Stable tameness of two-dimensional polynomial automorphisms over a regular ring. (English) Zbl 1246.14075

Berson, Joost; van den Essen, Arno; Wright, David

Adv. Math. 230, No. 4-6, 2176-2197 (2012).

From the abstract: We establish 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 \(\mathrm{spec}(R)\).
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 \(\mathbb Q\)-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.

Reviewer: Stefan Maubach (Bremen)

MSC:

14R10	Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
13F30	Valuation rings

Keywords:

polynomial automorphisms; stable tameness

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