The automorphism theorem and additive group actions on the affine plane

Transcription

1 The automorphism theorem and additive group actions on the affine plane arxiv: v1 [math.ac] 14 Apr 2016 Shigeru Kuroda Abstract Due to Rentschler, Miyanishi and Kojima, the invariant ring for a G a -action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk. 1 Introduction Let k be a field, A a k-domain, A[T] the polynomial ring in one variable over A, and σ : A A[T] a homomorphism of k-algebras. Then, σ defines an action of the additive group G a = Speck[T] on SpecA if and only if the following holds for each a A, where we write σ(a) = i 0 a it i with a i A, and U is a new variable: (A1) a 0 = a. (A2) i 0 σ(a i)u i = i 0 a i(t +U) i in A[T,U]. If this is the case, we call σ a G a -action on A. The σ-invariant ring A σ := {a A σ(a) = a} is equal to σ 1 (A) by (A1). We say that σ is nontrivial if A σ A. Now, let k[x 1,x 2 ] be the polynomial ring in two variables over k, and Aut k k[x 1,x 2 ] the automorphism group of the k-algebra k[x 1,x 2 ]. We often express φ Aut k k[x 1,x 2 ] as (φ(x 1 ),φ(x 2 )). We call f k[x 1,x 2 ] a coordinate of k[x 1,x 2 ] if there exists g k[x 1,x 2 ] such that (f,g) belongs to Aut k k[x 1,x 2 ], that is, k[f,g] = k[x 1,x 2 ]. The following theorem is a fundamental result for G a -actions on k[x 1,x 2 ]. Theorem 1.1. For every nontrivial G a -action σ on k[x 1,x 2 ], there exists a coordinate f of k[x 1,x 2 ] such that k[x 1,x 2 ] σ = k[f] Mathematics Subject Classification. Primary 14R10; Secondary 13A50, 14R20. Partly supported by JSPS KAKENHI Grant Number 15K

2 This theorem was first proved by Rentschler [13] when chark = 0 in 1968, and then by Miyanishi [11] when k is algebraically closed in Recently, Kojima [7] proved the general case by making use of Russell-Sathaye [14] (see also [9]). For each f k[x 1,x 2 ], we denote by degf the total degree of f, and by f or (f) the highest homogeneous part of f for the standard grading on k[x 1,x 2 ]. The following well-known theorem was first proved by Jung [5] when chark = 0 in The general case was proved by van der Kulk [8] in 1953 (see also the proof of Makar-Limanov [10] and its modifications by Dicks [3] and Cohn [1, Thm. 8.5]). Theorem 1.2. For every (f 1,f 2 ) Aut k k[x 1,x 2 ] with degf 1 2 or degf 2 2, there exist (i,j) {(1,2),(2,1)}, α k and l 1 such that f i = α f j l The purpose of this note is to give a new short proof of Theorem 1.1 based on Theorem 1.2 (cf. 2). We should mention that, if k is an infinite field, Theorem 1.1 can be derived from Theorem 1.2 by a group-theoretic approach (cf. [6]). Our approach is different from this approach, and is valid for an arbitrary k. Conversely, Theorem 1.2 can be derived easily from Theorem 1.1. This seems known to experts, at least when chark = 0 (cf. e.g. [4, 5.1] for related discussion). For completeness, we also give a proof for this implication (cf. 3). The author thanks Prof. Ryuji Tanimoto (Shizuoka University) for discussion. 2 G a -action Recall that, if σ is a nontrivial G a -action on A, then A has transcendence degree one over A σ (cf. [12, 1.5]). For each t A σ, the map σ t : A σ A[T] f(t) f(t) A is an automorphism of the k-algebra A. Actually, we have σ 0 = id A by (A1), and σ t σ u = σ t+u for each t,u A σ by (A2). Now, wederive Theorem 1.1fromTheorem 1.2. Foreach q = i 0 q it i k[x 1,x 2 ][T]\{0} with q i k[x 1,x 2 ], we define the Z 2 -degree of q by deg Z 2q := max{(i,degq i ) Z 2 i 0, q i 0}, where Z 2 isorderedlexicographically, i.e., (a,b) (a,b )if andonly ifa < a, ora = a andb b. Letσ beanynontrivialg a -actiononk[x 1,x 2 ]. Itsuffices to find (f 1,f 2 ) Aut k k[x 1,x 2 ] for which σ(f 1 ) or σ(f 2 ) belongs to k[x 1,x 2 ]. Suppose that such an (f 1,f 2 ) does not exist. Choose (f 1,f 2 ) Aut k k[x 1,x 2 ] 2

3 so that deg Z 2σ(f 1 ) + deg Z 2σ(f 2 ) is minimal, and write σ(f i ) = q i (T) = mi j=1 q i,jt j for i = 1,2, where q i,j k[x 1,x 2 ] with q i,mi 0. By supposition, we have m 1,m 2 1. Since k[x 1,x 2 ] σ k, we may take g k[x 1,x 2 ] σ with g k. Then, σ g r = (q 1 (g r ),q 2 (g r )) belongs to Aut k k[x 1,x 2 ] for each r 0 as mentioned above. Since degg 1, there exists r 0 > 0 such that, for each r r 0 and i = 1,2, we have (q i (g r )) = q i,mi ḡ rm i and degq i (g r ) 2. By Theorem 1.2, for each r r 0, there exist (i,j) {(1,2),(2,1)}, α k and l 1 such that q i,mi ḡ rm i = α( q j,mj ḡ rm j ) l. This equality implies that r(m i lm j )degg = ldegq j,mj degq i,mi. (2.1) Wenotethat(i,j),αandlabovedependonr. By(2.1),weseethatm i = lm j holds for sufficiently large r. Take such an r. Then, we have q i,mi = α q l j,m j, and hence deg(q i,mi αq l j,m j ) < degq i,mi. Thus, the Z 2 -degree of σ(f i αf l j ) = q i(t) αq j (T) l = (q i,mi αq l j,m j )T m i +(terms of lower degree in T) is strictly less than that of σ(f i ). Since (f i αf l j,f j) belongs to Aut k k[x 1,x 2 ], this contradicts the minimality of deg Z 2σ(f 1 )+deg Z 2σ(f 2 ), completing the proof. 3 Automorphism Theorem We derive Theorem 1.2 from Theorem 1.1. Let f = i 1,i 2 0 u i 1,i 2 x i 1 1 x i 2 2 be an element of k[x 1,x 2 ]\{0}, where u i1,i 2 k. For each w = (w 1,w 2 ) R 2, we define deg w f := max{i 1 w 1 +i 2 w 2 i 1,i 2 0, u i1,i 2 0} and f w := ui1,i 2 x i 1 1 x i 2 2, where the sum is taken over i 1,i 2 0 with i 1 w 1 +i 2 w 2 = deg w f. We say that f is w-homogeneous if f w = f, and non-univariate if f k[x 1 ] k[x 2 ]. We define w(f) := (deg (0,1) f,deg (1,0) f). We remark that f w(f) is non-univariate if f is non-univariate. The following lemma is a consequence of Theorem 1.1. Lemma 3.1. If σ is a nontrivial G a -action on k[x 1,x 2 ], and f k[x 1,x 2 ] σ is non-univariate, then there exist a,b k, (i,j) {(1,2),(2,1)} and l,m 1 such that f w(f) = a(x i bx l j) m. (3.1) 3

4 Proof. Since f is non-univariate, so is f w(f) as remarked. By Derksen Hadas Makar-Limanov[2, Prop. 2.2], thereexists anontrivialg a -actionτ on k[x 1,x 2 ] such that f w(f) belongs to k[x 1,x 2 ] τ. By Theorem 1.1, k[x 1,x 2 ] τ = k[h] holds for some coordinate h of k[x 1,x 2 ]. We may assume that h has no constant term. Then, since f w(f) belongs to k[h] \ k and f w(f) is w(f)- homogeneous, we see that h is w(f)-homogeneous, and f w(f) = αh m for some α k and m 1. This implies that h is non-univariate. Since h is a w(f)-homogeneous coordinate, h must have the form βx i +γx l j for some β,γ k and l 1. Therefore, f w(f) is written as in (3.1). Now, we prove Theorem 1.2. Take φ = (f 1,f 2 ) Aut k k[x 1,x 2 ]. Set w i := degf i and g i := φ 1 (x i ) for i = 1,2. Assume that w 1 2 or w 2 2. Then, there exists t {1,2} such that g t is non-univariate. Note that a nontrivialg a -actionσ onk[x 1,x 2 ]isdefinedbyσ(g t ) = g t andσ(g u ) = g u +T, where u t. Since g t belongs to k[x 1,x 2 ] σ, we may write g w(gt) t as in (3.1) by Lemma 3.1. Set w := (w 1,w 2 ). Then, we have deg w g t = mmax{w i,lw j } max{w 1,w 2 } 2 > 1 = degx t = degφ(g t ). This implies that a( f i b f l j )m = 0. Therefore, we get f i = b f l j, proving Theorem 1.2. References [1] P. M. Cohn, Free rings and their relations, Second edition, Academic Press, London, [2] H. Derksen, O. Hadas and L. Makar-Limanov, Newton polytopes of invariants of additive group actions, J. Pure Appl. Algebra 156 (2001), no. 2-3, [3] W. Dicks, Automorphisms of the polynomial ring in two variables, Publ. Sec. Mat. Univ. Autònoma Barcelona 27 (1983), [4] A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, Vol. 190, Birkhäuser, Basel, Boston, Berlin, [5] H. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), [6] T. Kambayashi, Automorphism group of a polynomial ring and algebraic group action on an affine space, J. Algebra 60 (1979),

5 [7] H. Kojima, Locally finite iterative higher derivations on k[x, y], Colloq. Math. 137 (2014), no. 2, [8] W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) 1 (1953), [9] S. Kuroda, A generalization of Nakai s theorem on locally finite iterative higher derivations, arxiv:math.ac/ [10] L. Makar-Limanov, On Automorphisms of Certain Algebras (Russian), PhD Thesis, Moscow, [11] M. Miyanishi, G a -action of the affine plane, Nagoya Math. J. 41 (1971), [12] M. Miyanishi, Curves on rational and unirational surfaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 60, Tata Inst. Fund. Res., Bombay, [13] R. Rentschler, Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), [14] P. Russell and A. Sathaye, On finding and cancelling variables in k[x, Y, Z], J. Algebra 57 (1979), no. 1, Department of Mathematics and Information Sciences Tokyo Metropolitan University 1-1 Minami-Osawa, Hachioji, Tokyo , Japan kuroda@tmu.ac.jp 5