following be a non-split exact sequence

(∗)  0 ⟶ C ⟶ B ⟶^{g} M ⟶ 0

(1) If C is artinian and (∗) is right almost split, then it is almost split;

(2) If M and C are artinian and

0 ⟶ C ⟶ g^{−1}(M′) ⟶ M′ ⟶ 0

is split exact for every proper submodule M′ of M, then (∗) is right almost split, hence almost split by (1).

Recall that an R-homomorphism f : M ⟶ N is irreducible if f is neither a split monomorphism nor a split epimorphism and if, whenever f factors as M ⟶^{j} X ⟶^{i} N, then either j is a split monomorphism or i is a split epimorphism.

Lemma 3.2. [AR, Prop. 2.7] A non-split exact sequence

(∗)  0 ⟶ M ⟶ ⨁ B_{i} ⟶ N ⟶ 0

in Mod(R) with M, N, and each B_{i} indecomposable with local endomorphism ring, is almost split if and only if each irreducible map M ⟶ X, with X indecomposable, occurs up to isomorphism as one of the maps M ⟶ B_{i} and each irreducible map Y ⟶ N occurs as one of the maps B_{i} ⟶ N.

Remark. Although [AR, Prop. 2.7] assumes that the modules are artinian, the same proof carries over to the non–artinian case. A categorical version of the proof is given in [Ri, Sect. 2.2].

4. The artinian case

In this section we compute the almost split sequences for the modules in the class A_{1}.

Let Λ′ = k[x, y]/(xy), a quotient of the polynomial ring k[x, y], with commuting variables x and y. Then Λ′ is a two-pullback ring defined by the augmentation maps v_{1} : k[x] ⟶ k and v_{2} : k[y] ⟶ k. The localization Λ of Λ′ at the maximal ideal (x, y) is then also a two-pullback ring with local coordinate rings. It is straightforward to check that Λ′ is the associated graded ring G(R) of R with respect to its maximal ideal ker(R ⟶ k). (Recall that, if A is a ring and I ⊂ A is an ideal, then the associated graded ring is defined to be the graded ring ⨁_{i≥0} I^{i}/I^{i+1}.)

If A is a ring and Mod(A) is the category of finitely generated A-modules, then define a quotient category of Mod(A) as follows. Let MOD(A) be the category with the same objects as Mod(A) but with morphism groups

HOM(M, N) = Hom_{A}(M, N)/Hom_{A}(M, rad(A)N) .

Here, rad(A) is the Jacobson radical of A. Then MOD(A) is an additive category with finite direct sums. In particular, M is a direct summand of N in MOD(A) if and only if there exists f ∈ Hom_{A}(M, N) and g ∈ Hom_{A}(N, M) such that gf −1 : M ⟶ rad(A)M. Obviously, there is a full functor Mod(A) ⟶ MOD(A), which is the identity on objects.

Lemma 4.1. Let R be a two-pullback ring with local coordinate rings and let M, N ∈ Mod(R).

(1) If h : M ⟶ rad(R)M, then 1 + h is an automorphism of M.

(2) M is indecomposable in Mod(R) if and only if M is indecomposable in MOD(R).

(3) M and N are isomorphic in Mod(R) if and only if they are isomorphic in MOD(R).

Proof. (1) Note that rad(R) = (π_{1}, 0)R+(0, π_{2})R, in particular, rad(Λ) = xΛ+yΛ. Now observe that h : (π_{1}, 0)M ⟶ (π_{1}^{2}, 0)M is an R_{1}-homomorphism. The module M_{1} = (π_{1}, 0)M is a finitely generated module over the discrete valuation ring R_{1} (with rad(R_{1}) = (π_{1})). It follows from Nakayama’s Lemma [CR, p. 105] that (1 + h) : M_{1} ⟶ M_{1} is onto. Similarly, the map (1 + h) : (0, π_{2})M ⟶ (0, π_{2})M is onto. Hence (1 + h) : M ⟶ M is onto since, if m ∈ M, then m + h(m) = m + (π_{1}, 0)n_{1} + (0, π_{2})n_{2} = m + (1 + h)((π_{1}, 0)m_{1}) + (1 + h)((0, π_{2})m_{2}) for some m_{i}, n_{i} ∈ M. Thus m = (1 + h)(m − (π_{1}, 0)m_{1} − (0, π_{2})m_{2}). Now suppose that 0 ≠ m ∈ ker(1+h). Choose i maximal such that m ∈ (rad(R))^{i}M. Then 0 = m+h(m) with h(m) ∈ (rad(R))^{i+1}M, which is a contradiction. This shows that 1 + h is an automorphism.

(2) Clearly, if M is indecomposable in MOD(R), then it is also indecomposable in Mod(R). Conversely, assume that N is a summand of M in MOD(R). Then there exists f : N ⟶ M and g : N ⟶ M with (gf − 1) : N ⟶ rad(R)N. By (1), gf is an automorphism of N, so that N is a summand of M in Mod(R).

(3) One direction is obvious. Assume that M ≅ N in MOD(R). Then there exist f : M ⟶ N and g : N ⟶ M such that (gf − 1) : M ⟶ rad(R)M and (fg − 1) : N ⟶ rad(R)N. Again, by (1), gf is an automorphism of M and fg is an automorphism of N. In particular, M ≅ N in Mod(R). □

For the proof of the next theorem we need to recall some facts about R–modules. Recall that an R–module is called separated if it is an R–submodule of a direct sum of an R_{1}–module and an R_{2}–module. A separated representation of an R–module M is an R–module epimorphism φ : S ↠ M with S a separated R–module, such that in any factorization S ⟶ S′ ↠ M of φ the first map must be one–to–one [L2, Sect. 1]. The main fact about separated representations is that every finitely generated R–module possesses one [L2, Theorem 2.8].

Theorem 4.2. The categories MOD(R) and MOD(Λ) are equivalent.

Proof. Let M ∈ MOD(R). Define M_{0} = M/(rad(R))M and

M_{i} = (rad(R))^{i}M/(rad(R))^{i+1}M

for i > 0. Now define a functor

(1)  F : MOD(R) ⟶ MOD(Λ)

as follows. On objects, F is defined by F (M) = ⨁_{i} M_{i}. Then F (M) is a k–vector space, since rad(R)M_{i} = 0 for all i and k = R/rad(R). The vector space F (M) becomes a Λ-module if we define the action of x, y on F (M) via multiplication by (π_{1}, 0), respectively (0, π_{2}), taking M_{i} to M_{i+1}. Note that xyF (M) = 0 since (π_{1}, 0)(0, π_{2}) = 0. Since M_{i+1} = xM_{i} + yM_{i}, F (M) is finitely generated as a Λ-module, with a k-basis of M_{0} as generators.

Given f ∈ Hom_{R}(M, N), define F (f) ∈ Hom_{Λ}(F (M), F (N)) by setting F (f) = ∑_{i} f |_{M_{i}}, observing that f : M_{i} ⟶ N_{i}. If f : M ⟶ rad(R)N, then F (f) = 0 in MOD(Λ), so that F factors through MOD(R). Then F is an additive functor.

We next show that each L ∈ MOD(Λ) is of the form F (M) for some M ∈ MOD(R). First observe that F (R) = Λ, so that for each n ≥ 1 there exists a finitely generated free R-module T such that F (T) = Λ^{n}. Then

(rad(R))^{i}T/(rad(R))^{i+1}T = (rad(Λ))^{i}Λ^{n}/(rad(Λ))^{i+1}Λ^{n}

for each i. Choose free Λ–bases u_{1}, …, u_{n} of Λ^{n} and x_{1}, …, x_{n} of T such that u_{i} + rad(Λ) = x_{i} + rad(R) for each i. Now write L = Λ^{n}/K′ for some K′ with n minimal. Since Λ^{n}/rad(Λ)Λ^{n} ≅ L/rad(Λ)L it follows that K′ ⊂ rad(Λ)Λ^{n}. Therefore K′ can be generated over k by elements of the form ∑ λ_{j}(x, y)u_{j} with each λ_{j}(x, y) ∈ rad(Λ) having constant term zero.

Let K be the R–submodule of the free module T generated by all elements of the form ∑ λ_{j}((π_{1}, 0), (0, π_{2}))x_{j}, where we view λ_{j}(x, y) as an element of R[x, y] by lifting a coefficient a ∈ k to an element (r, s) ∈ R with v_{1}(r) = a = v_{2}(s). Such a lifting is not unique, but it is unique modulo rad(R) since R/rad(R) = k. Then K ⊂ rad(R)T, and we define M = T/K. As a consequence we have M_{0} = M/rad(R)M = L_{0} = L/rad(Λ)L. Moreover, the choice of T and K guarantee that

(K ∩ (rad(R)^{i}T) + (rad(R)^{i+1}T ) /rad(R)^{i+1}T =

(K′ ∩ (rad(Λ)^{i}Λ^{n}) + (rad(Λ)^{i+1}Λ^{n}) /rad(Λ)^{i+1}Λ^{n} .

Hence, M_{i} = rad(R)^{i}M/rad(R)^{i+1}M = rad(Λ)^{i}L/rad(Λ)^{i+1}L = L_{i}. It follows that F (M) = ⊕M_{i} = ⊕L_{i} = L, as desired.

Now consider the homomorphism

F : Hom_{R}(M, N) ⟶ Hom_{Λ}(F (N), F (M))/Hom_{Λ}(F (M), rad(Λ)F (N)) ,

induced by F (by abuse of notation). If F (f) = 0, then f(F (M)) is contained in rad(Λ)F (N) = N_{1}⊕N_{2}⊕…. Therefore F (f)(M_{0}) = f(M/rad(R)M) = 0 ∈ N_{0} = N/rad(R)N and f : M ⟶ rad(R)N. Consequently, the functor F is one–to–one on morphisms in MOD(R).

To show that it is onto as well, first note that, if [e, f] = R/(℘_{1}^{e} ⊕ ℘_{2}^{f}) is a basic building block, then F ([e, f]) = [e, f] = Λ/ ((x)^{e} ⊕ (y)^{f}), since F (R) = Λ. Now, if both M and N are basic building blocks, then F in (1) above is onto. This is because f ∈ Hom_{Λ}(F (M), F (N)) is determined by f(1, 1), F (M) and F (N) are cyclic, and Λ/rad(Λ) = k = R/rad(R). (Note that none of the non–zero elements of Hom_{Λ}(F (M), rad(Λ)F (N)) are in the image of F.)

Now let M, N ∈ Mod(R) with separated representations h_{M} : S_{M} ↠ M and h_{N} : S_{N} ↠ N. It follows from the classification in [L1] that every separated module is a direct sum of basic building blocks. Since F is additive we get from the above that

F : Hom_{R}(S_{M}, S_{N}) ⟶ Hom_{Λ}(F (S_{M}), F (S_{N}))/Hom_{Λ}(F (S_{M}), rad(Λ)F (S_{N})) is onto.

Next we show that F (h_{M}) : F (S_{M}) ⟶ F (M) is a separated representation. The construction of F shows that the map is onto. Furthermore, F (S_{M}) is a direct sum of basic building blocks, hence separated. Note that h_{M} is one–to–one on (π_{1}, 0)S_{M} and (0, π_{2})S_{M}, and that ker(h_{M}) is contained in rad(R)S_{M} [L2, Prop. 2.3]. Moreover, ker(h_{M}) is a k–module annihilated by rad(R) [L2, Prop. 2.4], hence lies in the socle of S_{M}. It follows that F (h_{M}) is one-to-one on xF (S_{M}) and yF (S_{M}), and ker (F (h_{M})) is contained in rad(Λ)F (S_{M}). This is because

ker (F (h_{M})) = ⨁_{i} [ker(h_{M}) ∩ (rad(R)^{i}S_{M}) + rad(R)^{i+1}S_{M}/(rad(R)^{i+1}S_{M}]

and, for example, xF (S_{M}) = (π_{1}, 0)S_{M}/(π_{1}^{2}, 0)S_{M} ⊕ . . .. Hence F (S_{M}) = S_{F (M)}, again by [L2, Prop. 2.3]. In particular, ker(F (h_{M})) is contained in the socle of S_{F (M)} and is annihilated by rad(Λ).

Finally, assume that g : F (M) ⟶ F (N) is a Λ-map with im(g) not contained in rad(Λ)F (N). Then g lifts to g : S_{F (M)} ⟶ S_{F (N)} [L2, Thm. 2.6] with im(g) not contained in rad(Λ)S_{F (N)}. Then, by the above, we can find a (non-zero) f : S_{M} ⟶ S_{N} such that

F (f) − g : S_{F (M)} ⟶ rad(Λ)S_{F (N)} .

Therefore, F (f) = g : ker(F (h_{M})) ⟶ ker(F (h_{N})). To see that f : ker(h_{M}) ⟶ ker(h_{N}), observe that there exists a j such that ker(h_{M}) ∩ rad(R)^{j}S_{M} = 0. Then use the filtration of ker(h_{M}) induced by

ker(h_{M}) ∩ rad(R)^{i}S_{M} ≅ (rad(R)^{i}S_{M}) + (rad(R)^{i+1}S_{M})/(rad(R)^{i+1}S_{M})

starting from the bottom. Hence, f ∈ Hom_{R}(M, N). This completes the proof of the theorem. □

Corollary 4.3. The functors

Mod(R) ⟶ MOD(R) ⟶^{≅} MOD(Λ) ⟵ Mod(Λ)

induce a one–to–one correspondence between isomorphism classes of objects in Mod(R) and Mod(Λ), preserving indecomposability, direct sums and irreducible homomorphisms.

Proof. This follows directly from Lemma 4.1 and Theorem 4.2. In particular, irreducible maps are preserved since all the functors in question preserve and reflect split maps. Furthermore, Λ is a two–pullback ring with local coordinate rings. □

Corollary 4.3 implies that, in order to study indecomposable R–modules, we only need to study Λ–modules instead. For artinian indecomposables we may further reduce to certain artinian quotients of Λ, which were studied in [BS]. They admit two classes of indecomposable modules, called string modules and band modules.

We will show that these correspond precisely to Levy’s deleted cycle and block cycle indecomposables.

Let p, q be positive integers. Define Λ_{p,q} to be the k-algebra with generators a, b and relations ab = 0 = ba, a^{p+1} − b^{q+1} = 0. Hence, Λ_{p,q} ≅ Λ/(x^{p+1} − y^{q+1}). We now define two types of modules over Λ_{p,q}. Observe that Λ_{p,q} has {1, a^{i}, b^{j} | 1 ≤ i ≤ p + 1, 1 ≤ j ≤ q} as a k-basis.

Define a word to be a symbol

W = a^{i_{1}}b^{j_{1}}a^{i_{2}}b^{j_{2}} … a^{i_{n}}b^{j_{n}} ,

made up of elements ≠ 1 in the above basis of Λ_{p,q}. A word W is called an irreducible band word if it is not a power W = (W′)^{m} of another word W′. Note that a cyclic permutation of the constituent pairs a^{i}b^{j} in an irreducible band word results in another irreducible band word.

Definition [BS, Sect. 1].

(1) Let W be a word (not necessarily irreducible) as above. Define the associated string module M(W) to be the Λ_{pq}-module with generators u_{1}, …, u_{n} and relations

a^{i_{1}}u_{1} = 0,  b^{j_{k}}u_{k} = a^{i_{k+1}}u_{k+1} (k = 1, …, n − 1),  b^{j_{n}}u_{n} = 0 .

(2) Let W be an irreducible band word, and let φ be an indecomposable automorphism of a finite dimensional k-vector space V. Define the band module M(W, V, φ) to be the Λ_{pq}–module generated by n vector subspaces u_{1} ⊗ V, …, u_{n} ⊗ V, subject to the relations

\begin{aligned}b^{j_{k}}u_{k} ⊗ v & = a^{i_{k+1}}u_{k+1} ⊗ v for k = 1, …, n − 1, v ∈ V; \\ b^{j_{n}}u_{n} ⊗ v & = a^{i_{1}}u_{1} ⊗ φ(v), v ∈ V . \end{aligned}

Theorem 4.4. [BS, pp. 112–113]

(1) Let W, W′ be words. Then M(W) ≅ M(W′) if and only if W = W′ or W′ = W^{−1}.

(2) Let W, W′ be irreducible band words. Then M(W, V, φ) ≅ M(W′, V′, φ′) if and only if W is obtained from W′ by a cyclic permutation of the constituent pairs of W′, and φ and φ′ have the same characteristic polynomial g(x)^{ϵ}, where ϵ ≥ 1 and x ≠ g(x) is a monic irreducible polynomial in k[x].

Theorem 4.5. [BS, p. 112] String and band modules over Λ_{p,q} are indecomposable and account for all indecomposable Λ_{p,q}-modules.

Now let N be a finitely generated indecomposable artinian R-module. First suppose that N is a deleted cycle indecomposable of the form [e_{1}, f_{1}; …; e_{n}, f_{n}]. Then, if R/ann(N) = [e, f], it is easy to see that F (N) is a Λ_{e−2,f−2}-module of the form M(W) for

W = a^{e_{1}−1}b^{f_{1}−1}a^{e_{2}−1} ⋯ b^{f_{n−1}−1}a^{e_{n}−1}b^{f_{n}−1} ,

that is, F (N) has generators u_{1}, …, u_{n} ∈ N_{0} and relations

b^{f_{k}−1}u_{k} = a^{e_{k+1}−1}u_{k+1} k = 1, …, n − 1 .

Likewise, suppose that N is a block cycle indecomposable

N = ⟨e_{1}, f_{1}; …; e_{n}, f_{n}⟩

with blocking polynomial g(x)^{ϵ} = a_{0} + a_{1}x + ⋯ + a_{r−1}x^{r−1} + x^{r}. Assume that N is the r-fold repetition of the first t building blocks. Then F (N) can be described as a band module over Λ_{e−2,f−2} as follows. The word

W = a^{e_{1}−1}b^{f_{1}−1}a^{e_{2}−1} ⋯ b^{f_{t−1}−1}a^{e_{t}−1}b^{f_{t}−1} ,

is an irreducible band word. Let V be an r-dimensional k-vector space with basis v_{1}, …, v_{r}, and φ the indecomposable automorphism of V given by φ(v_{i}) = v_{i+1} for 1 ≤ i < r and φ(v_{r}) = ∑_{i=1}^{r} a_{i}v_{i}. Then F (N) = M(W, V, φ).

Observe that the classification of (artinian) deleted cycle and block cycle indecomposables in Mod(R) is the “same” as that of string and band modules given in Theorem 4.4. Corollary 4.3 and the preceding remarks provide an explanation.

We are now ready to describe the almost split sequences for indecomposable block cycle R–modules, via Corollary 4.3 and Lemma 3.2.

Theorem 4.6. Let M = ⟨e_{1}, f_{1}; …; e_{n}, f_{n}⟩ = M(W, V, φ) ∈ A_{1} be a block cycle indecomposable with blocking polynomial g(x)^{ϵ} = a_{0}+a_{1}x+⋯ a_{r−1}x^{r−1}+x^{r}. There exists an almost split sequence

0 ⟶ M ⟶^{μ} B ⟶^{ϕ} M ⟶ 0 ,

where B is the direct sum of at most two block cycle indecomposables (as described below).

Proof. The module F (M) = M(W, V, φ) ∈ Mod(Λ) is actually a module over Λ_{p,q} for suitable p, q. It is easily seen that an irreducible map in Mod(Λ_{p,q}) remains irreducible when viewed as a map in Mod(Λ), therefore an almost split sequence in Mod(Λ_{p,q}) is also almost split in Mod(Λ). We can then apply the functor F^{−1} to the almost split sequence for F (M) in Mod(Λ_{p,q}) to obtain the almost split sequence for F^{−1}F (M) ≅ M in Mod(R).

Indecomposable band modules in Mod(Λ_{p,q}) are of two types [BS, p. 118]:

(1) (W, V, g(x)), where W is an irreducible band word, g(x) ≠ x is an irreducible polynomial, and g(x) is the characteristic polynomial of an indecomposable automorphism of V;

(2) (W, V^{i}, g(x)^{i}) i ≥ 2, where W is an irreducible band word, g(x) ≠ x is an irreducible polynomial, and g(x)^{i} is the characteristic polynomial of an indecomposable automorphism of V^{i}.

The almost split sequences of band modules in Mod(Λ_{p,q}) for the two cases are given as follows [BS, p. 118].

(1)

0 ⟶ (W, V, g(x)) ⟶ (W, V^{2}, g(x)^{2}) ⟶ (W, V, g(x)) ⟶ 0 ;

(2)

0 ⟶ (W, V^{i}, g(x)^{i}) ⟶ (W, V^{i−1}, g(x)^{i−1})⊕
(W, V^{i+1}, g(x)^{i+1}) ⟶ (W, V^{i}, g(x)^{i}) ⟶ 0 .

In the first sequence, the middle term may or may not be indecomposable, while in the second sequence the summands in the middle term are necessarily indecomposable, by Lemma 3.2 and the computation of all irreducible maps in [BS, p. 115]. This completes the proof of the theorem. □

5. The non–artinian case

The non–artinian case is disposed of by explicitly computing the left and middle term of the almost split sequence from Theorem 1.1.

First let M be an indecomposable artinian R–module. Then the endomorphism ring End(M) is a local ring [CR, Prop. 6.10]. Let T be the unique simple End(M)– module, and let I be the End(M)–injective envelope of T. Define

D(M) = Hom_{End(M)}(M, I).

Then D(M) is again an R–module. Now suppose that M = T r(M′) for some indecomposable R–module M′. Then the unique simple \underline{End}(M′)^{op}–module is also the unique simple End(M)–module (see the discussion preceding Theorem 1.1). Hence D(M) = DT r(M′) is isomorphic to the left–hand term in the almost split sequence for M′ in Theorem 1.1. Thus, we can focus on D(M) instead.

We now return to the terminology of Levy. Let S = [e, f] be a basic building block with 2 ≤ e, f < ∞. Recall that [1, f; e, 1] = ([1, f]⊕[e, 1])/(0, π_{2}^{f−1}; π_{1}^{e−1}, 0)·k is a deleted cycle indecomposable (with a_{1} = b_{1} = 1).

The ring S ≅ R/(℘_{1}^{e} ⊕ ℘_{2}^{f}) is commutative and artinian. Multiplication is given by (r_{1}, s_{1})(r_{2}, s_{2}) = (r_{1}s_{1}, r_{2}s_{2}). Recall that (r, s) = (r + π_{1}^{e}, s + π_{2}^{f}) ∈ S, by abuse of notation. Let rad(S) denote the Jacobson radical of S. Then S/rad(S) ≅ k ≅ R/(℘_{1} ⊕ ℘_{2}). Note that [1, f; e, 1] is an S–module via (s_{1}, s_{2})(r, s; u, v) = (s_{1}r, s_{2}s; s_{1}u, s_{2}v). Note also that the socle of [1, f; e, 1] is isomorphic to k.

Lemma 5.1. Let S = [e, f], with 2 ≤ e, f < ∞. Then I = [1, f; e, 1] is an S– injective envelope of k ≅ S/rad(S).

Proof. We first prove that I is S–injective. Let J be an ideal of S and g : J ⟶ I an S-homomorphism. Then J is of the form ℘_{1}^{i} ⊕ ℘_{2}^{j} or S · y for y = (r_{1}π_{1}^{i}, r_{2}π_{2}^{j}) in S with 1 ≤ i ≤ e, 1 ≤ j ≤ f, and r_{1} ∈ R_{1} \ ℘_{1}, r_{2} ∈ R_{2} \ ℘_{2}.

In the first case, since (π_{1}, 0) ∈ S and g is an S-homomorphism, we have that g(π_{1}^{i}, 0) = (0, 0; π_{1}^{i}c′′, 0) for some c′′ ∈ R_{1}. To see this let g(π_{1}^{i}, 0) = (u, v; c, d). If i < e (otherwise (π_{1}^{i}, 0) = (0, 0)), then

0 = g(0, 0) = (π_{1}^{e−i}, 0)g(π_{1}^{i}, 0) = (π_{1}^{e−i}, 0)(u, v; c, d) = (0, 0; π_{1}^{e−i}c, 0) .

It follows that c = π_{1}^{i}c′ and d = 0, since d ∈ R_{1}/℘_{1} and d = v_{2}(c). Furthermore, 0 = (0, π_{2})g(π_{1}^{i}, 0) = (0, π_{2}v; 0, 0), which implies that v = π_{2}^{f−1}v′ and u = 0.

However, the deleted cycle relation (0, π_{2}^{f−1}; 0, 0) = (0, 0; −π_{1}^{e−1}, 0) in I implies that

g(π_{1}^{i}, 0) = (0, π_{2}^{f−1}v′; π_{1}^{i}c′, 0) = (0, 0; π_{1}^{i}c′′, 0)

for some c′′. A symmetric argument shows that g(0, π_{2}^{j}) = (0, π_{2}^{j}v′′; 0, 0) for some v′′, if j < f. If J is not the zero ideal, then at least one of i < e, j < f has to hold, so that g : J ⟶ I extends to a well defined S-homomorphism h : S ⟶ I where

h(1_{S}) = h(1, 1) = (v′′ + ℘_{1}, v′′; c′′, c′′ + ℘_{2}) .

In the second case, a similar argument shows that g(y) = (0, π_{2}^{j}b′′; π_{1}^{i}c′′, 0) for some b′′ and c′′, and again g extends. We conclude that I is S-injective. On the other hand, the socle of I is isomorphic to k. Thus, the S-injective envelope of k is a summand of I, hence equal to I, since I is indecomposable. □

The next proposition gives a way to compute D(M) for an indecomposable artinian R-module without computing any further injective envelopes (which is very difficult, in general). We will denote the length of an artinian R-module M by λ_{R}(M). If M is an S-module, then I_{S}(M) denotes the S-injective envelope of M.

Proposition 5.2. Suppose that M is an indecomposable artinian R-module. Let S = R/ann_{R}(M). Then D(M) ≅ Hom_{S}(M, I_{S}(k)) as R-modules.

Proof. As remarked in the proof of Lemma 5.1, the ideal ann_{R}(M) is of the form ℘_{1}^{e} ⊕ ℘_{2}^{f} or Rx for x = (aπ_{1}^{e−1}, π_{2}^{f−1}) with a ∈ R_{1} \ ℘_{1}. Thus, R/ann_{R}(M) ≅ S = [e, f] or, respectively, S = ⟨e, f⟩. Recall that the R-endomorphism ring End(M) is a local artinian S-algebra with T = End(M)/rad(End(M)) a simple End(M)- module.

The asserted isomorphism will be induced by an S-homomorphism

γ : I_{End(M)}(T) ⟶ I_{S}(k)

defined as follows. Let α : k ⟶ I_{End(M)}(T) be an S-embedding of k whose image is contained in the S-socle of T, and let β : k ⟶ I_{S}(k) be an S-embedding. Since I_{S}(k) is the S-injective envelope of k, the map β extends to an S-homomorphism γ such that γ ∘ α = β. Now define

μ : D(M) = Hom_{End(M)}(M, I_{End(M)}(T)) ⟶ Hom_{S}(M, I_{S}(k))

by μ(f) = γ ∘ f. Since S is contained in End(M), μ(f) is an S-homomorphism, so that μ is well defined.

To show that μ is a monomorphism, assume that f ≠ 0 and μ(f) = 0. Then f is an End(M)-homomorphism such that 0 ≠ f(M) = End(M) · f(M). Then f(M) ∩ α(k) = 0 since μ(f) = γ ∘ f = 0 and γ ∘ α = β is a monomorphism. On the other hand, since T is End(M)-simple, we have T = End(M) · α(k). Also, T is essential in its injective envelope I_{End(M)}(T), so that f(M) ∩ (End(M) · α(k)) ≠ 0. Now let m ∈ M such that 0 ≠ f(m) = h · x for some h ∈ End(M) and x ∈ α(k). Since rad(End(M)) annihilates T and therefore α(k) and End(M) is local, it follows that h is a unit. Therefore x = h^{−1}f(m) = f(h^{−1}m) ∈ f(M) ∩ α(k), which is a contradiction.

Finally, to show that μ is an epimorphism, we compute the lengths of both modules. Observe that λ_{S}(Hom_{S}(M, I_{S}(k))) = λ_{S}(M) and λ_{End(M)}(D(M)) = λ_{End(M)}(M) [Ka, Theorem 220]. Thus

\begin{aligned}λ_{S}(D(M)) & = λ_{S}(End(M))λ_{End(M)}(D(M)) = λ_{S}(End(M))λ_{End(M)}(M) \\ & = λ_{S}(M) = λ_{S}(Hom_{S}(M, I_{S}(k))) \end{aligned}

and μ is onto, as desired. □

Next we need to compute D of some modules, needed in the proofs of the main theorems of this section. We use the convention that [1, 1; m, n] = [m, n] = [m, n; 1, 1].

Corollary 5.3. If S = [e, f], with 1 ≤ e, f < ∞, then D(S) ≅ I_{S}(k) = [1, f; e, 1], and D(I_{S}(k)) ≅ S.

Proof. In this case, S ≅ R/(℘_{1}^{e} ⊕ ℘_{2}^{f}) = R/ann_{R}(S) = R/ann_{R}([1, f; e, 1]). The first assertion follows immediately from Proposition 5.2. To see the second assertion, observe that S is contained in D(I_{S}(k)) and λ_{S}(S) = λ_{S}(I_{S}(k)) = λ_{S}(D(I_{S}(k))). □

If M is indecomposable artinian, then so is D(M). This follows from the fact that T r preserves indecomposables, is a duality, and DT r(M) is the left hand term of an almost split sequence (Theorem 1.1), hence has a local endomorphism ring. The following corollary shows that D is also a duality.

Corollary 5.4. Suppose that M is indecomposable and artinian. Then D^{2}(M) ≅ M.

Proof. By Proposition 5.2 we have that D(M) ≅ F (M) = Hom_{S}(M, I_{S}(k)) with S = R/ann_{R}(M). But

ann_{R}(M) ≅ ann_{R}(F (M)) ≅ ann_{R}(D(M)) ,

therefore R/ann_{R}(M) ≅ R/ann_{R}(D(M)). Since M embeds in F^{2}(M), and

λ_{S}(M) = λ_{S}(F^{2}(M))

[Ka, Theorem 220], we obtain D^{2}(M) ≅ F^{2}(M) ≅ M, as desired. □

Theorem 5.5. Let M = [∞, f_{1}; e_{2}, f_{2}; …; e_{n}, ∞] ∈ N A_{1}. Then (1) T r(M) = [e_{n} − 1, f_{n−1}; e_{n−1}, f_{n−2}; …; e_{3}, f_{2}; e_{2}, f_{1} − 1] ∈ A_{∞};

(2) DT r(M) = [1, f_{1} − 1; e_{2}, f_{2}; …; e_{n−1}, f_{n−1}; e_{n} − 1, 1] ∈ A_{∞}.

Proof. (1) The module M is the cokernel of a homomorphism 0 ⟶ R^{n−1} ⟶ R^{n} described by the following n × (n − 1)-matrix [AL1, proof of Prop. 3.2 ]

H = ( \begin{matrix}f_{1} & 0 & & … & & 0 \\ e_{2} & f_{2} & 0 & … & & 0 \\ 0 & e_{3} & f_{3} & … & & 0 \\ . & . & . \\ 0 & 0 & & ... & e_{n−1} & f_{n−1} \\ 0 & 0 & & … & 0 & e_{n} \end{matrix} )

where, in order to simplify notation, f_{i} denotes (0, π_{2}^{f_{i}−1}) and e_{i} stands for (π_{1}^{e_{1}−1}, 0), etc.. The columns of H are the deleted cycle relations for M and the rows specify basic building blocks. Then, by definition, T r(M) is the cokernel of the map R^{n} ⟶ R^{n−1} with matrix H^{t}. Since the columns of H^{t} are the deleted cycle relations for T r(M) and the rows specify basic building blocks, it follows that T r(M) is as claimed (reading columns from right to left).

(2) In view of (1), it is sufficient to assume that

M = [e_{1}, f_{1}; …; e_{n}, f_{n}] ,

and to show that

D(M) ≅ M′ = [1, f_{n}; e_{n}, f_{n−1}; …; e_{2}, f_{1}; e_{1}, 1] .

The case n = 1 follows from Corollary 5.3, so we may assume n ≥ 2. Let S = R/ann(M) = [e, f], where e = max{e_{i} | 1 ≤ i ≤ n}, f = max{f_{i} | 1 ≤ i ≤ n}. Then I_{S}(k) = [1, f; e, 1], by Lemma 5.1.

The S–socle of M is an n + 1–dimensional k–vector space, thus

I_{S}(M) ≅ I_{S}(socle(M)) ≅ I_{S}(k)^{n+1}.

Therefore, we have an injective envelope

0 ⟶ M ⟶^{α} I_{S}(k)^{n+1} ⟶ L ⟶ 0

in Mod(S). Applying the functor D(−) ≅ Hom_{S}(−, I_{S}(k)) to this sequence gives the exact sequence

0 ⟶ D(L) ⟶ D(I_{S}(k)^{n+1}) ⟶^{D(α)} D(M) ⟶ 0 .

Since an element in D(I_{S}(k)) is just given by multiplication with an element of S, we can make the identification S ≅ D(I_{S}(k)) (Corollary 5.3).

We now define a specific embedding

α : M ⟶ I_{S}(k)^{n+1} ≅ [1, f; e, 1]^{n+1}

by

α(r_{1}, s_{1}; …; r_{n}, s_{n}) = (x_{n}, s_{n}π_{2}^{f−f_{n}}; 0, 0) ⊕ (x_{n−1}, s_{n−1}π_{2}^{f−f_{n−1}}; r_{n}π_{1}^{e−e_{n}}, y_{n−1})⊕

(x_{n−2}, s_{n−2}π_{2}^{f−f_{n−2}}; r_{n−1}π_{1}^{e−e_{n−1}}, y_{n−2}) ⊕ ⋯

⊕ (x_{1}, s_{1}π_{2}^{f−f_{1}}; r_{2}π_{1}^{e−e_{2}}, y_{1}) ⊕ (0, 0; r_{1}π_{1}^{e−e_{1}}, y_{n+1}) .

Here, x_{i}, resp. y_{i} denotes the image in k of the element to the right, resp. left, of it. A straightforward computation shows that α is a well-defined homomorphism and injective. We now show that ker(D(α)) contains the column space of the (n + 1) × (n + 2)-matrix (with entries in S):

A = ( \begin{matrix}1 & f_{n} & 0 & … & 0 & 0 & 0 \\ 0 & e_{n} & f_{n−1} & … & 0 & 0 & 0 \\ 0 & 0 & e_{n−1} & … & & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & … & e_{2} & f_{1} & 0 \\ 0 & 0 & 0 & … & 0 & e_{1} & 1 \end{matrix} )

In order to see this, identify an element of D(I_{S}(k))^{n+1} with an (n + 1)-tuple (a_{1}, …, a_{n+1}) in S^{n+1}. Then

D(α)(a_{1}, …, a_{n+1})(r_{1}, s_{1}; …; r_{n}, s_{n}) =

a_{1}(x_{n}, s_{n}π_{2}^{f−f_{n}}; 0, 0) + ⋯ + a_{n}(0, 0; r_{1}π_{1}^{e−e_{1}}, y_{n+1}) ∈ I_{S}(k).

Here, an element a_{i} ∈ S is viewed as an element of [1, f; e, 1] via the identification (r, s) = (x, s; r, y), where x, resp. y, is the image in k of the element to the right, resp. left, of it. If we now consider, for instance, the second column of A, then we compute

D(α)((0, π_{2}^{f_{n}−1}), (π_{1}^{e_{n}−1}, 0), 0, …, 0))(r_{1}, s_{1}; …; r_{n}, s_{n}) =

(0, π_{2}^{f_{n}−1}; 0, 0)·(x_{n}, s_{n}π_{2}^{f−f_{n}}; 0, 0)+(0, 0; π_{1}^{e_{n}−1}, 0)·(x_{n−1}, s_{n−1}π_{2}^{f−f_{n−1}}; r_{n}π_{1}^{e−e_{n}}, y_{n−1})

= (r_{n}, s_{n})(0, π_{2}^{f−1}; π_{1}^{e−1}, 0) = 0 ∈ I_{S}(k) .

The computation for the other columns is similar.

The columns of A are the relations for M′ and the rows represent the basic building blocks. Thus, we obtain a commutative diagram

0 ⟶  N  ⟶ S^{n+1} ⟶ M′ ⟶ 0
↓  ^{=}↓  ↓
0 ⟶ ker(D(α)) ⟶ S^{n+1} ⟶ D(M) ⟶ 0.

The right vertical map is onto by the Snake–Lemma. Both, M′ and D(M) are artinian S–modules. We now compute the S–lengths of both of these modules.

Recall from the proof of Prop. 5.2 that λ_{S}(M) = λ_{S} (D(M)). Furthermore, recall from Section 2 that M is defined by the short exact sequence

0 ⟶ k^{n−1} ⟶ ⨁_{i}[e_{i}, f_{i}] ⟶ M ⟶ 0.

Therefore,

λ_{S} (D(M)) = ∑_{i} λ_{S}([e_{i}, f_{i}]) − (n − 1).

A straightforward computation shows that λ_{S}([e_{i}, f_{i}]) = e_{i} + f_{i} − 1. Hence,

λ_{S} (D(M)) = ∑ e_{i} + ∑ f_{i} − 2n + 1.

A similar computation for M′ shows that λ_{S}(M′) = λ_{S} (D(M)). Hence the right vertical map in the above diagram is actually an isomorphism. This completes the proof of the theorem. □

Theorem 5.6. Let

M = [∞, f_{1}; e_{2}, f_{2}; …; e_{n−1}, f_{n−1}; e_{n}, ∞] ∈ N A_{1} .

There is an almost split sequence

0 ⟶ DT r(M) ⟶ B ⟶ M ⟶ 0

in Mod(R), where

DT r(M) = [1, f_{1} − 1; e_{2}, f_{2}; …; e_{n−1}, f_{n−1}; e_{n} − 1, 1] ∈ A_{∞} ,

as in Theorem 5.5.2, and

B = B_{1} ⊕ B_{2} = [∞, f_{1}; e_{2}, f_{2}; …; e_{n} − 1, 1] ⊕ [1, f_{1} − 1; e_{2}, f_{2}; …; e_{n}, ∞]

is the direct sum of two indecomposables in N A_{∞}.

Proof. The map μ : DT r(M) ⟶ B in the above sequence is defined by

μ(r_{1}, s_{1}; …; r_{n}, s_{n}) = (0, π_{2}s_{1}; r_{2}, s_{2}; …; r_{n}, s_{n}) ⊕ (−r_{1}, −s_{1}; …; −π_{1}r_{n}, 0) ,

and the map ϕ : B ⟶ M is given by

ϕ ((r_{1}, s_{1}; …; r_{n}, s_{n}) ⊕ (u_{1}, v_{1}; …; u_{n}, v_{n})) = (r_{1}, s_{1} + v_{1}π_{2}; r_{2} + u_{2}, s_{2} + v_{2}; …; r_{n}π_{1} + u_{n}, v_{n}) .

A routine verification shows that μ and ϕ are well defined, that μ is one-to-one, ϕ is onto and ϕ ∘ μ = 0. A slightly more complicated computation using the deleted cycle relations of B_{1} or B_{2} shows that ker(ϕ) = im(μ). Thus the sequence

0 ⟶ DT r(M) ⟶^{μ} B ⟶^{ϕ} M ⟶ 0

is exact. It is not split by the Krull-Schmidt-Azumaya Theorem, since M, DT r(M), B_{1} and B_{2} are indecomposable, pairwise non-isomorphic, and both M and DT r(M) have local isomorphism rings.

Since DT r(M) is artinian, it is sufficient, by Lemma 3.1 (1), to show that, if N is indecomposable and h : N ⟶ M is not a split epimorphism, then there exists α : N ⟶ B such that ϕ ∘ α = h. Let ϕ = ϕ_{1} ⊕ ϕ_{2} : B = B_{1} ⊕ B_{2} ⟶ M.

First, let N = [e, f] be a basic building block. If the [e_{n}, ∞]-coordinate of h(N) in M is contained in (π_{1}, 0) · M, then there exists an α : N ⟶ B_{1} such that ϕ ∘ α = ϕ_{1} ∘ α = h |_{N}, as a consequence of the definition of ϕ. Likewise, if the [∞, f_{1}]-coordinate of h(N) is contained in (0, π_{2}) · M, then h can be lifted through B_{2}. In particular, if e, f < ∞, then h can be lifted through either B_{1} or B_{2}, recalling that R_{i}/℘_{i}^{∞} = R_{i} is torsion free while R_{i}/℘_{i}^{e} is torsion. If N = [∞, f], resp. [e, ∞], then h factors through B_{1}, resp. B_{2}.

For the general case, let N = S_{N}/K_{N}, where S_{N} is a direct sum of basic building blocks, and K_{N} is a k-space with basis the deleted cycle relations and, possibly, a block cycle relation of N. From the above it now follows that h factors through one of B_{1} or B_{2}, unless N = [∞, f′_{1}; e′_{2}, …; e′_{n}, ∞]. This is because μ is a diagonal map and ϕ is a codiagonal map except at the first and last basic building blocks.

In the latter case, first assume that N = [∞, f; e, ∞]. Then the submodule K_{N} = k · (0, π_{2}^{f−1}, π_{1}^{e−1}, 0). Let

h(1, 1; 0, 0) = (r_{1}, s_{1}; …; r_{n}, s_{n}) and h(0, 0; 1, 1) = (r′_{1}, s′_{1}; …; r′_{n}, s′_{n}) .

Then

0 = h(0) = h(0, π_{2}^{f−1}; π_{1}^{e−1}, 0) = (0, π_{2}^{f−1})h(1, 1; 0, 0) + (π_{1}^{e−1}, 0)h(0, 0; 1, 1) .

Since the deleted cycle relations of M have 0 in the first and last coordinates, this implies that π_{2}^{f−1}s_{n} = 0 and π_{1}^{e−1}r′_{1} = 0. But M has an ∞ at the left and right end, so that s_{n} = 0 and r′_{1} = 0. This implies in turn that r_{n} is divisible by π_{1}, and s′_{1} is divisible by π_{2}. The above equation furthermore shows that

π_{2}^{f−1}s_{i} = a_{i}π_{2}^{f_{i}−1}  and  π_{1}^{e−1}r′_{i+1} = a_{i}π_{1}^{e_{i+1}−1}

for some a_{i} ∈ k and 1 ≤ i ≤ n − 1, which can be seen using the deleted cycle relations of M.

If n is odd, then define a lifting α : N ⟶ B = B_{1} ⊕ B_{2}by

α(a, b; c, d) = x_{1} ⊕ x_{2} ,

where

x_{1} = (ar_{1}, bs_{1} + ds′_{1}; cr′_{2}, ds′_{2}; ar_{3}, bs_{3}; …; cr′_{n−1}, ds′_{n−1}; 0, 0) ,
x_{2} = (0, 0; ar_{2}, bs_{2}; cr′_{3}, ds′_{3}; …; ar_{n−1}, bs_{n−1}; ar_{n} + cr′_{n}, ds′_{n}) .

The properties of the entries of h(1, 1; 1, 1), derived above, now imply that α is well defined. Furthermore, it now follows that

ϕ ∘ α(a, b; c, d) = (a, b)h(1, 1; 0, 0) + (c, d)h(0, 0; 1, 1) = h(a, b; c, d) ,

as desired.

If n is even, then define α(a, b; c, d) = x_{1} ⊕ x_{2}, where

x_{1} = (ar_{1}, bs_{1} + ds′_{1}; cr′_{2}, ds′_{2}; …; cr′_{n−2}, ds′_{n−2}; 0, 0; 0, 0) ,
x_{2} = (0, 0; ar_{2}, bs_{2}; …; ar_{n−1} + cr′_{n−1}, bs_{n−1} + ds′_{n−1}; ar_{n} + cr′_{n}, ds′_{n}) .

Once again, α is well defined, and ϕ ∘ α = h.

For the general case N = [∞, f′_{1}; …; e′_{n}, ∞], examine h applied to elements of N with (1, 1) in the [e′_{i}, f′_{i}] block of N and 0’s everywhere else, for 1 ≤ i ≤ n, and then construct α as above. □

References

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David M. Arnold, Baylor University, Department of Mathematics, Waco, TX 76798-7328

E-mail address: arnoldd@baylor.edu

Reinhard C. Laubenbacher, New Mexico State University, Department of Mathematics, Las Cruces, NM 88003

E-mail address: rlaubenb@nmsu.edu