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The tame automorphism group in two variables over basic Artinian rings

In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk Theorem, which deals with the case that R is a field (of any characteristic). Here we will show that for tameness over an Artinian ring, the Q-algebra assumption is really needed: we will give, for local Artinian rings with square-zero principal maximal ideal, a complete description of the tame automorphism subgroup. This will lead to an example of a non-tame automorphism, for any characteristic p>0.


Introduction
All two-dimensional polynomial automorphisms over a field are tame, as stated in the famous theorem by Jung and Van der Kulk ( [7], [8]). For fields of characteristic zero this was proved by Jung, and Van der Kulk generalized it to arbitrary characteristic. As is well-known, the statement fails to be true over a domain R which is not a field. The most common example of a non-tame automorphism is the one by Nagata ([9]), which is defined over R = k[Z] (a univariate polynomial ring), but can be transformed into an example over any domain which is not a field. For any domain R, [4, Corollary 5.1.6] even yields an algorithm to decide whether or not an automorphism in two variables over R is tame. To continue with the description of the tame automorphism groups over commutative rings in general, it is very convenient to start with Artinian rings. Namely, when it is clear which automorphisms in two variables over Artinian rings are tame, we can use this information to decribe the automorphisms over rings with higher Krull dimension, by lifting the former automorphisms (see for example Theorem 3.4). Moreover, the problem of describing the structure of the automorphism group over a general Artinian ring can be reduced to the case of a local Artinian ring (as will be explained in section 4).
One of the main results of the recent paper [2] by Van den Essen, Wright and the author is the fact that, over an Artinian ring R, all two-dimensional automorphisms with Jacobian determinant one are tame in case R is a Q-algebra. This is a generalization of Jung's Theorem. We show that in a non-Q-algebra setting, tameness is not guaranteed. In fact, for every characteristic p > 0 we give an example of a non-tame automorphism over a local Artinian ring having that characteristic. To show that these automorphisms are not tame, we provide a description of the structure of the tame automorphism groups over local Artinian rings of the most basic type to be found: the ones with square-zero principal maximal ideal.
This paper is set up as follows: In the next section we introduce the general automorphism group and its best-known subgroups. We describe classic results, explain which questions are still unanswered and in what way this paper contributes to the development of the theory on polynomial automorphism groups. In section 3, we review one of the results of the recent paper [2], saying that over an Artinian Q-algebra R, any two-dimensional automorphism is tame, provided that the Jacobian determinant is equal to one (Theorem 3.5). The preparations for this result, which will be done in that section, are also important for the remainder of this paper: most techniques also work in the non-Q-algebra setting. Lemma 3.1 is in fact the only tool that requires a Q-algebra. Section 4 examines the structure of the elementary automorphism subgroup EA2(R) for rings R of the form R = A[T ]/(T 2 ), where A is another ring. It essentially reduces the description of the elementary subgroup over R to the description of the elementary subgroup over A. This result can immediately be applied to the case of local Artinian rings with square-zero principal maximal ideal. This is done in the last section. It yields an example for every prime number p of a non-tame automorphism in two variables over Fp[T ]/(T 2 ).

Automorphism subgroups and their relations
In this paper, every ring is assumed to be commutative and to have an identity element. We will restrict ourselves to polynomial rings in two variables over a ring R, denoted as R[X, Y ]. This section describes the usual subgroups of the general polynomial automorphism group, and what is already known about how they are related.
A polynomial map over R is an ordered pair (F, G) of polynomials of R[X, Y ]. We can view polynomial maps as maps R 2 → R 2 , defined by (x, y) → (F (x, y), G(x, y)), but also as R-endomorphisms R[X, Y ] → R[X, Y ], given by the substitution h(X, Y ) → h(F, G). F and G are called the coordinates of (F, G).
We write Jϕ for the Jacobian matrix of an automorphism ϕ. By the chain rule, for any automorphism ϕ we have Jϕ ∈ GL2(R), whence |Jϕ| ∈ R[X] * . (Throughout this paper, the operator | | takes the determinant of a matrix.) Here is an overview of the usual subgroups of GA2(R): 1. SA2(R), the special automorphism group, is the subgroup of all ϕ for which |Jϕ| = 1.
2. The group GL2(R) of invertible matrices is usually viewed as a subgroup of GA2(R).
3. EA2(R) is the subgroup generated by the elementary automorphisms. An elementary automorphism is one of the form (X + f (Y ), Y ) or (X, Y + f (X)) for some univariate polynomial f . Note that EA2(R) ⊆ SA2(R).
In the case of a field we have the following classic theorem, which was already mentioned in the introduction.
So over a field the only examples of polynomial automorphisms are the tame ones. However, this doesn't hold for a domain which is not field. But there exists an algorithm to decide whether or not an automorphism in two variables over a domain R is tame in [4,Corollary 5.1.6]). This algorithm can be used to show that any non-unit r ∈ R\{0} produces a non-tame automorphism, or a polynomial ring R = k[Z] and r = Z, k a field, this is Nagata's famous example ( [9]). But for a general commutative ring little is known about which automorphisms in GA2(R) are tame. This paper is meant to extend our knowledge on this subject.
If R → S is a surjective ring homomorphism, then the induced group homomorphism EA2(R) → EA2(S) is also surjective. Note that this fails to hold for TA2(R) → TA2(S), because of the following: if M ∈ GL2(S), then there doesn't necessarily exist an N ∈ GL2(R) such that N → M . This is why tame automorphisms appearing in this paper are usually elements of EA2(−): we can lift these to automorphisms over rings with higher Krull dimension. Therefore, we would like to know the connection between TA2(R) and EA2(R). The following lemma (a special version of [2, Proposition 3.20]) and corollary decribe this connection, which applies to most coefficient rings considered in this paper.
In the following, SL2(R) denotes the group of all matrices with determinant one, D2(R) is the group of all invertible diagonal matrices, and E2(R) is the group generated by all elementary matrices.
The hypothesis holds when R is a local ring. For the second statement, consider an element M of SL2(R), where R is local. Since det(M ) = 1, there must at least be one entry of M which is in R * . We can use this entry to clear the other entries of the row and column to which this entry belongs, through multiplication by 2 elementary matrices. If the resulting matrix isn't diagonal, we can make it so by multiplying it with the matrix Hence, we may assume that the resulting matrix is diagonal, and since it is still an element of SL2(R), we can use the fact that, for any ring R and any a ∈ R * , If R is not assumed to be local, then the hypothesis of Lemma 2.2 still holds if R has a special structure, e.g. when R is a Euclidean domain (the proof of this well-known fact is very much like the proof of the second part of Lemma 2.2). It is important to note, however, that not all Principal Ideal Domains have this property. Let R be the ring of integers of Q( √ −19), one of the finitely many imaginary quadratic number fields of which the ring of integers is a Principal Ideal Domain, by the Stark-Heegner Theorem ( [10], [6]). In [3, Theorem 6.1] it was shown that, if α := 1 2 + 1 2 √ −19, then the following matrix is in SL2(R), but not in E2(R): Proof. For the first statement, let ϕ ∈ GA2(R) with |Jϕ| ∈ R * (since R is reduced). Then there exists an α ∈ GL2(R) such that αϕ ∈ SA2(R) = EA2(R). Thus, ϕ ∈ TA2(R).
The second statement follows directly from Lemma 2.2.

The Artinian Q-algebra result
Throughout this section (except for Lemma 3.2), we assume that R is a Q-algebra. We will restate and give a quick proof of one of the results from [2]: for an Artinian Q-algebra R, every special automorphism in two variables over R is tame (Theorem 3.5). The fact that this is also true for any reduced Artinian ring (Q-algebra or not) had already been observed in [9, Corollary 0.6] and [1, Proposition 3.10]. One of the basic tools of Theorem 3.5 is Lemma 3.2. This lemma is also useful for the general (non-Q-algebra) case in the subsequent sections. Lemma 3.1 is taken from [2], and its statement also appeared in [4, § 5.2, Exercise 7]. It is the only ingredient of Theorem 3.5 that requires R to be a Q-algebra.
can be written as a Q-linear combination of polynomials of the form (X + aY ) n+m , with a ∈ Q.
The following lemma also appears (in some form) in [2] and [5] and is a basic property of the type of automorphisms considered in this paper (also over non-Q-algebras).
The type of tame automorphisms considered in the following proposition provides a foundation on which we can build many other tame automorphisms.
Proof. Since a 2 = (0), |J(ϕ)| = 1 + ∂g ∂X + ∂h ∂Y . Then ∂g ∂X + ∂h ∂Y = 0, and since R is a Q-algebra, this implies that there exists a polynomial p ∈ a[X, Y ] such that g = ∂p ∂Y and h = − ∂p ∂X . Using Lemma 3.2, we may assume that p = rX n Y m for some r ∈ a, n, m ≥ 0 and n + m ≥ 1. With Lemma 3.1, we can write X n Y m as a Q-linear combination of polynomials of the form (X + aY ) n+m , with a ∈ Q. Applying Lemma 3.2 again, we may assume that where k = n + m and a, b ∈ Q. But then ϕ = α −1 βα, where α = (X + aY, Y ) and β = (X, Y − kbrX k−1 ). Therefore ϕ ∈ EA2(R).
The following theorem is a special case of [2, Theorem 4.1].
Proof. Since the assumption that ϕ ∈ EA2(R) can be expressed using only finitely many coefficients in the ideal a, we may assume that a is finitely generated. Hence it is a nilpotent ideal, say a m = (0) for some m ≥ 1. We will prove by induction on m that ϕ is a composition of elementary automorphisms.
Proof. The special case of a field follows from Corollary 2.4 and Theorem 2.1. For the general case, let η be the nilradical of R. Since R is Artinian, it is well-known that R/η is a product of fields. The statement now follows from Theorem 3.4 and the fact that, for any direct product of rings R = R1 × R2, the group EA2(R) is canonically isomorphic to the direct product of groups EA2(R1) × EA2(R2). (And the same for SA2(−).)

The square-zero principal ideal setting
To find the structure of the general polynomial automorphism group over an Artinian ring R, we can restrict ourselves to the case of local Artinian rings. Namely, it is well-known that R ∼ = R1×R2×· · ·×Rm, a direct product of local Artinian rings. And then GA2(R) is canonically isomorphic to the direct product of groups GA2(R1) × GA2(R2) × · · · × GA2(Rm). One can readily check that this also holds if GA2(−) is replaced by one of its mentioned subgroups.
The remainder of this paper will be focused on the case of a specific type of local Artinian rings, namely the ones for which the maximal ideal is principal and has its square equal to zero. The question of tameness over any Artinian ring can be reduced to this setting. We will see that the automorphism group has a clear structure in this case. To describe the basic aspects of this structure, we can use a more general setting: we suppose (for the moment) that R is any ring containing an element t satisfying t 2 = 0. In specific examples, such a ring is usually obtained as a factor ring of a univariate polynomial ring: R = A[T ]/(T 2 ), and t = T + (T 2 ). We often use the notation A[t] 2 to denote this ring. For this kind of ring we will give an explicit description of the group EA2(R) in terms of the group EA2(A). This will be very useful in the next section, when we apply this to the situation that R is local Artinian.
The conjugation formulas below are crucial properties of the structure of the automorphism group SA2(R).
Remark 4.3. The automorphisms of the form ϕ (h) have the following properties: In particular, if m ∈ N * satisfies m ∈ R * , and if a ∈ R, and f ∈ R[X, Y ] is one of the coordinates of an automorphism α ∈ EA2(R), then ϕ ( a m f m ) ∈ EA2(R). Combining this with property 1. yields many tame automorphisms: if we let H = a 1 m 1 f m 1 1 + · · · + ar mr f m 1 1 , where ai ∈ R, mi ∈ N * ∩ R * and each fi is a coordinate of an automorphism in EA2(R), then ϕ (H) ∈ EA2(R). In case R = A[t] 2 , where the ring A is contained in a Q-algebra, we have a reverse statement, displayed in Theorem 4.6.
Theorem 4.6. Let A be a ring which is contained in a Q-algebra Q. Let R := A[t] 2 . Then, for any ϕ1 ∈ EA2(R), there exist a ϕ0 ∈ EA2(A) and an where ai ∈ A, mi ∈ N * and each fi is a coordinate of an automorphism in EA2(A).
Proof. Let ϕ1 ∈ EA2(R). R = A ⊕ At, so EA2(R) = EA2(A), EA2(tA) . Since we've also seen that EA2(tA) ⊳ EA2(R) and as it is clear that EA2(A) ∩ EA2(tA) = {id}, we may conclude that EA2(R) = EA2(A) ⋉ EA2(tA). So, write ϕ1 = ϕ0 • ϕt, with ϕ0 ∈ EA2(A) and ϕt ∈ EA2(tA). Now define S ⊆ EA2(tA) by where each τi ∈ EA2(A) and each εi ∈ S (note that S −1 = S). Then, using Proposition 4.1, Note that fi is a coordinate of an automorphism in EA2(A). Now we can define H(X, Y ) := h1 + · · · + hr, and we derive , whence ϕ1 has the prescribed form. In case the coefficient ring is of the form B[t] 2 for a ring B which is not contained in a Qalgebra, the above theorem can still be used to unravel the structure of the group EA2(R), as is shown in Corollary 4.7.
where ai ∈ A, mi ∈ N * and each fi is one of the coordinates of an automorphism in EA2(A).

The case of a local Artinian ring with square-zero principal maximal ideal
In the previous section we examined the structure of EA2(R) in the general setting of a ring with a square-zero principal ideal. Now we specialize to the situation that the ring is local Artinian and the ideal is maximal. Whereas every automorphism over an Artinian Q-algebra is tame (Theorem 3.5), this is not true anymore in prime characteristic, as is shown by the following theorem.
Theorem 5.1. Let p be any prime number and R = Fp[t] 2 . Then SA2(R) ⊆ TA2(R). More precisely, the following automorphism over R is not tame: So the monomial X p−1 Y p occurs in ∂H ∂X , but this contradicts the fact that ∂H ∂X = 0 ! (since ϕ1 = (X + tX p Y p−1 , Y )) So ϕ1 cannot be tame. The next example shows, that for p = 2, a slightly modified version of the automorphism in Theorem 5.1 is tame. It is unknown to the author if, for all other primes p, the corresponding modified automorphism is tame.
is not tame according to Theorem 5.1, it became apparent at the end of the proof that this is because the monomial XY 2 doesn't occur in the second component of this automorphism. Then the following question arises: is the special automorphism (X + tX 2 Y, Y − tXY 2 ) tame over R? Yes, it is! Since X 2 Y and XY 2 are the partial derivatives of 1 2 X 2 Y 2 (over Q), we establish the tameness by writing this term as a linear combination of powers, in the style of (the end of ) Remark 4.3: Applying Proposition 4.1 to each of the terms appearing in this linear combination (taking R = Q[t] 2 ), we get that (X + tX 2 Y, Y − tXY 2 ) equals the composition Note that this is actually a composition over Z[t] 2 . Viewing this composition over R by calculating modulo 2, we obtain Let p be a prime number. From Corollary 4.7 it follows that, if R := Z[t] 2 and R := Fp[t] 2 , then any automorphism in EA2(R) is (up to an automorphism in EA2(Fp)) of the form is not of this form (which has been shown in the proof of Theorem 5.1), so it cannot be tame. It is still unknown to the author whether tameness (more precisely: 'being an element of EA2(R)') is guaranteed for all automorphisms over R of the form ϕ (H) . By Corollary 4.7, this question is equivalent to the following (Question 5.4). A more general version is Question 5.3. ? If Question 5.4 also has a negative answer, then the next challenge is to find an algorithm to decide for a given p and H whether such an H ′ exists. Such an algorithm would thus also be an algorithm for tameness in SA2(Fp[t] 2 ).
We conclude with an example of a monomial H ∈ Q[X, Y ] which has the property that ϕ (H) ∈ EA2(Fp[t] 2 ) for all primes p = 2. It is unknown to the author whether this also holds for p = 2.