Differents in modular invariant theory |
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Authors: | Abraham Broer |
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Institution: | (1) Departement de mathematiques et de statistiques, Universite de Montreal, C.P. 6128, succursale Centre-ville, Montreal (Quebec), Canada H3C 3J7 |
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Abstract: | Let
be a graded polynomial
algebra over a field k, such that each variable is homogeneous of positive degree. No restrictions are made with respect to
the field. Let the finite group G act on A by graded algebra automorphisms and denote the subalgebra of invariants by B. In
this paper the various "different ideals" of the extension
are studied that define the ramification locus. We prove, for example, that the subring of invariants is itself a polynomial
ring if and only if the ramification locus is pure of height one. Here the ramification locus is defined by either the Kahler
different, the Noether different or the Galois different. As a consequence we prove that the invariant ring is itself a polynomial
ring if and only if there are invariants
whose Jacobian determinant does not vanish and is of degree δ, where δ is the degree of the Dedekind different. Using this
criterion we give a quick proof of Serre's result that if the invariant ring is a polynomial algebra, then the group is generated
by generalized reflections. |
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Keywords: | |
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