Degenerations of finite dimensional modules are given by extensions

GrzegorzZwara (gzwara@mat.uni.torun.pl)

Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Abstract. Let A be a finite dimensional k-algebra over algebraically closed field k and mod A be the category of finite dimensional left A-modules. We show that a module M in mod A degenerates to another module N in mod A if and only if there is an exact sequence 0 → N → M ⊕ Z → Z → 0 in mod A for some A-module Z. Moreover, we give a description of minimal degenerations of finite dimensional A-modules.

Keywords: modules, degenerations

Mathematics Subject Classifications: 14L30, 16G10, 16G70

1. Introduction and main results

Let A be a finite dimensional associative k-algebra with an identity over an algebraically closed field k and modA be the category of finite dimensional left A-modules. If a_{1} = 1, …, a_{α} is a basis of A over k, we have the structure constants a_{ijk} defined by a_{i}a_{j} = ∑ a_{ijk} a_{k}. The affine variety mod _{A}^{d}(k) of d-dimensional unital left A-modules consists of α-tuples m = (m_{1}, …, m_{α}) of d × d-matrices with coefficients in k such that m_{1} is the identity matrix and m_{i}m_{j} = ∑ a_{ijk} m_{k} holds for all indices i and j. Any such α-tuple m corresponds to a d-dimensional module M ∈ modA in the obvious way. The general linear group Gl_{d}(k) acts on mod _{A}^{d}(k) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules in modA (see [7]). We denote by O(m) the Gl_{d}(k)-orbit of a point m in mod _{A}^{d}(k). By abuse of notation N is a degeneration of M if n belongs to the Zariski closure \overline{O(m)} of O(m) in mod _{A}^{d}(k), and we denote this fact by M ≤_{deg} N. Thus ≤_{deg} is a partial order on the set of isomorphism classes of A-modules of a given (finite) dimension. It was not clear how to characterize ≤_{deg} in terms of representation theory.

Let M, N, Z be modules in modA such that there is an exact sequence in modA of one of the forms

0 → N → M ⊕ Z → Z → 0  or  0 → Z → Z ⊕ M → N → 0.

In [11] Riedtmann proved that then M ≤_{deg} N. We shall show that the reverse implication is also true.

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THEOREM 1. Let m, n be points in mod _{A}^{d}(k), d ≥ 1. Then the following conditions are equivalent:

(1) M ≤_{deg} N.

(2) There is a short exact sequence 0 → N → M ⊕ Z → Z → 0 in modA for some module Z in modA.

(3) There is a short exact sequence 0 → Z → Z ⊕ M → N → 0 in modA for some module Z in modA.

As a direct consequence of Theorem 1 and the proof of Proposition 3.4 in [11] we get

COROLLARY 2. Let m, n be points in mod _{A}^{d}(k) such that M ≤_{deg} N. Then there is a nonempty open subset C of k, a morphism μ : C → \overline{O(m)} and a point c_{0} in C, such that μ(c_{0}) = n and μ(c) ∈ O(m) for all c ≠ c_{0}.

Following Abeasis and del Fra [1] we may consider another partial order ≤ext defined as follows:

• M ≤ext N: ⇔ there are modules M_{i}, U_{i}, V_{i} and short exact sequences 0 → U_{i} → M_{i} → V_{i} → 0 in modA such that M = M_{1}, M_{i+1} = U_{i} ⊕ V_{i}, 1 ≤ i ≤ s, and N = M_{s+1} for some natural number s.

Then for modules M and N in modA the following implication holds:

M ≤ext N ⟹ M ≤_{deg} N

(see [3], [11]). Observe that for any modules M, N in modA with M <ext N, the module N is decomposable. Since there exist proper degenerations to indecomposable modules even for very simple representation-finite algebras (see [11]), the reverse implication is not true in general. Our next result concerns degenerations M <_{deg} N which are not given by a sequence of the form 0 → N′→ M → N′′→ 0 with N = N′⊕ N′′.

THEOREM 3. Let M, N, N′, N′′ be modules in modA such that M ≤_{deg} N and N ≃ N′⊕ N′′. If every exact sequence in modA of the form

0 → N′→ W → N′′→ 0 or 0 → N′′→ W → N′→ 0

with M ≤_{deg} W is splittable (W ≃ N′⊕N′′), then there are modules M′, M′′in modA such that M′≤_{deg} N′, M′′≤_{deg} N′′and M ≃ M′⊕ M′′.

It will give us the following theorem about minimal degenerations.

THEOREM 4. Let M, N be modules in modA such that N is a minimal degeneration of M. Then M <ext N or there are modules W, \overline{M}, \overline{N} in modA such that M ≃ W ⊕ \overline{M}, N ≃ W ⊕ \overline{N}, \overline{M} <_{deg} \overline{N} and the module \overline{N} is indecomposable.

As a direct consequence of Theorem 4 we get the following fact.

COROLLARY 5. The orders ≤ext and ≤_{deg} are equivalent for all modules in modA if and only if for any modules M, N in modA with M <_{deg} N, the module N is decomposable.

It is known (see Corollary 2 in [15]) that, if A is an algebra and for any proper degeneration M <_{deg} N of A-modules the module N is decomposable, then A is tame, that is, the indecomposable Amodules occur, in each dimension d, in a finite number of discrete and a finite number of one-parameter families. Recall also that an algebra A is called quasi-tilted if A is of global dimension at most 2 and each indecomposable finite dimensional A-module has projective dimension at most one or injective dimension at most one. The structure of tame quasi-tilted algebras and their module categories has been described by Skowroński in [13]. Then, applying results of [14], we proved in our joint paper ([15], Theorem 3) that for any proper degeneration M <_{deg} N of modules over a tame quasi-tilted algebra A, the module N is decomposable. Applying Corollary 5 we may reformulate it now as follows.

COROLLARY 6. Let A be a tame quasi-tilted algebra. Then the orders ≤ext and ≤_{deg} are equivalent for all modules in modA.

The paper is organized as follows. In Section 2 we give characterisations of splittable exact sequences and introduce the notion of an affine scheme mod _{A}^{d}, playing a fundamental role in our proofs of Theorems 1 and 3. Sections 3, 4 and 5 are devoted to the proofs of Theorems 1, 3 and 4, respectively.

For basic background on the topics considered here we refer to [3], [4], [7] and [12]. The author would like to thank C. M. Ringel and A. Skowroński for helpful suggestions and comments during the preparation of this paper. The author also gratefully acknowledges support from the Polish Scientific Grant KBN No. 2 PO3A 012 14 and Sonderforschungsbereich 343 (Universität Bielefeld).

2. Preliminary results

2.1. Throughout the paper A denotes a fixed finite dimensional associative k-algebra with an identity over an algebraically closed field k. We denote by modA the category of finite dimensional left A-modules.

Let R be a ring and d′, d′′be two natural numbers. We denote by M_{d′×d′′}^{α}(R) the set of all α-tuple of d′× d′′matrices with coefficients in R (so mod _{A}^{d}(k) ⊆ M_{d×d}^{α} (k)). Clearly, M_{d′×d′′}^{α} is a functor from the category of rings to the category of sets. Let m = (m_{1}, …, m_{α}) belongs to M_{d′×d′′}^{α}(R) and h′ (resp. h′′) be any c′× d′ (resp. d′′× c′′) matrix with coefficients in R, for some natural numbers c′, c′′. Then we define

h′⋆ m = (h′m_{1}, …, h′m_{α}) ∈ M_{c′×d′′}^{α}(R),
m ⋆ h′′ = (m_{1}h′′, …, m_{α}h′′) ∈ M_{d′×c′′}^{α}(R).

In particular, if g ∈ Gl_{d}(k) and m ∈ mod _{A}^{d}(k), then g · m = g ⋆ m ⋆ g^{−1} defines the action of Gl_{d}(k) on mod _{A}^{d}(k) that was mentioned in the introduction. 2.2. For an exact sequence Σ : 0 → U → W → V → 0 in modA we define an additive function δ_{Σ} from modA to the set of integers as follows:

δ_{Σ} (X) = dim_{k} Hom_{A}(U ⊕ V, X) − dim_{k} Hom_{A}(W, X),

for any module X in modA. Then the following fact holds: LEMMA. Let Σ : 0 → U → W → V → 0 be an exact sequence in modA. Then δ_{Σ} (X) ≥ 0 for any module X in modA. Moreover, the following conditions are equivalent:

(1) the sequence Σ is splittable,

(2) W ≃ U ⊕ V,

(3) δ_{Σ} (X) = 0 for any module X in modA.

Proof. The exact sequence Σ : 0 → U−→^{f} W −→^{g} V → 0 induces the following exact sequence

0 → Hom_{A}(V, X)−→^{g^{⋆}} Hom_{A}(W, X)−→^{f^{⋆}} Hom_{A}(U, X),

what leads to δ_{Σ} (X) ≥ 0, for any module X in modA. Clearly, the condition (1) implies (2) and the condition (2) implies (3).

Assume that δ_{Σ} (X) = 0 for any module X in modA. In particular, δ_{Σ} (U) = 0 and hence the sequence

0 → Hom_{A}(V, U)−→^{g^{⋆}} Hom_{A}(W, U)−→^{f^{⋆}} Hom_{A}(U, U) → 0

is exact. This implies that 1_{U} = f^{⋆}(γ) = γ ∘f, for some homomorphism γ in Hom_{A}(W, U). Then the sequence Σ is splittable and the condition (1) holds.

2.3. Now we give another characterisation of splittable sequences. Let d, d′, d′′be natural numbers with d = d′+d′′and let u, w, v be points in mod _{A}^{d′}(k), mod _{A}^{d}(k), mod _{A}^{d′′}(k), respectively. Then the following lemma holds (see II.2.7 in [8]):

LEMMA. Assume that w = [ \begin{matrix}u & z \\ 0 & v \end{matrix} ] for some z ∈ M_{d′×d′′}^{α}(k). Then

there is an exact sequence Σ in modA of the form 0 → U → W → V →

0. Moreover, the sequence Σ is splittable if and only if z = u ⋆ h − h ⋆ v, for some d′× d′′matrix h with coefficients in k. 2.4. The affine variety mod _{A}^{d}(k) extends to an affine algebraic k-scheme mod _{A}^{d} (see, for example [2], [4], [10]). The scheme mod _{A}^{d} may be described in the functorial point of view as follows: for any commutative k-algebra R with an identity, the set mod _{A}^{d}(R) consists of α-tuples m = (m_{1}, m_{2}, …, m_{α}) in M_{d×d}^{α} (R), such that m_{1} is the identity matrix and m_{i}m_{j} = ∑ a_{ijk} m_{k} holds for all indices i and j. Clearly, any αtuple m ∈ mod _{A}^{d}(R) gives the A-R-bimodule structure on R^{d} with the natural action of R. The affine algebraic group scheme Gl_{d} acts on mod _{A}^{d} by conjugation.

Let u be a point in mod _{A}^{d}(R) and U be the corresponding A-R-bimodule. Observe that, for any homomorphism φ : R → S of kalgebras, the A-S-bimodule corresponding to the point mod _{A}^{d}(φ)(u) in mod _{A}^{d}(S) is isomorphic to U ⊗_{R} S. 2.5. Since any two points on an irreducible variety can be connected by an irreducible curve (see A.I.4.5 in [8]), then we get the following characterization of orbit closure (see Theorem 1.2 in [5], also [9]). PROPOSITION. Let m and n be any points in mod _{A}^{d}(k). Then M ≤_{deg} N if and only if there is a discrete valuation k-algebra R with the maximal ideal 𝔪 and residue field R/𝔪 = k, whose quotient field K is finitely generated over k of transcendence degree one, and an element y ∈ mod _{A}^{d}(R) such that

• mod _{A}^{d}(τ)(y) = g · (mod _{A}^{d}(τη)(m)), for some g ∈ Gl_{d}(K)

• mod _{A}^{d}(π)(y) = n,

where η, τ and π are the canonical homomorphisms

k−→^{η} R −→^{τ} K
↓^{π}
R/𝔪 = k.

There is a geometric interpretation of this characterization. Such a k-algebra R and a field K correspond to the local ring of a point c_{0} of some nonsingular affine curve C and the field of rational function on C, respectively (see, for example, I.6 in [6]). Hence, M ≤_{deg} N if and only if there is a nonsingular affine curve C, a point c_{0} in C and a morphism μ : C → \overline{O(m)} such that μ(c_{0}) = n and μ(c) ∈ O(m) for c in an open dense subset of C (compare with the proof of Theorem 1.2 in [5]).

3. The proof of Theorem 1

3.1. (2) implies (1) and (3) implies (1), by Proposition 3.4 in [11]. Let m, n be points in mod _{A}^{d}(k), d ≥ 1. By transposing all matrices in m and n we get points m′and n′, respectively, in mod _{A′}^{d}(k), where A′denotes an opposite algebra of A. One sees that the module M degenerates to N if and only if the same holds for dual modules M′ and N′ over A′. Hence, it remains to show that M ≤_{deg} N implies that there is an exact sequence in modA of the form 0 → N → M ⊕ Z → Z → 0, for some A-module Z.

Thus assume that M ≤_{deg} N. We apply Proposition 2.5 and use the notation introduced there. Let Y be an A-R-bimodule on R^{d} corresponding to y. Denote by finR the category of finite dimensional (over k) R-modules. Since Y_{R} is a free R-module of finite rank, we have the exact functor F = _{A}Y ⊗_{R} (−) : finR → modA. For i ≥ 1, let N_{i} = F(R/𝔪^{i}) = Y/Y 𝔪^{i} ∈ modA. Since Mod_{A}^{d}(π)(y) = n, then

AN_{1} = _{A}(Y/Y 𝔪) = _{A}(Y ⊗_{R} R/𝔪) = _{A}N.

We fix an element f ∈ 𝔪\𝔪^{2}. Since R is a discrete valuation ring, then 𝔪 = (f) and consequently, 𝔪^{i} = (f^{i}), for any i ≥ 1. Since F is exact, the following exact sequences:

0 → R/𝔪−→^{β_{i}} R/𝔪^{i+1} −→^{γ_{i}} R/𝔪^{i} → 0, i ≥ 1,

where β_{i}(r + 𝔪) = f^{i} · r + 𝔪^{i+1} , γ_{i}(r + 𝔪^{i+1} ) = r + 𝔪^{i}, for all r ∈ R, gives the exact sequence

0 → N_{1} −→^{F(β_{i})}N_{i+1} −→^{F(γ_{i})}N_{i} → 0, i ≥ 1,

in modA. Since N_{1} = N, it remains to show that N_{i+1} ≃ M ⊕ N_{i} for some i ≥ 1. The remaining part of this section is devoted to the proof that, for sufficiently large h, there is an A-module isomorphism N_{h+1} ≃ M ⊕ N_{h}. 3.2. The assumption that mod _{A}^{d}(τ)(y) = g · (mod _{A}^{d}(τη)(m)), for some g ∈ Gl_{d}(K), means that the function g : M ⊗_{k} K → Y ⊗_{R} K is an isomorphism of A-K-bimodules. Let g = [g_{ij}]_{i,j≤d} , with g_{ij} ∈ K. Since K is the quotient field of the discrete valuation ring R, there is a number b ≥ 0 such that f^{b} · g_{ij} ∈ R, for all 1 ≤ i, j ≤ d. Let g̃ = f^{b} · g and observe that

g̃·((τη)_{∗}(m)) = g̃⋆((τη)_{∗}(m))⋆g̃^{−1} = g⋆((τη)_{∗}(m))⋆g^{−1} = g·((τη)_{∗}(m)),

where (τη)_{∗} = mod _{A}^{d}(τη). Thus we may assume that b = 0, g̃ = g, and consequently g_{ij} ∈ R, for all 1 ≤ i, j ≤ d. Then we get the monomorphism

φ = g|_{M⊗_{k}R} : M ⊗_{k} R → Y

of A-R-bimodules.

We note that results in the remaining part of this section and their proofs extend without any changes to an arbitrary field k. 3.3. Let Y be an A-R-bimodule which is, as an R-module, free of rank d. Assume that there is a monomorphism φ : M ⊗_{k} R → Y of A-R-bimodules, for some d-dimensional A-module M. We set X = imφ. LEMMA. There is a natural number t with Y 𝔪^{t} ⊆ X. In particular, the module Y/X is finite-dimensional.

Proof. Since φ is a monomorphism between free R-modules of rank d (and since R is a principal ideal domain), then Y/X, as an R-module, is isomorphic to

R/𝔪^{t_{1}} ⊕ R/𝔪^{t_{2}} ⊕ … ⊕ R/𝔪^{t_{d}}.

Thus, the R-module Y/X is annihilated by 𝔪^{t}, where t is the maximum of t_{1}, t_{2}, …, t_{d}. Hence, Y 𝔪^{t} ⊆ X.

LEMMA 3.4. There exists a natural number s such that X𝔪^{s} is a direct summand of the A-module Y.

Proof. Given a subset C of R, we denote by ⟨C⟩ the k-subspace generated by C. Let B be a k-basis of R. For b ∈ B, let M_{b} = φ(M⊗_{k}⟨b⟩). This is an A-submodule of X and is isomorphic to _{A}M. Of course, X = ⨁_{b∈B} M_{b}, and this is a direct decomposition of A-modules.

We devide the proof into several steps.

(1) There exists an A-submodule Z of Y which satisfies Z + X = Y and Z ∩ X = ⨁_{b∈V} M_{b} for some finite subset V of B.

Take a projective cover ξ : P → Y/X of the A-module Y/X and lift this map to Y. We obtain an A-module homomorphism ξ′ : P → Y, say with image Z_{0}, and such that Z_{0}+X = Y. By construction, Z_{0} is a finite dimensional A-submodule of Y. Consider Z_{0}∩X = Z_{0}∩⨁_{b∈B} M_{b}.

Since Z_{0} is finite dimensional, there is a finite subset V of B such that Z_{0} ∩ ⨁_{b∈B} M_{b} = Z_{0} ∩ ⨁_{b∈V} M_{b}. Let Z = Z_{0} + ⨁_{b∈V} M_{b}. Then Z is finite dimensional and Z ∩ X = ⨁_{b∈V} M_{b}.

(1’) As a consequence: Y = Z ⊕ C, where C = ⨁_{b∈B\V} M_{b}.

We are going to replace the direct summand C by an A-submodule C′ of Y which contains X𝔪^{s} as a submodule. Actually, X𝔪^{s} will be required to be even a direct summand of C′. This exchange will be done inside X: both C and C′ will be A-submodules of X, they will be direct complements of (Z ∩ X) in X. First, we deal with the ring R itself.

(2) Let V be a finite subset of B. Then there is a natural number s and a finite subset W of B such that

𝔪^{s} ⊕ ⟨W⟩ ⊕ ⟨V⟩ = R.

Since ⟨V⟩ is finite dimensional and ⋂_{i≥1} 𝔪^{i} = 0, then ⟨V⟩ ∩ 𝔪^{s} = 0, for some natural number s. Consider the subspace 𝔪^{s} ⊕ ⟨V⟩ of R. Since B is a K-basis of R, we can find a subset W of B as required.

(3) Of course, we may tensor the above decomposition of R with M over k and apply the monomorphism φ. Note first that φ(M ⊗_{k} 𝔪^{s}) = X𝔪^{s}. Second, denote φ(M ⊗_{k} ⟨W⟩) = ⨁_{b∈W} M_{b} by W. Third, recall that φ(M ⊗_{k} ⟨V⟩) = ⨁_{b∈V} M_{b} = Z ∩ X. Altogether we get

X𝔪^{s} ⊕ W ⊕ (Z ∩ X) = X.

As mentioned above we denote ⨁_{b∈B\V} M_{b} = C. Let C′ = X𝔪^{s} ⊕ W. Then we see that

C ⊕ (Z ∩ X) = X = C′⊕ (Z ∩ X).

Note that Y = C ⊕ Z, according to (1’).

(4) It follows that

Y = C′⊕ Z = X𝔪^{s} ⊕ W ⊕ Z,

thus W ⊕ Z is a direct complement to X𝔪^{s}.

Namely, we have C′∩ Z = C′∩ Z ∩ X = 0, since C′ ⊆ X. Also, C′+ Z = C′+ Z + (Z ∩ X) = Z + X = Y, since Z ∩ X ⊆ Z. This completes the proof.

PROPOSITION 3.5. Let Y be an A-R-bimodule which is, as an Rmodule, free of rank d. Assume that there is a monomorphism φ : M ⊗_{k} R → Y of A-R-bimodules, for some d-dimensional A-module M. Then, for a sufficiently large h, there is an A-module isomorphism

Y/Y 𝔪^{h+1} ≃ Y/Y 𝔪^{h} ⊕ M.

Proof. Recall that X = imφ. It follows from Lemma 3.4 that there exists a natural number s such that X𝔪^{s} is a direct summand of Y. Fix a direct complement Z′ of X𝔪^{s} in Y. Next observe that, for each i ≥ 0, X𝔪^{i+1} is a direct summand of X𝔪^{i}, with a direct complement isomorphic to M. Further, by Lemma 3.3 there is a natural number t with Y 𝔪^{t} ⊆ X. Write Z′′= X/Y 𝔪^{t}. For any natural number j, we have Y 𝔪^{t+j} ⊆ X𝔪^{j}. The multiplication by f^{j} (recall that 𝔪^{j} = (f^{j})) induces an isomorphism Z′′−→^{∼} X𝔪^{j}/Y 𝔪^{t+j} .

For any natural number i, consider now the following chain of inclusions:

Y 𝔪^{s+t+i} ⊆ X𝔪^{s+i} ⊆ X𝔪^{s} ⊆ Y.

It follows from the above remarks that the last two inclusions have direct complements, namely M^{i} and Z′, while the first factor X𝔪^{s+i} /Y 𝔪^{s+t+i} is isomorphic to Z′′. Thus we get

Y/Y 𝔪^{s+t+i} ≃ Z′′⊕ M^{i} ⊕ Z′.

Therefore,

Y/Y 𝔪^{s+t+i} ≃ Y/Y 𝔪^{s+t} ⊕ M^{i}.

In particular, for h = s + t and i = 1, we obtain Y/Y 𝔪^{h+1} ≃ Y/Y 𝔪^{h} ⊕ M. This completes the proof of Proposition 3.5 and also the proof of Theorem 1.

4. The proof of Theorem 3

4.1. Let d, d′, d′′ be natural numbers with d = d′ + d′′. Let m, n be points in mod _{A}^{d}(k) such that M ≤_{deg} N, and n′, n′′be points in mod _{A}^{d′}(k), mod _{A}^{d′′}(k), respectively, such that N ≃ N′⊕ N′′. Assume that every exact sequence in modA of the form

0 → N′→ W → N′′→ 0 or 0 → N′′→ W → N′→ 0

with M ≤_{deg} W, is splittable. We apply Proposition 2.5 for points m, n, and we use the notation introduced there. Denote by π_{i} : R → R/𝔪^{i} and ϵ_{i} : R/𝔪^{i+1} → R/𝔪^{i}, i ≥ 1, the natural epimorphisms of kalgebras. Clearly, then π_{i} = ϵ_{i}π_{i+1} , for any i ≥ 1. As in (3.1), let Y be the A-R-bimodule on R^{d} corresponding to y, let F be the exact functor AY ⊗_{R} (−) : finR → modA, and finally let N_{i} = F(R/𝔪^{i}) ∈ modA, for any i ≥ 1. Then N_{1} = N and the module N_{i} corresponds to the point mod _{A}^{d}(π_{i})(y). Then the following fact holds. LEMMA 4.2. For any i ≥ 1, there is a point y_{i} ∈ mod _{A}^{d}(R) such that (1) y_{i} = g_{i} · y, for some g_{i} ∈ Gl_{d}(R),

(2) mod _{A}^{d}(π_{i})(y_{i}) = ( \begin{matrix}n′_{i} & 0 \\ 0 & n′′_{i} \end{matrix}), for some elements n′_{i} in mod _{A}^{d′}(R/𝔪^{i})

and n′′_{i}in mod _{A}^{d′′}(R/𝔪^{i}),

(3) if i > 1, then mod _{A}^{d′}(ϵ_{i−1} )(n′_{i}) = n′_{i−1} and mod _{A}^{d′′}(ϵ_{i−1} )(n′′_{i}) = n′′_{i−1} .

Moreover, n′_{1} = n′ and n′′_{1}= n′′.

Proof. Since R/𝔪 = k, we may assume that πη = 1_{k}. The isomorphism N ≃ N′⊕ N′′means that there is an element g ∈ Gl_{d}(k) such

that g · n = ( \begin{matrix}n′ & 0 \\ 0 & n′′ \end{matrix}). Let g_{1} = Gl_{d}(η)(g) and y_{1} = g_{1} · y. Then

mod _{A}^{d}(π_{1})(y_{1}) = mod _{A}^{d}(π)(Gl_{d}(η)(g) · y)
= Gl_{d}(π)(Gl_{d}(η)(g)) · mod _{A}^{d}(π)(y)
= Gl_{d}(πη)(g) · n = g · n = ( \begin{matrix}n′ & 0 \\ 0 & n′′ \end{matrix}) .

Assume now that there is a point y_{i} ∈ mod _{A}^{d}(R) satisfying the condi-

tions (1), (2) and (3), for some i ≥ 1. Then y_{i} = [ \begin{matrix}y′_{i} & ũ \\ ṽ & y′′_{i} \end{matrix}], for some

points y′_{i}, ũ, ṽ, y′′_{i}in M_{d′×d′}^{α} (R), M_{d′×d′′}^{α}(R), M_{d′′×d′}^{α} (R), M_{d′′×d′′}^{α}(R), respectively. Applying (2) we get

M_{d′×d′}^{α} (π_{i})(y′_{i}) = n′_{i}  and  M_{d′′×d′′}^{α}(π_{i})(y′′_{i}) = n′′_{i}.

Moreover, since kerπ_{i} = 𝔪^{i} = (f^{i}), then ũ = f^{i} · u and ṽ = f^{i} · v, for

some u ∈ M_{d′×d′′}^{α}(R) and v ∈ M_{d′′×d′}^{α} (R). Take g′= [ \begin{matrix}f^{i} · 1_{d′} & 0 \\ 0 & 1_{d′′} \end{matrix} ] ∈

Gl_{d}(K) and l = [ \begin{matrix}y′_{i} & u \\ f^{2i} · v & y′′_{i} \end{matrix}] ∈ mod _{A}^{d}(R). Then mod _{A}^{d}(τ)(l) equals

g′ · mod _{A}^{d}(τ)(y_{i}). We set w = mod _{A}^{d}(π)(l). Then w = [ \begin{matrix}n′ & û \\ 0 & n′′ \end{matrix}] ∈

mod _{A}^{d}(k), where û = M_{d′×d′′}^{α}(π)(u). Applying Lemma 2.3 we obtain an exact sequence Σ in modA of the form 0 → N′→ W → N′′→ 0. Observe that mod _{A}^{d}(τ)(l) = (g′Gl_{d}(τ)(g_{i}))·mod _{A}^{d}(τ)(y) = (g′Gl_{d}(τ)(g_{i})g)· mod _{A}^{d}(τη)(m), by (1). Then M ≤_{deg} W, by Proposition 2.5. Hence, by our assumptions, the sequence Σ is splittable. Consequently, by Lemma 2.3, there is a d′× d′′matrix ĥ′ with coefficients in k, such that û = n′⋆ĥ′−ĥ′⋆n′′. This implies that u−y′_{i}⋆h′+h′⋆y′′_{i}= f ·z′, for some z′∈ M_{d′×d′′}^{α}(R), where h′is a d′×d′′matrix with coefficients in R, and these coefficients are the images of the corresponding coefficients of ĥ′ via the homomorphism η. Dually, we conclude that there is a d′′× d′ matrix h′′with coefficients in R such that v − y′′_{i}⋆ h′′+ h′′⋆ y′_{i}= f · z′′, for some z′′∈ M_{d′′×d′}^{α} (R). Consider the d × d matrix

h = [ \begin{matrix}1_{d′} & f^{i} · h′ \\ 0 & 1_{d′′} \end{matrix}] · [ \begin{matrix}1_{d′} & 0 \\ f^{i} · h′′ & 1_{d′′} \end{matrix} ]

with coefficients in R. Then h ∈ Gl_{d}(R), since deth = 1_{R} ∈ R \ 𝔪. We set y_{i+1} = h · y_{i} and g_{i+1} = h · g_{i}. Then y_{i+1} = g_{i+1} · y and the

condition (1) holds for i + 1. Observe that y_{i+1} equals [ \begin{matrix}1_{d′} & f^{i} · h′ \\ 0 & 1_{d′′} \end{matrix}] ·

[ \begin{matrix}1_{d′} & 0 \\ f^{i} · h′′ & 1_{d′′} \end{matrix} ]⋆[ \begin{matrix}y′_{i} & f^{i} · u \\ f^{i} · v & y′′_{i} \end{matrix} ]⋆[ \begin{matrix}1_{d′} & 0 \\ −f^{i} · h′′ & 1_{d′′} \end{matrix} ]·[ \begin{matrix}1_{d′} & −f^{i} · h′ \\ 0 & 1_{d′′} \end{matrix} ] . Mul-

tiplying these matrices we get

y_{i+1} = [ \begin{matrix}y′_{i} & f^{i}(u − y′_{i}⋆ h′+ h′⋆ y′′_{i}) \\ f^{i}(v − y′′_{i}⋆ h′′+ h′′⋆ y′_{i}) & y′′_{i} \end{matrix} ] + f^{2i} · z,

for some z ∈ M_{d×d}^{α} (R). Invoking the above equalities we obtain

y_{i+1} = [ \begin{matrix}y′_{i} & 0 \\ 0 & y′′_{i} \end{matrix}] + f^{i+1} · [ \begin{matrix}0 & z′ \\ z′′ & 0 \end{matrix}] + f^{2i} · z.

Consequently,

mod _{A}^{d}(π_{i+1} )(y_{i+1} ) = [ \begin{matrix}n′_{i+1} & 0 \\ 0 & n′′_{i+1} \end{matrix} ] ,

where n′_{i+1} = M_{d′×d′}^{α} (π_{i+1} )(y′_{i}) and n′′_{i+1} = M_{d′′×d′′}^{α}(π_{i+1} )(y′′_{i}). Clearly, n′_{i+1} and n′′_{i+1} are elements in mod _{A}^{d′}(R/𝔪^{i+1} ) and mod _{A}^{d′′}(R/𝔪^{i+1} ), respectively, and hence the condition (2) for i + 1 holds. Moreover,

mod _{A}^{d′}(ϵ_{i})(n′_{i+1} ) = mod _{A}^{d′}(ϵ_{i})M_{d′×d′}^{α} (π_{i+1} )(y′_{i}) = M_{d′×d′}^{α} (ϵ_{i}π_{i+1} )(y′_{i})
= M_{d′×d′}^{α} (π_{i})(y′_{i}) = n′_{i}.

Similarly, mod _{A}^{d′′}(ϵ_{i})(n′′_{i+1} ) = n′′_{i}. Hence, the condition (3) holds for i + 1. This finishes the proof.

LEMMA 4.3. For any i ≥ 1, the A-modules N_{i} and N′_{i} ⊕ N′′_{i} are isomorphic.

Proof. Applying the conditions (1) and (2) of Lemma 4.2, we get

[ \begin{matrix}n′_{i} & 0 \\ 0 & n′′_{i} \end{matrix}] = mod _{A}^{d}(π_{i})(g_{i} · y) = Gl_{d}(π_{i})(g_{i}) · mod _{A}^{d}(π_{i})(y)

= Gl_{d}(π_{i})(g_{i}) · n_{i},

for some g_{i} ∈ Gl_{d}(R). But this implies that even as A-R/𝔪^{i}-bimodules N_{i} and N′_{i}⊕ N′′_{i}are isomorphic.

LEMMA 4.4. For any i ≥ 1, there are in modA exact sequences

\begin{matrix}Σ′_{i}: & 0 → N′_{i+1} → N′_{i}⊕ N′_{i+2} → N′_{i+1} → 0, \\ Σ′′_{i}: & 0 → N′′_{i+1} → N′′_{i}⊕ N′′_{i+2} → N′′_{i+1} → 0. \end{matrix}

Proof. Take a natural number i ≥ 1 and set j = i + 2. Since n′_{j} ∈ mod _{A}^{d′}(R/𝔪^{j}), then N′_{j} is an A-R/𝔪^{j}-bimodule, which as an R/𝔪^{j}- module is free of finite rank. Then the functor F′ = N′_{j} ⊗_{R/𝔪^{j}} (−) : fin(R/𝔪^{j}) → modA is exact. Hence the exact sequence

0 → R/𝔪^{i+1} −→^{β_{i}} R/𝔪^{i} ⊕ R/𝔪^{i+2} −→^{γ_{i}} R/𝔪^{i+1} → 0

in fin(R/𝔪^{j}), where β_{i}(r + 𝔪^{i+1} ) = (r + 𝔪^{i}, f · r + 𝔪^{i+2} ), γ_{i}(r + 𝔪^{i}, r′+ 𝔪^{i+2} ) = f · r − r′+ 𝔪^{i+1} , for any r, r′∈ R, gives the exact sequence:

0 → F′(R/𝔪^{i+1} ) −→^{F′(β_{i})}F′(R/𝔪^{i}) ⊕ F′(R/𝔪^{i+2} ) −→^{F′(γ_{i})}F′(R/𝔪^{i+1} ) → 0

in modA. Applying the condition (3) of Lemma 4.2, we obtain

mod _{A}^{d′}(ϵ_{i+1} )(n′_{j}) = n′_{i+1}  and  mod _{A}^{d′}(ϵ_{i}ϵ_{i+1} )(n′_{j}) = n′_{i}.

This implies that

F′(R/𝔪^{i}) = N′_{j}⊗_{R/𝔪^{j}} R/𝔪^{i} ≃ N′_{i},
F′(R/𝔪^{i+1} ) = N′_{j}⊗_{R/𝔪^{j}} R/𝔪^{i+1} ≃ N′_{i+1} ,
F′(R/𝔪^{i+2} ) = N′_{j}⊗_{R/𝔪^{j}} R/𝔪^{j} ≃ N′_{j} = N′_{i+2} .

Thus there exists an exact sequence Σ′_{i} in modA of the form 0 → N′_{i+1} → N′_{i}⊕ N′_{i+2} → N′_{i+1} → 0, and, by symmetry, also the exact sequence Σ′′_{i}: 0 → N′′_{i+1} → N′′_{i}⊕ N′′_{i+2} → N′′_{i+1} → 0.

4.5. Applying Proposition 3.5 we get a number h ≥ 1 such that N_{i+2} ⊕ N_{i} ≃ N_{i+1} ⊕ N_{i+1} for any i ≥ h. Moreover, we have the following lemma. LEMMA. N′_{i+2} ⊕ N′_{i} ≃ N′_{i+1} ⊕ N′_{i+1} and N′′_{i+2} ⊕ N′′_{i}≃ N′′_{i+1} ⊕ N′′_{i+1} , for any i ≥ h.

Proof. From Lemma 4.4, there are in modA exact sequences

\begin{aligned}Σ′_{i} & : 0 → N′_{i+1} → N′_{i}⊕ N′_{i+2} → N′_{i+1} → 0, \\ Σ′′_{i} & : 0 → N′′_{i+1} → N′′_{i}⊕ N′′_{i+2} → N′′_{i+1} → 0. \end{aligned}

Then δ_{Σ′_{i}} (X) ≥ 0 and δ_{Σ′′_{i}}(X) ≥ 0, for any module X in modA, by Lemma 2.2. Applying Lemma 4.3 we get

δ_{Σ′_{i}} (X) + δ_{Σ′′_{i}}(X) = dim_{k} Hom_{A}(N_{i+1} ⊕ N_{i+1} , X)
− dim_{k} Hom_{A}(N_{i+2} ⊕ N_{i}, X) = 0,

and consequently δ_{Σ′_{i}} (X) = 0 = δ_{Σ′′_{i}}(X), for any module X in modA. Then the claim follows from Lemma 2.2.

LEMMA 4.6. There are modules M′ and M′′ in modA such that N′_{h+1} ≃ M′⊕ N′_{h} and N′′_{h+1} ≃ M′′⊕ N′′_{h}.

Proof. Applying Lemma 4.5 several times we obtain that

⨁_{0≤i<l} (N′_{h+i+2} ⊕ N′_{h+i} ) ≃ ⨁_{0≤i<l} (N′_{h+i+1} ⊕ N′_{h+i+1} ),

for any l ≥ 1. If we cancel the common direct summand ⨁_{1≤i<l} (N′_{h+i+1} ⊕N′_{h+i} ), then we get an isomorphism of A-modules N′_{h+l+1} ⊕ N′_{h} ≃ N′_{h+l} ⊕ N′_{h+1} , for any l ≥ 1. Hence, there is an isomorphism

⨁_{1≤l<j} (N′_{h+l+1} ⊕ N′_{h}) ≃ ⨁_{1≤l<j} (N′_{h+l} ⊕ N′_{h+1} ),

for any j ≥ 2. After the cancellation of the common direct summand ⨁_{2≤l<j} N′_{h+l} , we get an isomorphism N′_{h+j} ⊕ (N′_{h})^{j−1} ≃ (N′_{h+1} )^{j} of A-modules, for any j ≥ 2. This implies that

μ(N′_{h+j} , X) + (j − 1) · μ(N′_{h}, X) = j · μ(N′_{h+1} , X),

for any indecomposable module X in modA and j ≥ 2, where μ(Y, X) denotes the multiplicity of an indecomposable module X ∈ modA as a direct summand of a module Y ∈ modA. Hence,

μ(N′_{h}, X) ≤ lim_{j→∞} \frac{j}{j − 1} · μ(N′_{h+1} , X) = μ(N′_{h+1} , X),

for any indecomposable module X in modA. This implies that the module N′_{h} is a direct summand of N′_{h+1} , so N′_{h+1} ≃ M′ ⊕ N′_{h} for some M′ ∈ modA. Similarly, we conclude that there is a module M′′ in modA such that N′′_{h+1} ≃ M′′⊕ N′′_{h}.

4.7. The following lemma completes the proof of Theorem 3. LEMMA. M′≤_{deg} N′, M′′≤_{deg} N′′and M ≃ M′⊕ M′′.

Proof. Applying Proposition 3.5 and Lemmas 4.3, 4.6 we get

M ⊕N_{h} ≃ N_{h+1} ≃ N′_{h+1} ⊕N′′_{h+1} ≃ M′⊕N′_{h}⊕M′′⊕N′′_{h}≃ M′⊕M′′⊕N_{h},

and consequently M ≃ M′⊕ M′′. Up to symmetry, it remains to show that M′ ≤_{deg} N′. We set j = h + 1. As in (4.4) we may consider the exact functor F′= N′_{j}⊗_{R/𝔪^{j}} (−) : fin(R/𝔪^{j}) → modA. Then the exact sequence

0 → R/𝔪−→^{β_{i}} R/𝔪^{i+1} −→^{γ_{i}} R/𝔪^{i} → 0

(see (3.1)) induces the following exact sequence

0 → F′(R/𝔪) −→^{F′(β_{i})}F′(R/𝔪^{h+1} ) −→^{F′(γ_{i})}F′(R/𝔪^{h}) → 0.

Applying the condition (3) of Lemma 4.2 several times, we obtain

\begin{matrix}mod _{A}^{d′}(ϵ_{h})(n′_{j}) = n′_{i}, \\ mod _{A}^{d′}(ϵ_{1} …ϵ_{h})(n′_{j}) = n′_{1} = n′. \end{matrix}

Together with Lemma 4.6 this implies that F′(R/𝔪^{h}) ≃ N′_{h}, F′(R/𝔪) ≃ N′ and F′(R/𝔪^{h+1} ) ≃ N′_{h+1} ≃ M′ ⊕ N′_{h}. Then we have an exact sequence

0 → N′→ M′⊕ N′_{h} → N′_{h} → 0

in modA, and so M′≤_{deg} N′, by Proposition 3.4 in [11].

5. The proof of Theorem 4

Let M, N be modules in modA such that N is a minimal degeneration of M and M ≮ext N. We proceed by induction on the number of direct summands of a decomposition of N into the direct sum of indecomposable modules. If N is indecomposable, then the claim follows for \overline{M} = M, \overline{N} = N and W = 0. Assume now that N ≃ N′⊕ N′′ for some nonzero modules N′, N′′in modA. Suppose that there is a nonsplittable exact sequence in modA of the form

0 → N′→ W → N′′→ 0 or 0 → N′′→ W → N′→ 0

with M ≤_{deg} W. Then M ≤_{deg} W <ext N and M ≃ W, because N is a minimal degeneration of M. Therefore M <ext N what contradicts our assumptions. Hence, by Theorem 3, there are modules M′, M′′in modA such that M′≤_{deg} N′, M′′≤_{deg} N′′and M ≃ M′⊕M′′. Observe that

M ≃ M′⊕ M′′≤_{deg} N′⊕ M′′≤_{deg} N′⊕ N′′≃ N.

Then either M′ ≃ N′ or M′′≃ N′′, since the degeneration M <_{deg} N is minimal. We may assume that M′′≃ N′′. Then we have the minimal degeneration M′ <_{deg} N′ and M′ ≮ext N′. Hence, the claim follows from our inductive assumption, applied to the minimal degeneration M′<_{deg} N′. This finishes the proof.

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