Relative Infinite Determinacy for Map-Germs

Abstract

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-left-equivalence 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 map-germs 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 ℝ^𝑛 and ℝ^𝑝, respectively, both containing the origin. Since this paper is concerned with a local study, without loss of generality, we may assume that 𝑁={0}×ℝ^𝑛−𝑛0⊂ℝ^𝑛,𝑃={0}×ℝ^𝑝−𝑝0⊂ℝ^𝑝𝑛₀,𝑝₀⩾1. (1)

Denote by 𝐶^𝑝_𝑛 the space of map-germs 𝑓∶(ℝ^𝑛,0)→ℝ^𝑝, with 𝑓(𝑁)⊂𝑃. Such map-germs are quite common in singularity theory and geometry.

Example 1. Let 𝑓∶(ℝ²,0)→ℝ³ be given by (right helicoid) 𝑓(𝑢,𝑣)=(𝑢cos𝑣,𝑢sin𝑣,𝑐𝑣). (2) It is clear that 𝑓(𝑁)⊂𝑃 for 𝑁={0}×ℝ¹⊂ℝ² and 𝑃={0}×ℝ¹⊂ℝ³.

Example 2. Let 𝑓∶(ℝ⁴,0)→ℝ⁴ be given by 𝑓𝑥₁,𝑥₂,𝑥₃,𝑥₄=𝑥₁,𝑥₂,𝑥²₃+𝑥₁𝑥₄,𝑥²₄+𝑥₂𝑥₃. (3)
Then 𝑓(𝑁)⊂𝑃 for 𝑁=𝑃={0}×ℝ²⊂ℝ⁴.

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 𝐸_𝑛 denote the ring of smooth function-germs at the origin in ℝ^𝑛, and let 𝑚_𝑛 denote its unique maximal ideal. For a germ 𝑓, let 𝑗^𝑘𝑓(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 ℝ^𝑛, and let ℛ_𝑁=𝜙∈ℛ∶𝜙_𝑁≡id_𝑁, (4) where id denotes the identity. Then ℛ_𝑁 is a subgroup of ℛ.

Let 𝐶_𝑁 be the local ring {𝑓∈𝐸_𝑛∶𝑓|_𝑁≡constant}, and let 𝑚_𝑁 denote the maximal ideal of 𝐶_𝑁.

Similarly, we can define the corresponding notation for (ℝ^𝑝,𝑃).

Let ℳ_𝑁={𝑀∣𝑀∶(ℝ^𝑛,0)→GL(𝑝,ℝ)a𝐶^∞map-germand𝑀|_𝑁=𝐼_𝑝}, where GL(𝑝,ℝ) denotes the general linear group, and 𝐼_𝑝 denotes the (𝑝×𝑝) identity matrix.

Now, we define two groups 𝒜_𝑁,𝑃=ℛ_𝑁×ℛ_𝑃,𝒦_𝑁=ℳ_𝑁×ℛ_𝑁. (5)

Obviously, 𝒜_𝑁,𝑃 and 𝒦_𝑁 are subgroups of the groups 𝒜 and 𝒦, respectively. In particular, when 𝑁=𝑃={0}, then 𝒜_𝑁,𝑃=𝒜. The two groups act on the space 𝐶^𝑝_𝑛 in the following way.

If 𝑓∈𝐶^𝑝_𝑛, let (𝜙,𝜓)∈𝒜_𝑁,𝑃 and (𝑀,ℎ)∈𝒦_𝑁; then (𝜙,𝜓)⋅𝑓 and (𝑀,ℎ)⋅𝑓 are defined by 𝜙,𝜓⋅𝑓=𝜓∘𝑓∘𝜙⁻¹,(𝑀,ℎ)⋅𝑓=𝑀⋅𝑓∘ℎ. (6)

Remark 3. (1) Let 𝑓,𝑔∈𝐶^𝑝_𝑛. If 𝑓 and 𝑔 are 𝒜_𝑁,𝑃-equivalent or 𝒦_𝑁-equivalent, then 𝑓|_𝑁≡𝑔|_𝑁. Set ℰ(𝑓,𝑁)={𝑔∈𝐶^𝑝_𝑛∶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 𝒜_𝑁,𝑃 and 𝒦_𝑁. For example, let 𝑓(𝑥₁,𝑥₂)=(𝑥₁,𝑥²₂) and 𝑔(𝑥₁,𝑥₂)=(𝑥₁,𝑥₁+𝑥²₂); where 𝑁=𝑃={0}×ℝ¹⊂ℝ². Set𝜙=id and 𝜓(𝑦₁,𝑦₂)=(𝑦₁,𝑦₁+𝑦₂), then (𝜙,𝜓)∈𝒜_𝑁,𝑃 and (𝜙,𝜓)⋅𝑓=𝑔. So 𝑓 is 𝒜_𝑁,𝑃-equivalent to 𝑔. But 𝑓 and 𝑔 are not 𝒦_𝑁-equivalent.

Let 𝑒₁,…,𝑒_𝑝 be the canonical basis of the vector space ℝ^𝑝, and they define a system of generators of 𝐶_𝑁-module 𝐶_𝑁^×𝑝=𝐶_𝑁𝑒₁,…,𝑒_𝑝. (7)

For any 𝑓∈𝐶^𝑝_𝑛, then the germ 𝑓 induces a ring homomorphism 𝑓^∗∶𝐶_𝑃⟶𝐶_𝑁 (8) defined by 𝑓^∗(ℎ)=ℎ∘𝑓, for any ℎ∈𝐶_𝑃.

This allows us to consider every 𝐶_𝑁-module as an 𝐶_𝑃-module via 𝑓^∗.

Let 𝑓^∗𝑚_𝑃=⟨𝑓₁,…,𝑓_𝑝0⟩ be the ideal generated by the components 𝑓₁,…,𝑓_𝑝0, and let 𝑓^∗(𝑚_𝑃) denote the image of 𝑚_𝑃 under 𝑓, which is not (in general) an ideal of 𝐶_𝑁.

For a map-germ 𝑓∈𝐶^𝑝_𝑛, define 𝑇𝒜_𝑁,𝑃𝑓=𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝑓^∗𝐶_𝑃𝑒₁,…,𝑒_𝑝,𝑇𝒦_𝑁𝑓=𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝑓^∗𝑚_𝑝⋅𝑚_𝑁^×𝑝. (9)

The notation 𝑇𝒜_𝑁,𝑃𝑓 and 𝑇𝒦_𝑁𝑓 are very nearly the tangent spaces to the orbit of germ 𝑓 under 𝒜_𝑁,𝑃-equivalence and 𝒦_𝑁-equivalence, respectively.

Definition 4. Let 𝑓∈𝐶^𝑝_𝑛; let 𝐺 be a group acting on 𝐶^𝑝_𝑛. We say that the map-germ 𝑓 is 𝑘-𝐺-determined if for any germ 𝑔∈ℰ(𝑓,𝑁) with 𝑗^𝑘𝑓(0)=𝑗^𝑘𝑔(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 ∞-𝒜_𝑁,𝑃-determinacy and ∞-𝒦_𝑁-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 ∞-𝒜_𝑁,𝑃-determined map-germs. In Section 3, we study the 𝒦_𝑁-determinacy of map-germs. We will give necessary and sufficient conditions for a map-germ to be finitely 𝒦_𝑁-determined or ∞-𝒦_𝑁-determined.

Throughout the paper, all map-germs will be assumed smooth.

2. The ∞-𝒜_𝑁,𝑃-Determinacy of Map-Germs

In this section, the main result is the following theorem.

Theorem 5. Let 𝑓∈𝐶^𝑝_𝑛 be a map-germ. Suppose that 𝑓 satisfies the following conditions. (1)For some 𝑟<∞, 𝑚^𝑟_𝑛𝑚_𝑁⊂𝑚_𝑁𝐽(𝑓)+𝑓^∗𝑚_𝑃⋅𝑚_𝑁, where 𝐽(𝑓) denotes the ideal in 𝐸_𝑛 generated by the determinants of (𝑝×𝑝) minors of the Jacobian matrix 𝑑𝑓 of 𝑓. (2)(𝑚^∞_𝑛𝑚_𝑁)^×𝑝⊂𝑇𝒜_𝑁,𝑃𝑓.
Then 𝑓 is ∞-𝒜_𝑁,𝑃-determined.

In order to prove this theorem, we need the following results.

Lemma 6 (see [10]). Let 𝑓∈𝐶^𝑝_𝑛 and 𝑀 be a finitely generated 𝐶_𝑁-module. Then 𝑀 is finitely generated as a 𝑓^∗(𝐶_𝑃)-module if and only if dim_ℝ(𝑀/𝑓^∗𝑚_𝑃⋅𝑀)<∞.

Lemma 7. Let 𝑓,𝑔∈𝐶^𝑝_𝑛. Let 𝑀 be a finitely generated (𝑓,𝑔)^∗(𝐶_𝑃×𝑃)-module. Then𝑀is finitely generated as a 𝑔^∗(𝐶_𝑃)-module if and only if dim_ℝ𝑀𝑔^∗𝑚_𝑃⋅𝑀<∞. (10)

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

Lemma 8. Let 𝑓,𝑔∈𝐶^𝑝_𝑛 and 𝑔−𝑓∈(𝑚^∞_𝑛𝑚_𝑁)^×𝑝. Then 𝑓,𝑔^∗𝐶_𝑃×𝑃⊂𝑓^∗𝐶_𝑃+𝑚^∞_𝑛𝑚_𝑁. (11)

Proof. For any ℎ∈𝐶_𝑃×𝑃, then (𝑓,𝑔)^∗(ℎ)−(𝑓,𝑓)^∗(ℎ) is in 𝑚^∞_𝑛𝑚_𝑁. So, 𝑓,𝑔^∗𝐶_𝑃×𝑃⊂𝑓,𝑓^∗𝐶_𝑃×𝑃+𝑚^∞_𝑛𝑚_𝑁. (12)
Let 𝜙∶(ℝ^𝑝,𝑃)→(ℝ^𝑝×ℝ^𝑝,𝑃×𝑃) be given by 𝜙(𝑦)=(𝑦,𝑦).
Since (𝑓,𝑓)^∗(ℎ)=𝑓^∗(𝜙^∗(ℎ)), it follows that 𝑓,𝑓^∗𝐶_𝑃×𝑃⊂𝑓^∗𝐶_𝑃. (13)
Thus, (11) holds.

Proof of Theorem 5. Suppose that 𝑓∈ℰ(𝑓,𝑁) and 𝑗^∞𝑓(0)=𝑗^∞𝑓(0). Let 𝑢=𝑓−𝑓; then 𝑢∈(𝑚^∞_𝑛𝑚_𝑁)^×𝑝. For any 𝑡₀∈[0,1], define 𝑔∶ℝ^𝑛×ℝ,0,𝑡₀⟶ℝ^𝑝 (14) by 𝑔(𝑥,𝑡)=𝑓(𝑥)+𝑡𝑢(𝑥) and 𝑔_𝑡(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: 𝐹^∗∶𝐶_{𝑃×{𝑡0}}⟶𝐶_{𝑁×{𝑡0}},ℎ⟼ℎ∘𝐹=𝐹^∗(ℎ), (15) where 𝐶_{𝑁×{𝑡0}} (resp., 𝐶_{𝑃×{𝑡0}}) denotes the ring of function-germs at (0,𝑡₀) which are constant when restricted to 𝑁×{𝑡₀} (resp., 𝑃×{𝑡₀}).
Set 𝑚^𝑘_𝑁𝐶_{𝑁×{𝑡0}}=𝑚^𝑘_𝑇 and 𝑚^𝑘_𝑃𝐶_{𝑃×{𝑡0}}=𝑚^𝑘_𝑇′.
Let 𝑇𝒜_𝑁,𝑃𝐹=𝑡𝐹+𝜔𝐹, where 𝑡𝐹 denotes 𝑚_𝑇⟨𝜕𝑓/𝜕𝑥₁,…,𝜕𝑓/𝜕𝑥_𝑛⟩ and 𝜔𝐹 denotes 𝐹^∗(𝐶_{𝑃×{𝑡0}}){𝑒₁,…,𝑒_𝑝}.
We are trying to show that 𝑓 is 𝒜_𝑁,𝑃-equivalent to 𝑓. It suffices to show that there exist germs 𝜙∶(ℝ^𝑛×ℝ,(0,𝑡₀))→(ℝ^𝑛,0) and 𝜓∶(ℝ^𝑝×ℝ,(0,𝑡₀))→(ℝ^𝑝,0) satisfying the following conditions. (1)𝜙(𝑥,𝑡)=𝑥 and 𝜓(𝑦,𝑡)=𝑦, for any 𝑡 sufficiently close to 𝑡₀, 𝑥∈𝑁 and 𝑦∈𝑃. (2)𝜙_𝑡0=id_ℝ^𝑛 and 𝜓_𝑡0=id_ℝ^𝑝. (3)𝑔_𝑡0=𝜓_𝑡∘𝑔_𝑡∘𝜙_𝑡⁻¹, for any 𝑡 sufficiently close to 𝑡₀.
By the method initiated by Mather in [12], it suffices to show that 𝑚^∞_𝑛𝑚_𝑇^×𝑝⊂𝑚_𝑇𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝐺^∗𝑚_𝑇′𝑒₁,…,𝑒_𝑝. (16)
To see this, it remains to show that (i)(𝑚^∞_𝑛𝑚_𝑇)^×𝑝⊂𝑇𝒜_𝑁,𝑃𝐹. (ii)𝑇𝒜_𝑁,𝑃𝐹=𝑇𝒜_𝑁,𝑃𝐺.
If (i) and (ii) hold, then 𝑚^∞_𝑛𝑚_𝑇^×𝑝⊂𝑇𝒜_𝑁,𝑃𝐺. (17)
Multiplying (17) by 𝐺^∗𝑚_𝑇′, we get 𝐺^∗𝑚_𝑇′⋅𝑚^∞_𝑛𝑚_𝑇^×𝑝⊂𝑚_𝑇𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝐺^∗𝑚_𝑇′𝑒₁,…,𝑒_𝑝. (18)
On the other hand, using condition (1) in Theorem 5, and 𝐽𝑓⋅𝑚_𝑁^×𝑝⊂𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛, (19) we get 𝑚^𝑟_𝑛𝑚_𝑁^×𝑝⊂𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝑓^∗𝑚_𝑃⋅𝑚_𝑁^×𝑝. (20)
Obviously, we have 𝑚^∞_𝑛𝑚_𝑁^×𝑝⊂𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝑓^∗𝑚_𝑃⋅𝑚_𝑁^×𝑝. (21)
Since 𝑔_𝑡−𝑓∈(𝑚^∞_𝑛𝑚_𝑁)^×𝑝, for each 𝑡∈ℝ, this gives that 𝑚_𝑁𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝑔^∗_𝑡𝑚_𝑃⋅𝑚_𝑁^×𝑝⊂𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝑓^∗𝑚_𝑃⋅𝑚_𝑁^×𝑝,𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝑓^∗𝑚_𝑃⋅𝑚_𝑁^×𝑝⊂𝑚_𝑁𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝑔^∗_𝑡𝑚_𝑃⋅𝑚_𝑁^×𝑝+𝑚_𝑛𝑚^∞_𝑛𝑚_𝑁^×𝑝. (22)
Hence, by Nakayama’s lemma we have 𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝑓^∗𝑚_𝑃⋅𝑚_𝑁^×𝑝=𝑚_𝑁𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝑔^∗_𝑡𝑚_𝑃⋅𝑚_𝑁^×𝑝. (23)
From (21), it follows that 𝑚^∞_𝑛𝑚_𝑁^×𝑝⊂𝑚_𝑁𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝑔^∗_𝑡𝑚_𝑃⋅𝑚_𝑁^×𝑝, (24) for any given 𝑡∈ℝ.
Multiplying (24) by 𝑚^∞_𝑛𝐶_{𝑁×{𝑡0}}, we get 𝑚^∞_𝑛𝑚_𝑇^×𝑝⊂𝑚_𝑇𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝐺^∗𝑚_𝑇′⋅𝑚^∞_𝑛𝑚_𝑇^×𝑝. (25)
Thus, (16) follows by substituting (18) in (25).
Now we prove the assertion (i).
By hypothesis, we have 𝑚^𝑟_𝑛𝑚_𝑁⊂𝑚_𝑁𝐽𝑓+𝑓^∗𝑚_𝑃⋅𝐶_𝑁, (26) for some 𝑟<∞.
This means that ideal 𝑚_𝑁𝐽(𝑓)+𝑓^∗𝑚_𝑃⋅𝐶_𝑁 has finite codimension in 𝐶_𝑁. Let 𝐶_𝑁𝑚_𝑁𝐽𝑓+𝑓^∗𝑚_𝑃⋅𝐶_𝑁=ℝ𝑎₁,…,𝑎_𝑠, (27) where 𝑎_𝑖∈𝐶_𝑁 and 𝑎_𝑖 is the projection of 𝑎_𝑖 in the quotient space, 𝑖=1,…,𝑠.
Now, set 𝑀=𝐶_{𝑁×{𝑡0}}/𝑚_𝑇𝐽(𝑓); then 𝑀 is a finitely generated 𝐶_{𝑁×{𝑡0}}-module. Since 𝐹^∗𝑚_{𝑃×{𝑡0}}⋅𝐶_{𝑁×{𝑡0}}=𝑓^∗𝑚_𝑃⋅𝐶_{𝑁×{𝑡0}}+(𝑡−𝑡₀)𝐶_{𝑁×{𝑡0}}; 𝑀𝐹^∗𝑚_{𝑃×{𝑡0}}≅𝐶⋅𝑀𝑚_𝑁𝐽𝑓+^𝑁𝑓^∗𝑚_𝑃⋅𝐶_𝑁. (28)
Thus, by Lemma 6, 𝑀 is finitely generated 𝐹^∗(𝐶_{𝑃×{𝑡0}})-module; that is, 𝑀=𝐹^∗𝐶_{𝑃×{𝑡0}}𝑎₁,…,𝑎_𝑠. (29)
In particular, 𝐶_{𝑁×{𝑡0}}=𝑚_𝑇𝐽𝑓+𝐹^∗𝐶_{𝑃×{𝑡0}}𝐶_𝑁. (30)
Since 𝑇𝒜_𝑁,𝑃𝐹⊃𝑚_𝑁𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑛+𝐹^∗𝐶_{𝑃×{𝑡0}}𝑒₁,…,𝑒_𝑝⋅𝐹^∗𝐶_{𝑃×{𝑡0}}⊃𝑇𝒜_𝑁,𝑃𝑓⋅𝐹^∗𝐶_{𝑃×{𝑡0}}⊃𝑚^∞_𝑛𝑚_𝑁^×𝑝⋅𝐹^∗𝐶_{𝑃×{𝑡0}}, (31) and 𝑚_𝑁𝐽(𝑓)⋅(𝐶_{𝑁×{𝑡0}})^×𝑝⊂𝑚_𝑇⟨𝜕𝑓/𝜕𝑥₁,…,𝜕𝑓/𝜕𝑥_𝑛⟩, it follows that 𝑇𝒜_𝑁,𝑃𝐹⊃𝑚^∞_𝑛𝑚_𝑁^×𝑝⋅𝑚_𝑇𝐽𝑓+𝐹^∗𝐶_{𝑃×{𝑡0}}𝐶_𝑁=𝑚^∞_𝑛𝑚_𝑇^×𝑝. (32)
So (i) holds.
Proof of Assertion (ii). Since 𝑔−𝑓∈(𝑚^∞_𝑛𝑚_𝑇)^×𝑝, by (i) we get 𝑇𝒜_𝑁,𝑃𝐺⊆𝑇𝒜_𝑁,𝑃𝐹=𝑇𝒜_𝑁,𝑃𝐺+𝑚^∞_𝑛𝑚_𝑇^×𝑝. (33)
Applying (25) and (i), we have 𝑚^∞_𝑛𝑚_𝑇^×𝑝⊂𝑚_𝑇𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝑚^∞_𝑛𝑚_𝑇^×𝑝⊂𝑚_𝑇𝜕𝑔𝜕𝑥₁,…,𝜕𝑔𝜕𝑥_𝑛+𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝑇𝒜_𝑁,𝑃𝐹. (34)
By (33) and (34), we have 𝑇𝒜_𝑁,𝑃𝐺⊆𝑇𝒜_𝑁,𝑃𝐹=𝑇𝒜_𝑁,𝑃𝐺+𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝑇𝒜_𝑁,𝑃𝐹. (35)
Set 𝐸=𝑇𝒜_𝑁,𝑃𝐹/𝑡𝐺, and (35) implies 𝐸=𝑇𝒜_𝑁,𝑃𝐺𝑡𝐺+𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝐸. (36)
Now it remains to show that
(a) 𝐸 is a finitely generated 𝐺^∗(𝐶_{𝑃×{𝑡0}})-module.
For if this holds, then by Nakayama’s lemma, 𝐸=𝑇𝒜_𝑁,𝑃𝐺.𝑡𝐺 (37)
Hence, 𝑇𝒜_𝑁,𝑃𝐹=𝑇𝒜_𝑁,𝑃𝐺.
To prove (a), by Lemma 7, it suffices to show that
(b) 𝐸 is a finitely generated (𝐹,𝐺)^∗(𝐶_{𝑃×{𝑡0}×𝑃×{𝑡0}})-module.
(c) dim_ℝ(𝐸/𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝐸)<∞.
Set 𝑅=(𝐹,𝐺)^∗(𝐶_{𝑃×{𝑡0}×𝑃×{𝑡0}}). By Lemma 8 and (i), we get 𝑅⋅𝑇𝒜_𝑁,𝑃𝐹⊂𝐹^∗𝐶_{𝑃×{𝑡0}}+𝑚^∞_𝑛𝑚_𝑇⋅𝑇𝒜_𝑁,𝑃𝐹⊂𝑇𝒜_𝑁,𝑃𝐹. (38)
Thus, 𝑇𝒜_𝑁,𝑃𝐹 is a 𝑅-module. Hence, 𝐸 is a 𝑅-module.
Obviously, 𝜔𝐹 is a finitely generated 𝐹^∗(𝐶_{𝑃×{𝑡0}})-module; hence 𝜔𝐹 is also finitely generated as a 𝑅-module.
Moreover, since 𝑡𝐹/𝑡𝐺∩𝑡𝐹 is a finitely generated 𝐶_{𝑁×{𝑡0}}-module and annihilated by 𝑚_𝑇𝐽(𝑔_𝑡). Thus, 𝑡𝐹/𝑡𝐺∩𝑡𝐹 is a finitely generated 𝐶_{𝑁×{𝑡0}}/𝑚_𝑇𝐽(𝑔_𝑡)-module. By the argument as equality (29), it follows that 𝐶_{𝑁×{𝑡0}}/𝑚_𝑇𝐽(𝑔_𝑡) is a finitely generated 𝐺^∗(𝐶_{𝑃×{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 dim_ℝ𝐸𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝐸=dim_ℝ𝑇𝒜_𝑁,𝑃𝐺+𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝑇𝒜_𝑁,𝑃𝐹𝑡𝐺+𝐺^∗𝑚_{𝑃×{𝑡0}}𝑇𝒜_𝑁,𝑃𝐺+𝑇𝒜_𝑁,𝑃𝐹=dim_ℝ𝜔𝐺+𝑡𝐺+𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝑇𝒜_𝑁,𝑃𝐹𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝜔𝐺+𝑡𝐺+𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝑇𝒜_𝑁,𝑃𝐹≤dim_ℝ𝜔𝐺𝐺^∗𝑚_{𝑃×{𝑡0}}⋅𝜔𝐺<∞. (39)
So (b) and (c) hold. This completes the proof.

3. The 𝒦_𝑁-Determinacy of Map-Germs

In this section, by a similar way as [13], we will give necessary and sufficient conditions for finitely 𝒦_𝑁-determined map-germs and ∞-𝒦_𝑁-determined map-germs.

Theorem 9. Suppose that 𝑓∈𝐶^𝑝_𝑛. The following conditions are equivalent. (1)𝑓 is finitely 𝒦_𝑁-determined. (2)(𝑚^𝑟_𝑛𝑚_𝑁)^×𝑝⊂𝑇𝒦_𝑁𝑓, for some 𝑟∈ℕ.

Proof. Let 𝐽^𝑟+1(𝑛,𝑝) denote the set of (𝑟+1)-jets at 0 of elements in ℰ(𝑓,𝑁). If 𝑓 is 𝑟-𝒦_𝑁-determined, then for any 𝑔∈ℰ(𝑓,𝑁) with the same 𝑟-jet as 𝑓, the 𝒦_𝑁-orbit of 𝑓 contains 𝑔; that is, 𝑔∈ℰ𝑓,𝑁∶𝑗^𝑟𝑔(0)=𝑗^𝑟𝑓(0⊂)𝑔∈ℰ𝑓,𝑁∶𝑔and𝑓are𝒦_𝑁-equivalent. (40)
Taking (𝑟+1)-jets on both sides, we have 𝑗^𝑟+1𝑔(0)∈𝐽^𝑟+1𝑛,𝑝∶𝑗^𝑟𝑔(0)=𝑗^𝑟𝑓(0⊂𝑗_𝑟+1)𝑔(0)∈𝐽^𝑟+1𝑛,𝑝∶𝑔and𝑓are𝒦_𝑁-equivalent. (41)
Taking tangent spaces at 𝑗^𝑟+1𝑓(0) on both sides, we have 𝑚^𝑟_𝑛𝑚_𝑁^×𝑝⊂𝑇𝒦_𝑁𝑓+𝑚_𝑛^𝑟+1𝑚_𝑁^×𝑝. (42)
By Nakayama’s lemma, this implies 𝑚^𝑟_𝑛𝑚_𝑁^×𝑝⊂𝑇𝒦_𝑁𝑓. (43)
Then (1) implies (2).
Conversely, let 𝑡₀∈[0,1] be fixed and 𝑔∈ℰ(𝑓,𝑁) with 𝑗^𝑟+1𝑓(0)=𝑗^𝑟+1𝑔(0). Define 𝐹∶ℝ^𝑛×ℝ,0,𝑡₀⟶ℝ^𝑝 (44) by 𝐹(𝑥,𝑡)=𝑓(𝑥)+𝑡(𝑔(𝑥)−𝑓(𝑥)) and 𝐹_𝑡(𝑥)=𝐹(𝑥,𝑡).
We are trying to show that 𝑔 is 𝒦_𝑁-equivalent to 𝑓. Since 𝐹₀=𝑓 and 𝐹₁=𝑔, and [0,1] is connected, it suffices to show that 𝐹_𝑡 is 𝒦_𝑁-equivalent to 𝐹_𝑡0, for all𝑡sufficiently close to 𝑡₀.
Since 𝐹_𝑡0(𝑥)−𝑓(𝑥)=𝑡₀(𝑔(𝑥)−𝑓(𝑥)) and 𝑔−𝑓∈(𝑚_𝑛^𝑟+1𝑚_𝑁)^×𝑝, from condition (2), it follows that 𝑇𝒦_𝑁𝐹_𝑡0⊂𝑇𝒦_𝑁𝑓, (45) and 𝑇𝒦_𝑁𝑓⊂𝑇𝒦_𝑁𝐹_𝑡0+𝑚_𝑛⋅(𝑇𝒦_𝑁𝑓).
Since 𝑇𝒦_𝑁𝑓 is a finitely generated 𝐸_𝑛-module, and 𝑚_𝑛 is the maximal ideal of 𝐸_𝑛, by Nakayama’s lemma, we get 𝑇𝒦_𝑁𝐹_𝑡0=𝑇𝒦_𝑁𝑓. (46)
Hence 𝑇𝒦_𝑁𝐹_𝑡0 also satisfies condition (2). So 𝑚^𝑟_𝑛𝑚_𝑁𝐸^∗_𝑛+1^×𝑝⊂𝑇𝒦_𝑁𝐹_𝑡0⋅𝐸^∗_𝑛+1, (47) where 𝐸^∗_𝑛+1 denotes the ring of smooth function-germs in variables (𝑥,𝑡) at the point (0,𝑡₀).
Let 𝑚^∗_𝑛+1 denote the maximal ideal of 𝐸^∗_𝑛+1.
Let 𝑇𝒦_𝑁𝐹=𝑚_𝑁𝐸^∗_𝑛+1⟨𝜕𝐹/𝜕𝑥₁,…,𝜕𝐹/𝜕𝑥_𝑛⟩+𝐹^∗𝑚_𝑝⋅(𝑚_𝑁𝐸^∗_𝑛+1)^×𝑝. Obviously, 𝑇𝒦_𝑁𝐹 is a finitely generated 𝐸^∗_𝑛+1-module.
By the same argument as (46), we have 𝑇𝒦_𝑁𝐹=𝑇𝒦_𝑁𝐹_𝑡0⋅𝐸^∗_𝑛+1. (48)
By (47) and (48), we get 𝑚^𝑟_𝑛𝑚_𝑁𝐸^∗_𝑛+1^×𝑝⊂𝑇𝒦_𝑁𝐹. (49)
Since 𝜕𝐹/𝜕𝑡=𝑔−𝑓∈(𝑚_𝑛^𝑟+1𝑚_𝑁)^×𝑝, this means that there exist germs 𝑋_𝑖 in 𝑚_𝑁𝐸^∗_𝑛+1, 𝑖=1,…,𝑛, such that 𝑔−𝑓+^𝑛_𝑖=1𝑋_𝑖(𝑥,𝑡)𝜕𝐹𝜕𝑥_𝑖∈𝐹^∗𝑚_𝑝⋅𝑚_𝑁𝐸^∗_𝑛+1^×𝑝. (50)
Thus, we can find a germ of vector field 𝑋 in ℝ^𝑛×ℝ of the following form: 𝜕+𝜕𝑡^𝑛_𝑖=1𝑋_𝑖()𝜕𝑥,𝑡𝜕𝑥_𝑖, (51) such that DF(𝑋)∈𝐹^∗𝑚_𝑝⋅(𝑚_𝑁𝐸^∗_𝑛+1)^×𝑝.
That is, we can find a (𝑝×𝑝) matrix 𝐴(𝑥,𝑡) with entries in 𝑚_𝑁𝐸^∗_𝑛+1 such that DF(𝑋)=𝐴(𝑥,𝑡)⋅𝐹(𝑥,𝑡). (52)
By integrating the vector field 𝑋, we get a one-parameter family of local diffeomorphisms 𝜙_𝑡 in ℛ_𝑁. Thus 𝑑𝐹𝜙𝑑𝑡^𝑡(𝑥)=𝜕,𝑡𝐹𝜙𝜕𝑡^𝑡(𝑥)+,𝑡_𝑛𝑖=1𝑋_𝑖𝜙_𝑡(𝑥),𝑡𝜕𝐹𝜕𝑥_𝑖𝜙_𝑡(𝑥)𝜙,𝑡=𝐴_𝑡(𝑥),𝑡⋅𝐹𝜙_𝑡(𝑥),𝑡. (53)
Hence, for fixed 𝑥∈ℝ^𝑛, 𝐹(𝜙_𝑡(𝑥),𝑡) is a solution of the differential equation 𝑦̇=𝐴(𝜙_𝑡(𝑥),𝑡)𝑦 with initial condition 𝑦(𝑥,𝑡₀)=𝐹_𝑡0(𝑥).
Since the solution of this differential equation is unique and of the form 𝑦(𝑥,𝑡)=𝐵(𝑥,𝑡)⋅𝑦𝑥,𝑡₀, (54) with 𝐵(𝑥,𝑡) as an invertible matrix, 𝐵(𝑥,𝑡₀)=𝐼_𝑝 and 𝐵(𝑥,𝑡)=𝐼_𝑝 for each 𝑥∈𝑁. Thus 𝐹_𝑡(𝑥)=𝐵𝜙_𝑡⁻¹(𝑥),𝑡⋅𝐹_𝑡0∘𝜙_𝑡⁻¹(𝑥). (55)
Thus, 𝐹_𝑡 and 𝐹_𝑡0 are 𝒦_𝑁-equivalent for all 𝑡 sufficiently close to 𝑡₀, which completes the proof.

Theorem 10. Suppose that 𝑓∈𝐶^𝑝_𝑛. Then 𝑓 is ∞-𝒦_𝑁-determined if and only if 𝑚^∞_𝑛𝑚_𝑁^×𝑝⊂𝑇𝒦_𝑁𝑓. (56)

Proof. “Only if”: if 𝑓 is ∞-𝒦_𝑁-determined, by the definition, we get 𝑓+𝑚^∞_𝑛𝑚_𝑁^×𝑝⊂𝒦_𝑁⋅𝑓. (57)
Taking tangent spaces at 𝑓 on both sides, we have 𝑇_𝑓𝑓+𝑚^∞_𝑛𝑚_𝑁^×𝑝⊂𝑇_𝑓𝒦_𝑁⋅𝑓. (58)
Note that (𝑚^∞_𝑛𝑚_𝑁)^×𝑝⊂𝑇_𝑓(𝑓+(𝑚^∞_𝑛𝑚_𝑁)^×𝑝), and 𝑇_𝑓(𝒦_𝑁⋅𝑓)⊂𝑇𝒦_𝑁𝑓. Hence, 𝑚^∞_𝑛𝑚_𝑁^×𝑝⊂𝑇𝒦_𝑁𝑓. (59)
“If”: the proof is the same as that of Theorem 9.

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.