A commuting derivations theorem on UFDs
| dc.creator | Derksen, Harm | |
| dc.creator | Essen, Arno van den | |
| dc.creator | Maubach, Stefan | |
| dc.date | 2008-06-12 | |
| dc.date.accessioned | 2026-07-07T09:44:10Z | |
| dc.date.available | 2026-07-07T09:44:10Z | |
| dc.description | Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy $A^{D_1,...,D_m}=k[f]$ where $f$ is a coordinate. The conjecture can be formulated as stating that a $(G_m)^n$-action on $k^{n+1}$ must have invariant ring $k[f]$ where $f$ is a coordinate. In this paper we prove a statement (theorem \ref{CDH2}) where we assume less on $A$ ($A$ is a {\sc UFD} over $k$ of transcendence degree $n+1$ satisfying $A^*=k$) and prove less ($A/(f-α)$ is a polynomial ring for all but finitely many $α$). Under certain additional conditions (the $D_i$ are linearly independent modulo $(f-α)$ for each $α\in k$) we prove that $A$ is a polynomial ring itself and $f$ is a coordinate. This statement is proven even more generally by replacing ``free unipotent action of dimension $n$'' for ``$G_a^n$-action''. We make links with the (Abhyankar-)Sataye conjecture and give a new equivalent formulation of the Sataye conjecture. | |
| dc.description | This draft was already written in 2006 | |
| dc.identifier | https://arxiv.org/abs/0806.2038 | |
| dc.identifier | http://arxiv.org/abs/0806.2038 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/162776 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | 4L17, 14R20, 20E45, 20G20, 14L35 | |
| dc.title | A commuting derivations theorem on UFDs | |
| dc.type | text |