A GENERALIZATION OF ZWARA’S THEOREM

ON DEGENERATION OF MODULES

YUJI YOSHINO

This is a survey of my recent works on degenerations of modules and their G-dimension. For the datail of the contents, the reader should refer to the original papers [4], [5].

1. Degenerations of modules

In this section k always denotes a field and R is a k-algebra. Note that R may not be commutative nor Noetherian.

Definition 1.1. For finitely generated left R-modules M and N, we say that M degenerates to N along a discrete valuation ring, or N is a degeneration of M along a DVR, if there is a discrete valuation ring (V, tV, k) that is a k-algebra (where t is a prime element) and a finitely generated left R ⊗_{k} V -module Q which satisfies the following conditions:

(1) Q is flat as a V -module

(2) Q/tQ ≅ N as a left R-module.

(3) Q[\frac{1}{t}] ≅ M ⊗_{k} V [\frac{1}{t}] as a left R ⊗_{k} V [\frac{1}{t}]-module.

We have considered a different kind of degenerations in the previous paper [3], mainly for maximal Cohen-Macaulay modules over a commutative Cohen-Macaulay local ring. To distinguish it from the degenerations defined in Definition 1.1, we make the following definition.

Definition 1.2. In this definition we assume that k is an algebraically closed field to identify the affine line with k. And let R be a k-algebra as before. For finitely generated left R-modules M and N, we say that M degenerates to N along an affine line, or N is a degeneration of M along an affine line, if there is a finitely generated left module Q over R ⊗_{k} k[t] which satisfies the following conditions:

(1) Q is flat as a k[t]-module.

(2) For any c ∈ k, let us denote Q/(t − c)Q by Q_{c}, which is a finitely generated left R-module. Then, Q_{0} ≅ N as a left R-module.

(3) There is a non-empty Zariski open subset U of 𝔸_{k}^{1} ≅ k such that if c ∈ U, then Q_{c} ≅ M as a left R-module.

The following is our main theorem from [5].

Theorem 1.1. The following conditions are equivalent for finitely generated left Rmodules M and N.

(1) N is a degeneration of M along a DVR.

(2) There is a short exact sequence of finitely generated left R-modules

0 → Z \underrightarrow{\binom{ϕ}{ψ}} M ⊕ Z → N → 0,

such that the endomorphism ψ on Z is nilpotent, i.e. ψ^{n} = 0 for n ≫ 1.

Remark 1.1. As G.Zwara has shown in [6], if R is an Artinian k-algebra, then the following conditions are equivalent for finitely generated left R-modules M and N:

(1) N is a degeneration of M along a DVR. (2′) There is a short exact sequence of finitely generated left R-modules

0 → Z \underrightarrow{\binom{ϕ}{ψ}} M ⊕ Z → N → 0.

Note here that we need not the nilpotency assumption for ψ. It is easy to see from Fitting Theorem that if R is an Artinian ring then the condition (2′) is equivalent to the condition (2). In this way, our theorem contains the theorem of Zwara.

By the proof of (2) ⇒ (1) of the theorem we get the following result as a corollary.

Corollary 1.2. Suppose that M degenerates to N along a DVR. Then as a discrete valuation ring V we can always take the ring k[t]_{(t)}.

Remark 1.2. Assume that there is an exact sequence of finitely generated left R-modules

0 → N′ →^{p} M →^{q} N′′ → 0.

Then it is easy to see that M degenerates to N′ ⊕ N′′ along a DVR. In fact, we have only to notice that there is an exact sequence

0 → N′ \underrightarrow{\binom{p}{0}} M ⊕ N′ \underrightarrow{(\begin{matrix}q & 0 \\ 0 & 1 \end{matrix})} N′′ ⊕ N′ → 0,

where the mapping ψ : N′ → N′ is the zero mapping, hence nilpotent.

We can prove an implication between degenerations as in the following theorem. But one should remark that the converse implication does not hold in general. See Remark 2.1.

Theorem 1.3. Assume that k is an algebraically closed field and that R is a left Noetherian k-algebra. Let M and N be finitely generated left R-modules. If M degenerates to N along a DVR, then M degenerates to N along an affine line.

2. Remarks for commutative Noetherian algebras

In the rest of the paper, we assume that R is a commutative Noetherian algebra over a field k. In this case we have the following result as a corollary of Theorem 1.1.

Corollary 2.1. Suppose that M and N are R-modules of finite length. Then the following conditions are equivalent.

(1) N is a degeneration of M along a DVR.

(2) There is an exact sequence

0 → Z \underrightarrow{\binom{ϕ}{ψ}} M ⊕ Z → N → 0,

where Z is also a module of finite length. In particular, if M degenerates to N along a DVR, then we must have an equality of the lengths; ℓ_{R}(M) = ℓ_{R}(N).

Remark 2.1. There is an example where the opposite direction of the implication in Theorem 1.3 does not hold.

For example, let R = k[[x]] be the formal power series ring over an algebraically closed field k and let M = R/(x) and N = R/(x^{2}). Since M and N have distinct lengths, N can never be a degeneration of M along a DVR by Corollary 2.1. On the other hand, consider the R[t]-module Q = R[t]/(x^{2} − tx). It is easy to see that Ass_{R[t]}Q = {(x), (x − t)}, hence any nonzero element of k[t] is a non-zero divisor on Q. This implies that Q is flat over k[t]. For any element c ∈ k, note that Q_{c} ≅ R/(x(x − c)), hence that Q_{0} ≅ R/(x^{2}) and Q_{c} ≅ R/(x) for c ≠ 0, since x − c is a unit in R. Therefore, from the definition, the module M degenerates to N along an affine line. Note that there is an exact sequence

0 → R \underrightarrow{\binom{1}{x}} R/(x) ⊕ R ⟶^{(x,−1)} R/(x^{2}) → 0,

however the endomorphism R →^{x} R is not nilpotent.

3. Openness of the G-dimension zero property

Let R be a commutative Noetherian ring as before. Auslander and Bridger [1] give a definition of G-dimension, which we denote by G-dim_{R}M for a finitely generated Rmodule M. In our subsequent work [4], we have shown the following fact:

Theorem 3.1. If there is an exact sequence of finitely generated R-modules

0 → Z → M ⊕ Z → N → 0,

then we have an inequality G-dim_{R}M ≦ G-dim_{R}N.

Combining this with Theorem 1.1, we have the following result as a corollary

Corollary 3.2. Assume that R is a Noetherian commutative algebra over a field k and let M and N be finitely generated R-modules. Suppose that N is a degeneration of M along a DVR. Then the inequality G-dim_{R}M ≦ G-dim_{R}N holds.

In particular, if N has G-dimension 0, then so does M in this case. We infer from this that if there is an algebraic set that parameterizes a family of finitely generated R-modules, then the set of points corresponding to modules with G-dimension 0 should form an open subset. By this property we may say that the property for a module having G-dimension 0 is an ‘open’ property. This generalizes a well-known fact that the maximal Cohen-Macaulay property for modules over a Gorenstein local ring is an open property.

References

[1] M.Auslander and M.Bridger, “Stable module theory”, Mem. Amer. Math. Soc. 94 (1969).

[2] Y.Yoshino, “Cohen-Macaulay modules over Cohen-Macaulay rings”, Lecture Note Series 146, Cambridge University Press, 1990.

[3] Y.Yoshino, On degenerations of Cohen-Macaulay modules, Journal of Algebra 248, 272–290 (2002).

[4] Y.Yoshino, Degeneration and G-dimension of modules, Lecture at Lisbon Conference of Commutative Algebra, June, 2003.

[5] Y.Yoshino, On degenerations of modules, to appear in Journal of Algebra.

[6] G.Zwara, Degenerations of finite dimensional modules are given by extensions, Composito Mathematica 121, 205–218 (2000).

Department of Mathematics Faculty of Science Okayama University Okayama, Okayama 700-8530 JAPAN

E-mail address: yoshino@math.okayama-u.ac.jp