ON DEGENERATIONS OF MODULES OVER GENERAL RINGS

YOSUKE OHNUKI AND MASAHISA SATO

In this paper, R is an associative ring with an identity and a module means a left R-module otherwise stated. We denote by ℕ the set of positive integers.

Recall [6], [8] that for a finite dimensional algebra A over an algebraically closed field k and (finite) d-dimensional A-modules M, N, M degenerates to N (i.e., N lies in the Gl_{d}(k)-orbit closure of M) if and only if there is some exact tube, called (M, N)-tube, if and only if there is a short exact sequence 0 → Z → Z ⊕ M → N → 0. Our aims are to define the degeneration for infinite dimensional modules and to characterize the non-split exact sequence 0 → N → N ⊕ N → N → 0.

1. Degenerations

Our aim in this section is to generalize Zwara’s results, that is, M degenerates to N if and only if there is an exact sequence 0 → Z → Z ⊕ M → N → 0.

Let N_{i} (i ∈ ℕ) be modules and

T : N_{1} ⟵^{β_{1}} N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} ⋯

a sequence of homomorphisms. T[1] is a shift of T, i.e.,

T[1] : N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} N_{4} ⟵^{β_{4}} ⋯

A morphism α : T → T[1] is the set of homomorphisms α_{i} N_{i} → N_{i+1}, (i ∈ ℕ) which makes a commutative diagram;

N_{1} ⟵^{β_{1}} N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} ⋯
α_{1}↓  ^{α_{2}}↓  ^{α_{3}}↓
N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} N_{4} ⟵^{β_{4}} · · · .

A morphism α : T → T[1] is said to be “exact” or “exact tube” if for each i, we have the short exact sequences

0 ⟶ N_{i+1} ⟶^{( \begin{matrix}β_{i} \\ α_{i+1} \end{matrix} )} N_{i} ⊕ N_{i+2} ⟶^{(α_{i},−β_{i+1})} N_{i+1} ⟶ 0

with a monomorphism α_{i} and an epimorphism β_{i}.

For an exact tube α : T → T[1] and N := Cok(α_{1}), we have a short exact sequence of inverse systems;

0 ⟶ T ⟶^{α} T[1] ⟶ (N, 1_{N}) ⟶ 0.

Here we consider that (N, 1_{N}) is a sequence of identity maps of N,

(N, 1_{N}) : N ⟵^{1_{N}} N ⟵^{1_{N}} N ⟵^{1_{N}} ⋯

The detailed version of this paper will be submitted for publication elsewhere.

Then, this induces the exact sequence of inverse limit;

0 ⟶ lim_{⟵} T ⟶^{lim_{⟵} α} lim_{⟵} T[1] ⟶ N ⟶ 0,

because the inverse limit functor lim_{⟵} is left exact and a right term N is the inverse limit of the inverse system (N, 1_{N}). For an exact tube α : T → T[1] and any module L, α ⊕ 1_{L} : T ⊕ L → T[1] ⊕ L is also an exact tube which makes a commutative diagram;

N_{1} ⊕ L ⟵^{β_{1}⊕1_{L}} N_{2} ⊕ L ⟵^{β_{2}⊕1_{L}} N_{3} ⊕ L ⟵^{β_{3}⊕1_{L}} ⋯
α_{1}⊕1_{L}↓  ^{α_{2}⊕1_{L}}↓  ^{α_{3}⊕1_{L}}↓
N_{2} ⊕ L ⟵^{β_{2}⊕1_{L}} N_{3} ⊕ L ⟵^{β_{3}⊕1_{L}} N_{4} ⊕ L ⟵^{β_{4}⊕1_{L}} · · · .

Lemma 1.1. Let α : T → T[1] be an exact tube. If an exact sequence 0 → lim_{⟵} T ⟶^{lim_{⟵} α} lim_{⟵} T[1] → N → 0 splits, then exact sequences 0 → N_{i} → N_{i+1} → N → 0 split for all i ≥ 1.

Definition 1.2. Let M and N be non-zero modules. An exact tube α : T → T[1] is called an (M, N)-tube if there is an isomorphism ( \begin{matrix}f \\ g \end{matrix} ) : lim_{⟵} T[1] → lim_{⟵} T ⊕ M with the property that lim_{⟵} T has no non-zero direct summand L such that lim_{⟵} α|L : L → lim_{⟵} α(L) is not an isomorphism. In this case N is called a “degeneration” of M.

The following theorem generalizes the results in [1, 7, 8] on general rings and modules.

Theorem 1.3. The following statements are equivalent; (1) N is a degeneration of M.

(2) There is an exact sequence 0 → Z \underrightarrow{( \begin{matrix}f \\ g \end{matrix} )} Z ⊕ M → N → 0 for some module Z which has no non-zero direct summand L such that f|L : L → f(L) is not an isomorphism.

For a degeneration N of M, we constructed an exact sequence 0 → Z → Z ⊕ M → N → 0 that the module Z was realized by an inverse limit for a chain of epimorphisms. However, we can also construct it by using a direct limit. In fact, for an exact sequence

0 → Z \underrightarrow{( \begin{matrix}f \\ g \end{matrix} )} Z ⊕ M → N → 0, we can construct a commutative diagram consisting of direct systems ;

T̄ :  Z ⟶^{ι_{0}} Z ⊕ M ⟶^{ι_{1}} Z ⊕ M^{2} ⟶^{ι_{2}} ⋯
α↓ ^{h_{0}}↓  ^{h_{1}}↓  ^{h_{2}}↓
T̄[1] : Z ⊕ M ⟶^{ι_{1}} Z ⊕ M^{2} ⟶^{ι_{2}} Z ⊕ M^{3} ⟶^{ι_{3}} · · · .

Here h_{i} : Z⊕M^{i} → Z⊕M^{i}⊕M is given by h_{i} ( \begin{matrix}z \\ x \end{matrix}) = ( \begin{matrix}f(z) \\ x \\ g(z) \end{matrix} ) and ι_{i} : Z⊕M^{i} → Z⊕M ⊕M^{i}

is given by ι_{i} ( \begin{matrix}z \\ x \end{matrix}) = ( \begin{matrix}z \\ 0 \\ x \end{matrix} ) for z ∈ Z, x ∈ M^{i}. Note that h_{0} = ( \begin{matrix}f \\ g \end{matrix} ).

Then it induces the exact sequence of direct limits of direct systems;

0 ⟶ lim_{⟶} T̄ ⟶^{lim_{⟶} α} lim_{⟶} T̄[1] ⟶ N ⟶ 0.

Concretely, we have lim_{⟶} T = Z ⊕ M^{(ℕ)}, where M^{(ℕ)} is the direct sum of ℕ-copies of M. Note that Z has no non-zero direct summand L such that f|L : L → f(L) is not an isomorphism. On our definition of a degeneration, it is satisfied the following basic properties.

Corollary 1.4. (1) If 0 → A → B → C → 0 is an exact sequence, then A ⊕ C is a

degeneration of B.

(2) If N is a degeneration of M, then N^{(r)} is a degeneration of M^{(r)}, where N^{(r)} is direct sum of r-copies of N and r is a cardinal number.

(3) If N is a degeneration of M, then N^{r} is a degeneration of M^{r}, where N^{r} is direct product of r-copies of N.

Definition 1.5. Let 0 → Z \underrightarrow{( \begin{matrix}f \\ g \end{matrix} )} Z ⊕ M → N → 0 be an exact sequence which gives a degeneration N of M. A degeneration is called “Fitting type” if f : Z → Z is nilpotent.

On a degeneration for finitely generated modules M ≇ N, the endomorphism f must

be nilpotent for the induced exact sequence 0 → Z \underrightarrow{( \begin{matrix}f \\ g \end{matrix} )} Z ⊕ M → N → 0 [1], [7], [8].

Theorem 1.6. Assume a module N is finitely generated and there is a Fitting type degeneration N of M. Then M is finitely generated and there is a finitely generated module

Z such that 0 → Z \underrightarrow{\binom{f}{g}} Z ⊕ M → N → 0 is exact and f is nilpotent.

2. Exact sequences of the form 0 → N → N ⊕ N → N → 0

In this section, we give a non-split exact sequence of the form 0 → N → N ⊕ N → N → 0 and these examples make strange degenerations which do not happen for finitely generated modules over finite dimensional algebras.

We give the preliminary lemma.

Lemma 2.1. The following statements are equivalent.

(1) Any exact sequence 0 → M → M ⊕ N → N → 0 splits for any modules M and N.

(2) Any exact sequence 0 → N → N ⊕ N → N → 0 splits for any modules N.

An exact tube:

N_{1} ⟵^{β_{1}} N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} ⋯
α_{1}↓  ^{α_{2}}↓  ^{α_{3}}↓
N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} N_{4} ⟵^{β_{4}} ⋯

induces an inductive systems;

0 ⟶ N_{i} \underrightarrow{( \begin{matrix}β_{i−1} \\ α_{i} \end{matrix} )} N_{i−1} ⊕ N_{i+1} ⟶^{(α_{i−1},−β_{i})} N_{i} ⟶ 0
↓^{α_{i}}  ↓( \begin{matrix}α_{i−1} & 0 \\ 0 & α_{i+1} \end{matrix} )  ↓^{α_{i}}
0 ⟶ N_{i+1} \overrightarrow{( \begin{matrix}β_{i} \\ α_{i+1} \end{matrix} )} N_{i} ⊕ N_{i+2} ⟶_{(α_{i},−β_{i+1})} N_{i+1} ⟶ 0.

Proposition 2.2. We set N_{ℵ_{0}} = lim_{⟶} N_{i}, ( \begin{matrix}β_{ℵ_{0}} \\ α_{ℵ_{0}} \end{matrix} ) = lim_{⟶} ( \begin{matrix}β_{i} \\ α_{i+1} \end{matrix} ), (α_{ℵ_{0}}, −β_{ℵ_{0}}) = lim_{⟶}(α_{i}, −β_{i+1}).

Then the short exact sequence

0 ⟶ N_{ℵ_{0}} \underrightarrow{( \begin{matrix}β_{ℵ_{0}} \\ α_{ℵ_{0}} \end{matrix} )} N_{ℵ_{0}} ⊕ N_{ℵ_{0}} ⟶^{(α_{ℵ_{0}},−β_{ℵ_{0}})} N_{ℵ_{0}} ⟶ 0

splits.

Aehle, Riedtmann and Zwara [1] defined a complexity for an (M, N)-tube where M ≇ N are finitely generated modules over a finite dimensional algebra, that is, a complexity for an (M, N)-tube is the minimal number such that exact sequences

0 ⟶ N_{i} \underrightarrow{( \begin{matrix}β_{i−1} \\ α_{i} \end{matrix} )} N_{i−1} ⊕ N_{i+1} ⟶^{(α_{i−1},−β_{i})} N_{i} ⟶ 0

split for all i ≥ m.

Their results hold for an exact tube if there is such an m that the above property is satisfied. So we can define the complexity of an exact tube over general rings.

By Proposition 2.2, the following fact holds.

Proposition 2.3. The complexity of an exact tube always exists and this is less than or equal to ℵ_{0}.

The following example shows that there is a non-split exact sequence 0 → N → N ⊕ N → N → 0 which induces the strange degeneration.

Example 2.4. Let k be a field and ℚ the set of rational numbers. We set a semi-group algebra R = ∑_{{q∈ℚ, \atop 0≤q≤1}} ⊕kz_{q}. The multiplication is given by z_{p}z_{q} = z_{pq}. Then idempotents of

R are only z_{0}, z_{1} and z_{1} is the identity of R.

Now we shall construct R-module N. It is defined as vector space with bases { w_{t}| t ∈ ℚ, −1 ≤ t ≤ 1 } in the following;

N = ∑_{t∈Q,_{−1≤q≤1}} ⊕Kw_{t}.

The R-operation is defined by z_{q}w_{t} = w_{qt}. The generators of N are w_{1}, w_{−1} and the socle of N is Kw_{0}.

We consider two R-homomorphisms f_{1}, f_{2} : N → N by

f_{1}(w_{t}) = \begin{cases}w_{t}, & (t ≤ 0), \\ w_{\frac{t}{2}} , & (t > 0), \end{cases} f_{2}(w_{t}) = \begin{cases}w_{\frac{t}{2}} , & (t ≤ 0), \\ w_{t}, & (t > 0). \end{cases}

Then we have the non-split exact sequence;

0 ⟶ N \underrightarrow{( \begin{matrix}f_{1} \\ f_{2} \end{matrix} )} N ⊕ N ⟶ N ⟶ 0.

In particular, N has no non-zero direct summand L such that f_{1}|L : L → f_{1}(L) is not an isomorphism.

We set M_{q} = ∑_{{x∈ℚ \atop 0≤x≤q}} ⊕Kw_{x} an ideal of R. Then we have the exact sequence ;

0 ⟶ M_{\frac{1}{2}} /M_{\frac{1}{4}} ⟶ M_{1}/M_{\frac{1}{4}} ⟶ M_{1}/M_{\frac{1}{2}} ⟶ 0.

For any rational numbers 0 ≤ r < s ≤ 1, all of the modules of the form M_{s}/M_{r} are isomorphic, thus we have a non-split exact sequence;

0 ⟶ M ⟶ M ⟶ M ⟶ 0.

Hence M^{2} is a degeneration of M by Corollary 1.4.

Crawley-Boevey and Krause studied a generic modules [2], [5]. This is determined to a homogeneous tube. In particular, for a tame concealed algebra, a generic module is uniquely constructed by a tubular family (it is a one-parameter family of tubes that almost all of them are homogeneous.)

Example 2.5 ([5]). Let Λ be an algebra over a field k. An exact tube T is called a homogeneous tube if it is a following exact diagrams

0 ⟵ N_{1} ⟵^{β_{1}} N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} ⋯
↓  ^{α_{1}}↓  ^{α_{2}}↓  ^{α_{3}}↓
N_{1} ⟵^{β_{1}} N_{2} ⟵^{β_{2}} N_{3} ⟵^{β_{3}} N_{4} ⟵^{β_{4}} · · · .

We have the exact sequence of inverse limits

0 ⟶ lim_{⟵} T ⟶^{ψ} lim_{⟵} T[1] ⟶ N ⟶ 0.

Then this also gives the following direct system of exact sequences

0 ⟶ lim_{⟵} T ⟶^{ψ^{i}} lim_{⟵} T ⟶ N_{i} ⟶ 0
∥  ^{ψ}↓  ^{α_{i}}↓
0 ⟶ lim_{⟵} T ⟶^{ψ^{i+1}} lim_{⟵} T ⟶ N_{i+1} ⟶ 0.

Note that lim_{⟵} T ⊕ N_{i} is a degeneration of lim_{⟵} T by Corollary 1.4. Otherwise, we also have the exact sequence of direct limits

0 ⟶ lim_{⟵} T ⟶ M ⟶ lim_{⟶} T ⟶ 0.

It is called the universal exact sequence corresponding to T. In fact, M is uniquely determined to a tubular family, and it is called a generic module. By Corollary 1.4, we have that lim_{⟶} T ⊕ lim_{⟵} T is a degeneration of M.

Now, we shall introduce the behavior of the homogeneous tube explaining in [5]. Let k[x] be a polynomial ring with one invariant and k(x) be the quotient field of k[x]. Note that k[x] ≅ lim_{⟵} k[x]/x^{i} and k(x) is the direct limit of the direct system k[x] ⟶^{·x} k[x] ⟶^{·x} · · ·. The short exact sequence 0 → k[x] → k(x) → k(x)/k[x] → 0 is the universal exact sequence corresponding to the tube (k[x]/x^{i})_{i∈ℕ}

Lemma 2.6 ([5]). Let T be a homogeneous tube. Then T is isomorphic to (k[x]/x^{i} ⊗_{k[x]}

(lim_{⟵} T))_{i∈ℕ}.

References

[1] R. Aehle,Ch. Riedtmann and G. Zwara, Complexity of degenerations of modules, Comment. Math. Helv. 76 (2001), 781–803.

[2] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. 63 (1991), 241– 264.

[3] F. Grunewald and J. O’Halloran, A characterization of orbit closure and applications, J. Algebra 116 (1988), 163–175.

[4] H. Kraft, Geometische Methoden in der Invarianten theorie , Vieweg (1984).

[5] H. Krause, Generic modules over artin algebra, Proc. London Math. Soc. 76 (1998), 276–306.

[6] Ch. Riedtmann, Degenarations for Representations of Quivers with Relations, Ann. Sci. École Normal Sup. 19 (1986), 275–301.

[7] Y. Yoshino, On Degenerations of modules, Preprint.

[8] G. Zwara, Degenerations and Finite-Dimensional Modules are Given by Extensions, Composito Math. 121 (2000), 205–218.

Tokyo University of Agriculture and Technology Koganei, Tokyo 184-1855 JAPAN

 E-mail address: ohnuki@cc.tuat.ac.jp

Mathematical section in Interdisciplinary Graduate School of Medicine and Engineering Yamanashi University Kofu, Yamanashi 400-8511 JAPAN

 E-mail address: msato@yamanashi.ac.jp