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Invariant Theory for Unipotent Groups and an Algorithm for Computing Invariants
Author(s) -
Salas Carlos Sancho De
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012508
Subject(s) - mathematics , unipotent , quotient , pure mathematics , algebraic group , algebraic extension , discrete mathematics , degree (music) , combinatorics , algebraic number , mathematical analysis , differential algebraic equation , ordinary differential equation , physics , acoustics , differential equation
Let X =Spec B be an affine variety over a field of arbitrary characteristic, and suppose that there exists an action of a unipotent group (possibly neither smooth nor connected). The fundamental results are as follows. (1) An algorithm for computing invariants is given, by means of introducing a degree in the ring of functions of the variety, relative to the action. Therefore an algorithmic construction of the quotient, in a certain open set, is obtained. In the case of a Galois extension, k ↪ B = K , which is cyclic of degree p =char k (that is, such that the unipotent group is G =Z/ p Z), an element of minimal degree becomes an Artin–Schreier radical, and the method for computing invariants gives, in particular, the expression for any element of K in terms of these radicals, with an explicit formula. This replaces the well‐known formula of Lagrange (which is valid only when the degree of the extension and the characteristic are relatively prime) in the case of an extension of degree p =char k . (2) In this paper we give an effective construction of a stable open subset where there is a quotient. In this sense we obtain an algebraic local criterion for the existence of a quotient in a neighbourhood. It is proved (provided the variety is normal) that, in the following cases, such an open set is the greatest one that admits a quotient: when the action is such that the orbits have dimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any action of the additive group G a ; in characteristic 0, when the action is proper (obtained from the results of Fauntleroy) or the group is abelian. 1991 Mathematics Subject Classification : primary 14L30; secondary 14D25, 14D20.