Discovery of the structure of A2-fibrations having fixed point free locally nilpotent derivations
Mathematics
A problem in affine algebraic geometry asks the following.
Problem 1. Let R be a ring containing Q, A an A2-fibration over R and D : A → A a fixed point free R-LND. Does D have a slice?
Contributions by Rentschler (1968), Daigle-Freudenburg (1998), Bhatwadekar-Dutta (1997), Berson-van den Essen-Maubach (2001) and van den Essen (2007) established that the above problem has affirmative answer when A is trivial, i.e., A is a polynomial algebra in two variables over R. However, it was not known whether the above result holds when A is a non-trivial A2-fibration over R. In 2016, a partial answer to the problem was given by Kahoui-Ouali showing that the above question has an affirmative answer under the assumption that A is a stably polynomial algebra. But the result of Kahoui-Ouali shows that the hypothesis essentially makes the A2-fibration to be trivial.
In 2021, Babu-Das completely solved Problem 1 when R is Noetherian. Further, the result of Babu-Das completely describes the structure of A2-fibrations over Noetherian rings containing Q having fixed point free locally nilpotent derivations. The main results of the article are the following.
Theorem A: Let R be a Noetherian ring containing Q and A an A2-fibration over R with a fixed point free R-LND D : A → A. Then, Ker(D) is an A1-fibration over R and D has a slice.
Theorem B: Let R be a Noetherian domain containing Q and A an A2-fibration over R having a fixed point free R-LND. Then, A has another irreducible R-LND D : A → A such that Ker(D) is a polynomial algebra in one variable over R and A is an A1-fibration over Ker(D). Further, the following are equivalent.
- D is fixed point free.
- A is stably polynomial over R.
- A is a polynomial algebra in two variables over R .
Ref: Janaki Raman Babu and Prosenjit Das. Structure of A2-fibrations having fixed point free locally nilpotent derivations. J. Pure Appl. Algebra, 225(12):Paper No. 106763, 12, 2021.