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Invariants of Finite Group Schemes
Author(s) -
Skryabin Serge
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610701002903
Subject(s) - mathematics , subring , reductive group , algebraically closed field , algebraic group , sheaf , ring (chemistry) , pure mathematics , quotient , complete intersection , finite group , algebraic closure , morphism , invariant (physics) , discrete mathematics , group (periodic table) , algebraic number , group theory , mathematical analysis , differential algebraic equation , ordinary differential equation , chemistry , organic chemistry , mathematical physics , differential equation
Let G be a finite group scheme operating on an algebraic variety X , both defined over an algebraically closed field k . The paper first investigates the properties of the quotient morphism X ‐ X / G over the open subset of X consisting of points whose stabilizers have maximal index in G . Given a G ‐linearized coherent sheaf on X , it describes similarly an open subset of X over which the invariants in the sheaf behave nicely in some way. The points in X with linearly reductive stabilizers are characterized in representation theoretic terms. It is shown that the set of such points is nonempty if and only if the field of rational functions k(X) is an injective G‐module. Applications of these results to the invariants of a restricted Lie algebra g operating on the function ring k[X] by derivations are considered in the final section. Furthermore, conditions are found ensuring that the ring k[X] g is generated over the subring of pth powers in k[X] , where p=char,k>0, by a given system of invariant functions and is a locally complete intersection.