Relative Infinite Determinacy for Map-Germs

Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 549845, 6 pages http://dx.doi.org/10.1155/2013/549845

Research Article Relative Infinite Determinacy for Map-Germs Changmei Shi1,2 and Donghe Pei1 1 2

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China School of Mathematics and Computer Science, Guizhou Normal College, Guiyang 550018, China

Correspondence should be addressed to Donghe Pei; [email protected] Received 5 May 2013; Revised 8 September 2013; Accepted 9 September 2013 Academic Editor: Stanislav Hencl Copyright © 2013 C. Shi and D. Pei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The infinite determinacy for smooth map-germs with respect to two equivalence relations will be investigated. We treat the space of smooth map-germs with a constraint, and the constraint is that a fixed submanifold in the source space is mapped into another fixed submanifold in the target space. We study the infinite determinacy for such map-germs with respect to a subgroup of right-left equivalence group and finite and infinite determinacy with respect to a subgroup of contact group and give necessary and sufficient conditions for the corresponding determinacy.

1. Introduction This work is concerned with the singularity theory of differentiable maps. Singularity theory of differentiable maps is a wide-ranging generalization of the theory of the maxima and minima of functions of one variable, and it is now an essential part of nonlinear analysis. This theory has numerous applications in mathematics and the natural sciences; see, for example, [1, 2]. In differentiable analysis, the local behavior of a differentiable map can be determined by the derivatives of the map at a point. Hence we have the theories of finitely and infinitely determined map-germs. We know that every finitely determined map-germ is equivalent to its Taylor polynomial of some degree, and infinite determinacy is a way to express the stability of smooth map-germs under flat perturbations. The analysis of the conditions for a map-germ to be finitely or infinitely determined involves the most important local aspects of singularity theory. Therefore, the study of finite and infinite determinacy of smooth map-germs is a very important subject in singularity theory. Now there are several articles treating the question of infinite determinacy with respect to the most frequently encountered and naturally occurring equivalence groups, for instance, the right-equivalence group R, left-equivalence group L, group C, contact group K, and right-leftequivalence group A. In [3, 4], Wilson characterized the

infinite determinacy of smooth map-germs with respect to one of the groups R, L, C, and K and the infinite determinacy for finitely K-determined map-germs with respect to A. Besides, for the case of A, Brodersen [5] showed that the results of [4] hold for map-germs without assuming K-finiteness. We can see [6] for detailed survey of all of these results. Recently, there appeared the notion of relative infinite determinacy for smooth function-germs with nonisolated singularities. Sun and Wilson [7] treated the smooth function-germs with real isolated line singularities, and this work was later generalized and modified in the case where germs have a nonisolated singular set containing a more general set; for instance, see [8, 9]. In addition, [10] studied the relative versality for map-germs and these mapgerms with the constraint that a fixed submanifold in the source space is mapped into another fixed submanifold in the target space. Inspired by the aforementioned papers, we will study the relative infinite determinacy for map-germs with respect to a subgroup of the group A and relative finite and infinite determinacy with respect to a subgroup of the group K by means of some algebraic ideas and tools, and our main results extend the part of the works in [3, 4]. Now, we consider the following map-germs. Let 𝑁 and 𝑃 be submanifolds without boundary of R𝑛 and R𝑝 , respectively, both containing the origin. Since this paper

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Journal of Function Spaces and Applications

is concerned with a local study, without loss of generality, we may assume that 𝑁 = {0} × R𝑛−𝑛0 ⊂ R𝑛 ,

𝑃 = {0} × R𝑝−𝑝0 ⊂ R𝑝 (𝑛0 , 𝑝0 ⩾ 1) .

(1)

Denote by 𝐶𝑛𝑝 the space of map-germs 𝑓 : (R𝑛 , 0) → R , with 𝑓(𝑁) ⊂ 𝑃. Such map-germs are quite common in singularity theory and geometry. 𝑝

Example 1. Let 𝑓 : (R2 , 0) → R3 be given by (right helicoid) 𝑓 (𝑢, V) = (𝑢 cos V, 𝑢 sin V, 𝑐V) .

(2)

It is clear that 𝑓(𝑁) ⊂ 𝑃 for 𝑁 = {0} × R1 ⊂ R2 and 𝑃 = {0} × R1 ⊂ R3 .

2

Let 𝑒1 , . . . , 𝑒𝑝 be the canonical basis of the vector space R , and they define a system of generators of 𝐶𝑁-module 𝑝

×𝑝

(𝐶𝑁)

(3)

𝑓∗ : 𝐶𝑃 󳨀→ 𝐶𝑁

4

Then 𝑓(𝑁) ⊂ 𝑃 for 𝑁 = 𝑃 = {0} × R ⊂ R . In this paper we want to characterize the relative infinite determinacy for such map-germs. To formulate the main results we need to introduce some notations and definitions. Let 𝐸𝑛 denote the ring of smooth function-germs at the origin in R𝑛 , and let 𝑚𝑛 denote its unique maximal ideal. For a germ 𝑓, let 𝑗𝑘 𝑓(𝑥) denote the Taylor expansion of 𝑓 of order 𝑘 at 𝑥. In the case 𝑘 = ∞, 𝑗∞ 𝑓(𝑥) can be identified with the Taylor series of 𝑓 at 𝑥. Let R denote the group of germs at the origin of local diffeomorphisms of R𝑛 , and let 󵄨 R𝑁 = { 𝜙 ∈ R : 𝜙󵄨󵄨󵄨𝑁 ≡ id𝑁} ,

(4)

where id denotes the identity. Then R𝑁 is a subgroup of R. Let 𝐶𝑁 be the local ring {𝑓 ∈ 𝐸𝑛 : 𝑓|𝑁 ≡ constant}, and let 𝑚𝑁 denote the maximal ideal of 𝐶𝑁. Similarly, we can define the corresponding notation for (R𝑝 , 𝑃). Let M𝑁 = {𝑀 | 𝑀 : (R𝑛 , 0) → GL(𝑝, R) a 𝐶∞ map-germ and 𝑀|𝑁 = 𝐼𝑝 }, where GL(𝑝, R) denotes the general linear group, and 𝐼𝑝 denotes the (𝑝 × 𝑝) identity matrix. Now, we define two groups A𝑁,𝑃 = R𝑁 × R𝑃 , K𝑁 = M 𝑁 × R𝑁 .

(5)

Obviously, A𝑁,𝑃 and K𝑁 are subgroups of the groups A and K, respectively. In particular, when 𝑁 = 𝑃 = {0}, then A𝑁,𝑃 = A. The two groups act on the space 𝐶𝑛𝑝 in the following way. If 𝑓 ∈ 𝐶𝑛𝑝 , let (𝜙, 𝜓) ∈ A𝑁,𝑃 and (𝑀, ℎ) ∈ K𝑁; then (𝜙, 𝜓) ⋅ 𝑓 and (𝑀, ℎ) ⋅ 𝑓 are defined by (𝜙, 𝜓) ⋅ 𝑓 = 𝜓 ∘ 𝑓 ∘ 𝜙−1 ,

= 𝐶𝑁 {𝑒1 , . . . , 𝑒𝑝 } .

(7)

For any 𝑓 ∈ 𝐶𝑛𝑝 , then the germ 𝑓 induces a ring homomorphism

Example 2. Let 𝑓 : (R4 , 0) → R4 be given by 𝑓 (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) = (𝑥1 , 𝑥2 , 𝑥32 + 𝑥1 𝑥4 , 𝑥42 + 𝑥2 𝑥3 ) .

Remark 3. (1) Let 𝑓, 𝑔 ∈ 𝐶𝑛𝑝 . If 𝑓 and 𝑔 are A𝑁,𝑃 -equivalent or K𝑁-equivalent, then 𝑓|𝑁 ≡ 𝑔|𝑁. Set E(𝑓, 𝑁) = {𝑔 ∈ 𝐶𝑛𝑝 : 𝑓|𝑁 ≡ 𝑔|𝑁}. (2) In general, if 𝑓 and 𝑔 are A-equivalent, then 𝑓 and 𝑔 also are K-equivalent. However, this result does not hold for A𝑁,𝑃 and K𝑁. For example, let 𝑓(𝑥1 , 𝑥2 ) = (𝑥1 , 𝑥22 ) and 𝑔(𝑥1 , 𝑥2 ) = (𝑥1 , 𝑥1 + 𝑥22 ); where 𝑁 = 𝑃 = {0} × R1 ⊂ R2 . Set 𝜙 = id and 𝜓(𝑦1 , 𝑦2 ) = (𝑦1 , 𝑦1 + 𝑦2 ), then (𝜙, 𝜓) ∈ A𝑁,𝑃 and (𝜙, 𝜓) ⋅ 𝑓 = 𝑔. So 𝑓 is A𝑁,𝑃 -equivalent to 𝑔. But 𝑓 and 𝑔 are not K𝑁-equivalent.

(𝑀, ℎ) ⋅ 𝑓 = 𝑀 ⋅ (𝑓 ∘ ℎ) . (6)

(8)

defined by 𝑓∗ (ℎ) = ℎ ∘ 𝑓, for any ℎ ∈ 𝐶𝑃 . This allows us to consider every 𝐶𝑁-module as an 𝐶𝑃 module via 𝑓∗ . Let 𝑓∗ 𝑚𝑃 = ⟨𝑓1 , . . . , 𝑓𝑝0 ⟩ be the ideal generated by the components 𝑓1 , . . . , 𝑓𝑝0 , and let 𝑓∗ (𝑚𝑃 ) denote the image of 𝑚𝑃 under 𝑓, which is not (in general) an ideal of 𝐶𝑁. For a map-germ 𝑓 ∈ 𝐶𝑛𝑝 , define 𝑇A𝑁,𝑃 𝑓 = 𝑚𝑁 ⟨ 𝑇K𝑁𝑓 = 𝑚𝑁 ⟨

𝜕𝑓 𝜕𝑓 ,..., ⟩ + 𝑓∗ (𝐶𝑃 ) {𝑒1 , . . . , 𝑒𝑝 } , 𝜕𝑥1 𝜕𝑥𝑛

𝜕𝑓 𝜕𝑓 ×𝑝 ,..., ⟩ + 𝑓∗ 𝑚𝑝 ⋅ (𝑚𝑁) . 𝜕𝑥1 𝜕𝑥𝑛

(9)

The notation 𝑇A𝑁,𝑃 𝑓 and 𝑇K𝑁𝑓 are very nearly the tangent spaces to the orbit of germ 𝑓 under A𝑁,𝑃 -equivalence and K𝑁-equivalence, respectively. Definition 4. Let 𝑓 ∈ 𝐶𝑛𝑝 ; let 𝐺 be a group acting on 𝐶𝑛𝑝 . We say that the map-germ 𝑓 is 𝑘-𝐺-determined if for any germ 𝑔 ∈ E(𝑓, 𝑁) with 𝑗𝑘 𝑓(0) = 𝑗𝑘 𝑔(0), 𝑔 is 𝐺-equivalent to 𝑓. If 𝑓 is 𝑘-𝐺-determined for some 𝑘 < ∞, then it is finitely 𝐺-determined. If 𝑘 = ∞, we say that 𝑓 is ∞-𝐺-determined. The purpose of this paper is to characterize the ∞-A𝑁,𝑃 determinacy and ∞-K𝑁-determinacy for map-germs. The main results in this paper are stated in Theorems 5 and 10. The rest of this paper is organized as follows. In Section 2, we will give a suffcient condition for ∞-A𝑁,𝑃 -determined map-germs. In Section 3, we study the K𝑁-determinacy of map-germs. We will give necessary and sufficient conditions for a map-germ to be finitely K𝑁-determined or ∞-K𝑁determined. Throughout the paper, all map-germs will be assumed smooth.

Journal of Function Spaces and Applications

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2. The ∞-A𝑁,𝑃-Determinacy of Map-Germs In this section, the main result is the following theorem. Theorem 5. Let 𝑓 ∈ 𝐶𝑛𝑝 be a map-germ. Suppose that 𝑓 satisfies the following conditions. (1) For some 𝑟 < ∞, 𝑚𝑛𝑟 𝑚𝑁 ⊂ 𝑚𝑁𝐽(𝑓) + 𝑓∗ 𝑚𝑃 ⋅ 𝑚𝑁, where 𝐽(𝑓) denotes the ideal in 𝐸𝑛 generated by the determinants of (𝑝 × 𝑝) minors of the Jacobian matrix 𝑑𝑓 of 𝑓. (2) (𝑚𝑛∞ 𝑚𝑁)×𝑝 ⊂ 𝑇A𝑁,𝑃 𝑓.

(1) 𝜙(𝑥, 𝑡) = 𝑥 and 𝜓(𝑦, 𝑡) = 𝑦, for any 𝑡 sufficiently close to 𝑡0 , 𝑥 ∈ 𝑁 and 𝑦 ∈ 𝑃.

Then 𝑓 is ∞-A𝑁,𝑃 -determined. In order to prove this theorem, we need the following results. 𝐶𝑛𝑝

and 𝑀 be a finitely generated Lemma 6 (see [10]). Let 𝑓 ∈ 𝐶𝑁-module. Then 𝑀 is finitely generated as a 𝑓∗ (𝐶𝑃 )-module if and only if dimR (𝑀/𝑓∗ 𝑚𝑃 ⋅ 𝑀) < ∞. Lemma 7. Let 𝑓, 𝑔 ∈ 𝐶𝑛𝑝 . Let 𝑀 be a finitely generated (𝑓, 𝑔)∗ (𝐶𝑃×𝑃 )-module. Then 𝑀 is finitely generated as a 𝑔∗ (𝐶𝑃 )-module if and only if dimR

where 𝐶𝑁×{𝑡0 } (resp., 𝐶𝑃×{𝑡0 } ) denotes the ring of functiongerms at (0, 𝑡0 ) which are constant when restricted to 𝑁 × {𝑡0 } (resp., 𝑃 × {𝑡0 }). 𝑘 Set 𝑚𝑁 𝐶𝑁×{𝑡0 } = 𝑚𝑇𝑘 and 𝑚𝑃𝑘 𝐶𝑃×{𝑡0 } = 𝑚𝑇𝑘 󸀠 . Let 𝑇A𝑁,𝑃 𝐹 = 𝑡𝐹 + 𝜔𝐹, where 𝑡𝐹 denotes 𝑚𝑇 ⟨𝜕𝑓/𝜕𝑥1 , . . . , 𝜕𝑓/𝜕𝑥𝑛 ⟩ and 𝜔𝐹 denotes 𝐹∗ (𝐶𝑃×{𝑡0 } ){𝑒1 , . . . , 𝑒𝑝 }. We are trying to show that 𝑓̃ is A𝑁,𝑃 -equivalent to 𝑓. It suffices to show that there exist germs 𝜙 : (R𝑛 × R, (0, 𝑡0 )) → (R𝑛 , 0) and 𝜓 : (R𝑝 × R, (0, 𝑡0 )) → (R𝑝 , 0) satisfying the following conditions.

𝑀 < ∞. 𝑔∗ 𝑚𝑃 ⋅ 𝑀

(10)

(2) 𝜙𝑡0 = idR𝑛 and 𝜓𝑡0 = idR𝑝 . (3) 𝑔𝑡0 = 𝜓𝑡 ∘ 𝑔𝑡 ∘ 𝜙𝑡−1 , for any 𝑡 sufficiently close to 𝑡0 . By the method initiated by Mather in [12], it suffices to show that ×𝑝

(𝑚𝑛∞ 𝑚𝑇 )

⊂ 𝑚𝑇 ⟨

𝜕𝑔 𝜕𝑔 ,..., ⟩ + 𝐺∗ (𝑚𝑇󸀠 ) {𝑒1 , . . . , 𝑒𝑝 } . 𝜕𝑥1 𝜕𝑥𝑛 (16)

To see this, it remains to show that (i) (𝑚𝑛∞ 𝑚𝑇 )×𝑝 ⊂ 𝑇A𝑁,𝑃 𝐹.

Proof. The proof is essentially the same as that of Corollary 1.17 in [11].

(ii) 𝑇A𝑁,𝑃 𝐹 = 𝑇A𝑁,𝑃 𝐺.

Lemma 8. Let 𝑓, 𝑔 ∈ 𝐶𝑛𝑝 and 𝑔 − 𝑓 ∈ (𝑚𝑛∞ 𝑚𝑁)×𝑝 . Then

If (i) and (ii) hold, then

∗

(𝑓, 𝑔) (𝐶𝑃×𝑃 ) ⊂ 𝑓∗ (𝐶𝑃 ) + 𝑚𝑛∞ 𝑚𝑁.

Proof. For any ℎ ∈ 𝐶𝑃×𝑃 , then (𝑓, 𝑔)∗ (ℎ) − (𝑓, 𝑓)∗ (ℎ) is in 𝑚𝑛∞ 𝑚𝑁. So, ∗

∗

(𝑓, 𝑔) (𝐶𝑃×𝑃 ) ⊂ (𝑓, 𝑓) (𝐶𝑃×𝑃 ) + 𝑚𝑛∞ 𝑚𝑁.

(12)

Let 𝜙 : (R𝑝 , 𝑃) → (R𝑝 × R𝑝 , 𝑃 × 𝑃) be given by 𝜙(𝑦) = (𝑦, 𝑦). Since (𝑓, 𝑓)∗ (ℎ) = 𝑓∗ (𝜙∗ (ℎ)), it follows that ∗

∗

(𝑓, 𝑓) (𝐶𝑃×𝑃 ) ⊂ 𝑓 (𝐶𝑃 ) .

×𝑝

(𝑚𝑛∞ 𝑚𝑇 )

(11)

×𝑝

𝐺∗ 𝑚𝑇󸀠 ⋅ (𝑚𝑛∞ 𝑚𝑇 )

𝐹∗ : 𝐶𝑃×{𝑡0 } 󳨀→ 𝐶𝑁×{𝑡0 } ,

ℎ 󳨃󳨀→ ℎ ∘ 𝐹 = 𝐹∗ (ℎ) ,

(18)

On the other hand, using condition (1) in Theorem 5, and ×𝑝

⊂ 𝑚𝑁 ⟨

𝜕𝑓 𝜕𝑓 ,..., ⟩, 𝜕𝑥1 𝜕𝑥𝑛

(19)

we get ×𝑝

(𝑚𝑛𝑟 𝑚𝑁)

⊂ 𝑚𝑁 ⟨

(14)

(15)

𝜕𝑔 𝜕𝑔 ,..., ⟩ 𝜕𝑥1 𝜕𝑥𝑛

+ 𝐺 (𝑚𝑇󸀠 ) {𝑒1 , . . . , 𝑒𝑝 } .

𝐽 (𝑓) ⋅ (𝑚𝑁)

by 𝑔(𝑥, 𝑡) = 𝑓(𝑥) + 𝑡𝑢(𝑥) and 𝑔𝑡 (𝑥) = 𝑔(𝑥, 𝑡). Let 𝐹(𝑥, 𝑡) = (𝑓(𝑥), 𝑡) and 𝐺(𝑥, 𝑡) = (𝑔(𝑥, 𝑡), 𝑡) denote the map-germs at (0, 𝑡0 ); then 𝐹 (or 𝐺) induces a ring homomorphism:

⊂ 𝑚𝑇 ⟨ ∗

Thus, (11) holds.

𝑔 : (R𝑛 × R, (0, 𝑡0 )) 󳨀→ R𝑝

(17)

Multiplying (17) by 𝐺∗ 𝑚𝑇󸀠 , we get

(13)

̃ = Proof of Theorem 5. Suppose that 𝑓̃ ∈ E(𝑓, 𝑁) and 𝑗∞ 𝑓(0) 𝑗∞ 𝑓(0). Let 𝑢 = 𝑓̃ − 𝑓; then 𝑢 ∈ (𝑚𝑛∞ 𝑚𝑁)×𝑝 . For any 𝑡0 ∈ [0, 1], define

⊂ 𝑇A𝑁,𝑃 𝐺.

𝜕𝑓 𝜕𝑓 ×𝑝 ,..., ⟩ + 𝑓∗ 𝑚𝑃 ⋅ (𝑚𝑁) . 𝜕𝑥1 𝜕𝑥𝑛 (20)

Obviously, we have ×𝑝

(𝑚𝑛∞ 𝑚𝑁)

⊂ 𝑚𝑁 ⟨

𝜕𝑓 𝜕𝑓 ×𝑝 ,..., ⟩ + 𝑓∗ 𝑚𝑃 ⋅ (𝑚𝑁) . 𝜕𝑥1 𝜕𝑥𝑛 (21)

4

Journal of Function Spaces and Applications Since 𝑔𝑡 − 𝑓 ∈ (𝑚𝑛∞ 𝑚𝑁)×𝑝 , for each 𝑡 ∈ R, this gives that 𝑚𝑁 ⟨

𝜕𝑔 𝜕𝑔 ×𝑝 ,..., ⟩ + 𝑔𝑡∗ 𝑚𝑃 ⋅ (𝑚𝑁) 𝜕𝑥1 𝜕𝑥𝑛

⊂ 𝑚𝑁 ⟨ 𝑚𝑁 ⟨

⊂ 𝑚𝑁 ⟨

𝐶𝑁×{𝑡0 } = 𝑚𝑇 𝐽 (𝑓) + 𝐹∗ (𝐶𝑃×{𝑡0 } ) 𝐶𝑁.

𝑇A𝑁,𝑃 𝐹 ⊃ (𝑚𝑁 ⟨ (22)

𝜕𝑓 𝜕𝑓 ,..., ⟩ 𝜕𝑥1 𝜕𝑥𝑛

+𝐹∗ (𝐶𝑃×{𝑡0 } ) {𝑒1 , . . . , 𝑒𝑝 } ) ⋅ 𝐹∗ (𝐶𝑃×{𝑡0 } )

𝜕𝑔 𝜕𝑔 ×𝑝 ,..., ⟩ + 𝑔𝑡∗ 𝑚𝑃 ⋅ (𝑚𝑁) 𝜕𝑥1 𝜕𝑥𝑛

⊃ 𝑇A𝑁,𝑃 𝑓 ⋅ 𝐹∗ (𝐶𝑃×{𝑡0 } )

×𝑝

+ 𝑚𝑛 (𝑚𝑛∞ 𝑚𝑁) .

×𝑝

⊃ (𝑚𝑛∞ 𝑚𝑁)

⋅ 𝐹∗ (𝐶𝑃×{𝑡0 } ) , (31)

Hence, by Nakayama’s lemma we have 𝑚𝑁 ⟨

𝜕𝑓 𝜕𝑓 ×𝑝 ,..., ⟩ + 𝑓∗ 𝑚𝑃 ⋅ (𝑚𝑁) 𝜕𝑥1 𝜕𝑥𝑛

𝜕𝑔 𝜕𝑔 ×𝑝 = 𝑚𝑁 ⟨ ,..., ⟩ + 𝑔𝑡∗ 𝑚𝑃 ⋅ (𝑚𝑁) . 𝜕𝑥1 𝜕𝑥𝑛

and 𝑚𝑁𝐽(𝑓) ⋅ (𝐶𝑁×{𝑡0 } )×𝑝 ⊂ 𝑚𝑇 ⟨𝜕𝑓/𝜕𝑥1 , . . . , 𝜕𝑓/𝜕𝑥𝑛 ⟩, it follows that (23)

×𝑝

𝑇A𝑁,𝑃 𝐹 ⊃ (𝑚𝑛∞ 𝑚𝑁)

⋅ [𝑚𝑇 𝐽 (𝑓) + 𝐹∗ (𝐶𝑃×{𝑡0 } ) 𝐶𝑁]

From (21), it follows that ×𝑝 (𝑚𝑛∞ 𝑚𝑁)

𝜕𝑔 𝜕𝑔 ×𝑝 ⊂ 𝑚𝑁 ⟨ ,..., ⟩ + 𝑔𝑡∗ 𝑚𝑃 ⋅ (𝑚𝑁) , 𝜕𝑥1 𝜕𝑥𝑛 (24)

for any given 𝑡 ∈ R. Multiplying (24) by 𝑚𝑛∞ 𝐶𝑁×{𝑡0 } , we get ×𝑝

(𝑚𝑛∞ 𝑚𝑇 )

⊂ 𝑚𝑇 ⟨

∗

⊂ 𝑚𝑁𝐽 (𝑓) + 𝑓 𝑚𝑃 ⋅ 𝐶𝑁,

𝑀 ⋅𝑀

≅

𝐶𝑁 . 𝑚𝑁𝐽 (𝑓) + 𝑓∗ 𝑚𝑃 ⋅ 𝐶𝑁

(33)

Applying (25) and (i), we have ×𝑝

(𝑚𝑛∞ 𝑚𝑇 )

⊂ 𝑚𝑇 ⟨

𝜕𝑔 𝜕𝑔 ,..., ⟩ 𝜕𝑥1 𝜕𝑥𝑛

𝜕𝑔 𝜕𝑔 ⊂ 𝑚𝑇 ⟨ ,..., ⟩ 𝜕𝑥1 𝜕𝑥𝑛

(26)

(27)

(28)

Thus, by Lemma 6, 𝑀 is finitely generated 𝐹∗ (𝐶𝑃×{𝑡0 } )module; that is, 𝑀 = 𝐹∗ (𝐶𝑃×{𝑡0 } ) {𝑎1 , . . . , 𝑎𝑠 } .

Proof of Assertion (ii). Since 𝑔 − 𝑓 ∈ (𝑚𝑛∞ 𝑚𝑇 )×𝑝 , by (i) we get

×𝑝

where 𝑎𝑖 ∈ 𝐶𝑁 and 𝑎𝑖 is the projection of 𝑎𝑖 in the quotient space, 𝑖 = 1, . . . , 𝑠. Now, set 𝑀 = 𝐶𝑁×{𝑡0 } /𝑚𝑇 𝐽(𝑓); then 𝑀 is a finitely generated 𝐶𝑁×{𝑡0 } -module. Since 𝐹∗ 𝑚𝑃×{𝑡0 } ⋅ 𝐶𝑁×{𝑡0 } = 𝑓∗ 𝑚𝑃 ⋅ 𝐶𝑁×{𝑡0 } + (𝑡 − 𝑡0 )𝐶𝑁×{𝑡0 } ; 𝐹∗ 𝑚𝑃×{𝑡0 }

So (i) holds.

+ 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ (𝑚𝑛∞ 𝑚𝑇 )

for some 𝑟 < ∞. This means that ideal 𝑚𝑁𝐽(𝑓) + 𝑓∗ 𝑚𝑃 ⋅ 𝐶𝑁 has finite codimension in 𝐶𝑁. Let 𝐶𝑁 = R {𝑎1 , . . . , 𝑎𝑠 } , (𝑚𝑁𝐽 (𝑓) + 𝑓∗ 𝑚𝑃 ⋅ 𝐶𝑁)

×𝑝

×𝑝

𝜕𝑔 𝜕𝑔 ×𝑝 ,..., ⟩ + 𝐺∗ 𝑚𝑇󸀠 ⋅ (𝑚𝑛∞ 𝑚𝑇 ) . 𝜕𝑥1 𝜕𝑥𝑛 (25)

(32)

= (𝑚𝑛∞ 𝑚𝑇 ) .

𝑇A𝑁,𝑃 𝐺 ⊆ 𝑇A𝑁,𝑃 𝐹 = 𝑇A𝑁,𝑃 𝐺 + (𝑚𝑛∞ 𝑚𝑇 ) .

Thus, (16) follows by substituting (18) in (25). Now we prove the assertion (i). By hypothesis, we have 𝑚𝑛𝑟 𝑚𝑁

(30)

Since

𝜕𝑓 𝜕𝑓 ×𝑝 ,..., ⟩ + 𝑓∗ 𝑚𝑃 ⋅ (𝑚𝑁) , 𝜕𝑥1 𝜕𝑥𝑛

𝜕𝑓 𝜕𝑓 ×𝑝 ,..., ⟩ + 𝑓∗ 𝑚𝑃 ⋅ (𝑚𝑁) 𝜕𝑥1 𝜕𝑥𝑛

In particular,

(29)

(34)

+ 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝑇A𝑁,𝑃 𝐹. By (33) and (34), we have 𝑇A𝑁,𝑃 𝐺 ⊆ 𝑇A𝑁,𝑃 𝐹 = 𝑇A𝑁,𝑃 𝐺 + 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝑇A𝑁,𝑃 𝐹. (35) Set 𝐸 = 𝑇A𝑁,𝑃 𝐹/𝑡𝐺, and (35) implies 𝐸=

𝑇A𝑁,𝑃 𝐺 + 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝐸. 𝑡𝐺

(36)

Now it remains to show that (a) 𝐸 is a finitely generated 𝐺∗ (𝐶𝑃×{𝑡0 } )-module. For if this holds, then by Nakayama’s lemma, 𝐸=

𝑇A𝑁,𝑃 𝐺 . 𝑡𝐺

(37)

Journal of Function Spaces and Applications

5

Hence, 𝑇A𝑁,𝑃 𝐹 = 𝑇A𝑁,𝑃 𝐺. To prove (a), by Lemma 7, it suffices to show that (b) 𝐸 is a finitely generated (𝐹, 𝐺)∗ (𝐶𝑃×{𝑡0 }×𝑃×{𝑡0 } )-module. (c) dimR (𝐸/𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝐸) < ∞. Set 𝑅 = (𝐹, 𝐺)∗ (𝐶𝑃×{𝑡0 }×𝑃×{𝑡0 } ). By Lemma 8 and (i), we get 𝑅 ⋅ 𝑇A𝑁,𝑃 𝐹 ⊂ [𝐹∗ (𝐶𝑃×{𝑡0 } ) + 𝑚𝑛∞ 𝑚𝑇 ] ⋅ 𝑇A𝑁,𝑃 𝐹 ⊂ 𝑇A𝑁,𝑃 𝐹. (38) Thus, 𝑇A𝑁,𝑃 𝐹 is a 𝑅-module. Hence, 𝐸 is a 𝑅-module. Obviously, 𝜔𝐹 is a finitely generated 𝐹∗ (𝐶𝑃×{𝑡0 } )-module; hence 𝜔𝐹 is also finitely generated as a 𝑅-module. Moreover, since 𝑡𝐹/𝑡𝐺 ∩ 𝑡𝐹 is a finitely generated 𝐶𝑁×{𝑡0 } module and annihilated by 𝑚𝑇 𝐽(𝑔𝑡 ). Thus, 𝑡𝐹/𝑡𝐺 ∩ 𝑡𝐹 is a finitely generated 𝐶𝑁×{𝑡0 } /𝑚𝑇 𝐽(𝑔𝑡 )-module. By the argument as equality (29), it follows that 𝐶𝑁×{𝑡0 } /𝑚𝑇 𝐽(𝑔𝑡 ) is a finitely generated 𝐺∗ (𝐶𝑃×{𝑡0 } )-module. Hence, 𝑡𝐹/𝑡𝐺 ∩ 𝑡𝐹 is a finitely generated 𝑅-module. The earlier argument shows that 𝐸 is finitely generated as a 𝑅-module. Besides, by (35), we have dimR

𝐸 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝐸

= dimR

= dimR ≤ dimR

𝑇A𝑁,𝑃 𝐺 + 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝑇A𝑁,𝑃 𝐹 𝑡𝐺 + 𝐺∗ 𝑚𝑃×{𝑡0 } (𝑇A𝑁,𝑃 𝐺 + 𝑇A𝑁,𝑃 𝐹)

Taking (𝑟 + 1)-jets on both sides, we have {𝑗𝑟+1 𝑔 (0) ∈ 𝐽𝑟+1 (𝑛, 𝑝) : 𝑗𝑟 𝑔 (0) = 𝑗𝑟 𝑓 (0)} ⊂ {𝑗𝑟+1 𝑔 (0) ∈ 𝐽𝑟+1 (𝑛, 𝑝) : 𝑔 and 𝑓 are K𝑁-equivalent} . Taking tangent spaces at 𝑗𝑟+1 𝑓(0) on both sides, we have ×𝑝

(𝑚𝑛𝑟 𝑚𝑁)

⋅ 𝜔𝐺 + (𝑡𝐺 +

𝐺∗ 𝑚𝑃×{𝑡0 }

⋅ 𝑇A𝑁,𝑃 𝐹)

𝜔𝐺 < ∞. 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝜔𝐺

×𝑝

(𝑚𝑛𝑟 𝑚𝑁)

𝐹 : (R𝑛 × R, (0, 𝑡0 )) 󳨀→ R𝑝

(43)

(44)

by 𝐹(𝑥, 𝑡) = 𝑓(𝑥) + 𝑡(𝑔(𝑥) − 𝑓(𝑥)) and 𝐹𝑡 (𝑥) = 𝐹(𝑥, 𝑡). We are trying to show that 𝑔 is K𝑁-equivalent to 𝑓. Since 𝐹0 = 𝑓 and 𝐹1 = 𝑔, and [0, 1] is connected, it suffices to show that 𝐹𝑡 is K𝑁-equivalent to 𝐹𝑡0 , for all 𝑡 sufficiently close to 𝑡0 . Since 𝐹𝑡0 (𝑥) − 𝑓(𝑥) = 𝑡0 (𝑔(𝑥) − 𝑓(𝑥)) and 𝑔 − 𝑓 ∈ (𝑚𝑛𝑟+1 𝑚𝑁)×𝑝 , from condition (2), it follows that 𝑇K𝑁𝐹𝑡0 ⊂ 𝑇K𝑁𝑓,

(45)

and 𝑇K𝑁𝑓 ⊂ 𝑇K𝑁𝐹𝑡0 + 𝑚𝑛 ⋅ (𝑇K𝑁𝑓). Since 𝑇K𝑁𝑓 is a finitely generated 𝐸𝑛 -module, and 𝑚𝑛 is the maximal ideal of 𝐸𝑛 , by Nakayama’s lemma, we get 𝑇K𝑁𝐹𝑡0 = 𝑇K𝑁𝑓.

∗ (𝑚𝑛𝑟 𝑚𝑁𝐸𝑛+1 )

In this section, by a similar way as [13], we will give necessary and sufficient conditions for finitely K𝑁-determined mapgerms and ∞-K𝑁-determined map-germs. Theorem 9. Suppose that 𝑓 ∈ 𝐶𝑛𝑝 . The following conditions are equivalent. (1) 𝑓 is finitely K𝑁-determined. (2) (𝑚𝑛𝑟 𝑚𝑁)×𝑝 ⊂ 𝑇K𝑁𝑓, for some 𝑟 ∈ N.

×𝑝

∗ ⊂ 𝑇K𝑁𝐹𝑡0 ⋅ 𝐸𝑛+1 ,

(46)

Proof. Let 𝐽 (𝑛, 𝑝) denote the set of (𝑟 + 1)-jets at 0 of elements in E(𝑓, 𝑁). If 𝑓 is 𝑟-K𝑁-determined, then for any 𝑔 ∈ E(𝑓, 𝑁) with the same 𝑟-jet as 𝑓, the K𝑁-orbit of 𝑓 contains 𝑔; that is, {𝑔 ∈ E (𝑓, 𝑁) : 𝑗𝑟 𝑔 (0) = 𝑗𝑟 𝑓 (0)} (40)

(47)

∗ where 𝐸𝑛+1 denotes the ring of smooth function-germs in variables (𝑥, 𝑡) at the point (0, 𝑡0 ). ∗ ∗ denote the maximal ideal of 𝐸𝑛+1 . Let 𝑚𝑛+1 ∗ Let 𝑇K𝑁𝐹 = 𝑚𝑁𝐸𝑛+1 ⟨𝜕𝐹/𝜕𝑥1 , . . . , 𝜕𝐹/𝜕𝑥𝑛 ⟩ + 𝐹∗ 𝑚𝑝 ⋅ ∗ ∗ (𝑚𝑁𝐸𝑛+1 )×𝑝 . Obviously, 𝑇K𝑁𝐹 is a finitely generated 𝐸𝑛+1 module. By the same argument as (46), we have ∗ 𝑇K𝑁𝐹 = 𝑇K𝑁𝐹𝑡0 ⋅ 𝐸𝑛+1 .

𝑟+1

are K𝑁-equivalent} .

⊂ 𝑇K𝑁𝑓.

Hence 𝑇K𝑁𝐹𝑡0 also satisfies condition (2). So

3. The K𝑁-Determinacy of Map-Germs

⊂ {𝑔 ∈ E (𝑓, 𝑁) : 𝑔 and 𝑓

(42)

Then (1) implies (2). Conversely, let 𝑡0 ∈ [0, 1] be fixed and 𝑔 ∈ E(𝑓, 𝑁) with 𝑗𝑟+1 𝑓(0) = 𝑗𝑟+1 𝑔(0). Define

(39) So (b) and (c) hold. This completes the proof.

×𝑝

⊂ 𝑇K𝑁𝑓 + (𝑚𝑛𝑟+1 𝑚𝑁) .

By Nakayama’s lemma, this implies

𝜔𝐺 + (𝑡𝐺 + 𝐺∗ 𝑚𝑃×{𝑡0 } ⋅ 𝑇A𝑁,𝑃 𝐹) 𝐺∗ 𝑚𝑃×{𝑡0 }

(41)

(48)

By (47) and (48), we get ∗ (𝑚𝑛𝑟 𝑚𝑁𝐸𝑛+1 )

×𝑝

⊂ 𝑇K𝑁𝐹.

(49)

Since 𝜕𝐹/𝜕𝑡 = 𝑔 − 𝑓 ∈ (𝑚𝑛𝑟+1 𝑚𝑁)×𝑝 , this means that there ∗ , 𝑖 = 1, . . . , 𝑛, such that exist germs 𝑋𝑖 in 𝑚𝑁𝐸𝑛+1 𝑛

(𝑔 − 𝑓) + ∑𝑋𝑖 (𝑥, 𝑡) 𝑖=1

𝜕𝐹 ×𝑝 ∗ ∈ 𝐹∗ 𝑚𝑝 ⋅ (𝑚𝑁𝐸𝑛+1 ) . 𝜕𝑥𝑖

(50)

6

Journal of Function Spaces and Applications

Thus, we can find a germ of vector field 𝑋 in R𝑛 × R of the following form: 𝑛 𝜕 𝜕 , + ∑𝑋𝑖 (𝑥, 𝑡) 𝜕𝑡 𝑖=1 𝜕𝑥𝑖

(51)

∗ such that DF(𝑋) ∈ 𝐹∗ 𝑚𝑝 ⋅ (𝑚𝑁𝐸𝑛+1 )×𝑝 . That is, we can find a (𝑝 × 𝑝) matrix 𝐴(𝑥, 𝑡) with entries ∗ such that in 𝑚𝑁𝐸𝑛+1

DF (𝑋) = 𝐴 (𝑥, 𝑡) ⋅ 𝐹 (𝑥, 𝑡) .

(52)

By integrating the vector field 𝑋, we get a one-parameter family of local diffeomorphisms 𝜙𝑡 in R𝑁. Thus 𝑑 𝜕 𝐹 (𝜙𝑡 (𝑥) , 𝑡) = 𝐹 (𝜙𝑡 (𝑥) , 𝑡) 𝑑𝑡 𝜕𝑡 𝑛

+ ∑𝑋𝑖 (𝜙𝑡 (𝑥) , 𝑡) 𝑖=1

𝜕𝐹 (𝜙 (𝑥) , 𝑡) 𝜕𝑥𝑖 𝑡

(53)

= 𝐴 (𝜙𝑡 (𝑥) , 𝑡) ⋅ 𝐹 (𝜙𝑡 (𝑥) , 𝑡) . Hence, for fixed 𝑥 ∈ R𝑛 , 𝐹(𝜙𝑡 (𝑥), 𝑡) is a solution of the differential equation 𝑦̇ = 𝐴(𝜙𝑡 (𝑥), 𝑡)𝑦 with initial condition 𝑦(𝑥, 𝑡0 ) = 𝐹𝑡0 (𝑥). Since the solution of this differential equation is unique and of the form 𝑦 (𝑥, 𝑡) = 𝐵 (𝑥, 𝑡) ⋅ 𝑦 (𝑥, 𝑡0 ) ,

(54)

with 𝐵(𝑥, 𝑡) as an invertible matrix, 𝐵(𝑥, 𝑡0 ) = 𝐼𝑝 and 𝐵(𝑥, 𝑡) = 𝐼𝑝 for each 𝑥 ∈ 𝑁. Thus 𝐹𝑡 (𝑥) = 𝐵 (𝜙𝑡−1 (𝑥) , 𝑡) ⋅ (𝐹𝑡0 ∘ 𝜙𝑡−1 (𝑥)) .

(55)

Thus, 𝐹𝑡 and 𝐹𝑡0 are K𝑁-equivalent for all 𝑡 sufficiently close to 𝑡0 , which completes the proof. Theorem 10. Suppose that 𝑓 ∈ 𝐶𝑛𝑝 . Then 𝑓 is ∞-K𝑁determined if and only if ×𝑝

(𝑚𝑛∞ 𝑚𝑁)

⊂ 𝑇K𝑁𝑓.

(56)

Proof. “Only if ”: if 𝑓 is ∞-K𝑁-determined, by the definition, we get ×𝑝

𝑓 + (𝑚𝑛∞ 𝑚𝑁)

⊂ K𝑁 ⋅ 𝑓.

(57)

Taking tangent spaces at 𝑓 on both sides, we have ×𝑝

𝑇𝑓 (𝑓 + (𝑚𝑛∞ 𝑚𝑁) ) ⊂ 𝑇𝑓 (K𝑁 ⋅ 𝑓) .

(58)

Note that (𝑚𝑛∞ 𝑚𝑁)×𝑝 ⊂ 𝑇𝑓 (𝑓+(𝑚𝑛∞ 𝑚𝑁)×𝑝 ), and 𝑇𝑓 (K𝑁 ⋅ 𝑓) ⊂ 𝑇K𝑁𝑓. Hence, ×𝑝

(𝑚𝑛∞ 𝑚𝑁)

⊂ 𝑇K𝑁𝑓.

“If ”: the proof is the same as that of Theorem 9.

(59)

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 11271063) and supported in part by Graduate Innovation Fund of Northeast Normal University of China (no. 12SSXT140). The authors would like to thank the referee for his/her valuable suggestions which improved the first version of the paper.

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Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

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Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

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Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014