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On the Castelnuovo-Mumford regularity of rings of polynomial invariants. (English) Zbl 1235.13005

Let \(k\) be any field and \(S=k[x_1,\dots,x_n]\) the polynomial ring. Let \(G\) be a finite group acting on \(S\) by homogeneous linear substitutions. The main theorem of the paper says that the invariant subring \(S^G\) has Castelnuovo-Mumford regularity at most zero. At first the author proves this theorem when \(k\) is a finite field. Then an argument by Burt Totaro permits to extend the result for all fields. As a consequence the invariant subring \(S^G\) is generated in degrees at most \(n(|G|-1).\)

MSC:

13A50 Actions of groups on commutative rings; invariant theory
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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