In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. This results from the main theorem of this paper, which asserts that an automorphism in any dimension n is stably tame if said condition holds point-wise over Spec R. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame.proved Nagata's conjecture. 1 Meanwhile it had been shown by Smith [15] and Wright (unpublished) that Nagata's example is stably tame, in fact tame with the addition of one more variable. 2 The matter of stable tameness is one of intrigue because no example has been produced (to the authors' knowledge) of a polynomial automorphism over a domain which cannot be shown to be stably tame.The remarkable result of Umirbaev and Shestakov mentioned above actually asserts that an automorphism in three variables T, X, Y over a field k which fixes T is tame (if and) only if it is tame as an automorphism over k [T ]. As there are known to be many non-tame two-dimensional automorphisms over k[T ], this establishes the existence of many non-tame three-dimensional automorphisms over k. However, it will follow from the main result of this paper (Corollary 4.9) that all three-dimensional automorphisms of this type are stably tame over k.The main result of this paper is Theorem 4.10 (Main Theorem), which asserts that all two-dimensional polynomial automorphisms over a regular ring are stably tame. It is proved by a somewhat delicate argument for which Theorem 4.1 plays an essential role. The latter result yields the consequence that all two-dimensional automorphisms over an Artinian ring A are stably tame, Theorem 4.3. Moreover, they are actually tame in the case A is a Qalgebra. The latter statement can be viewed as a generalization of Jung's Theorem, and it yields a stronger version of the Main Theorem for the case of a Dedekind Q-algebra (Theorem 4.6). Another keystone in the proof of the Main Theorem is Theorem 4.14, which reveals stable tameness to be a local property.Also used in the proof of the Main Theorem are the Jung-Van der Kulk Theorem, a number of technical results, and a theorem of Suslin, all of which appear in §3. Stable tameness has the flavor of K-theory, and some of the tools are suggestive of those used to prove results about the behavior of the functor K 1 under polynomial extensions (compare Lemma 3.12, for example, with Suslin's Lemma 3.3 in [16]).We here note that the appeal to Suslin's theorem (Theorem 3.23) is precisely where the hypothesis A is regular is required. This is evoked to conclude the proof of Theorem 4.5, on which the Main Theorem depends. The Main Theorem certainly fails for non-reduced rings, over which there exist automor-1 The...
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