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Subrings invariant under endomorphisms
Let S and R be the rings of regular functions on affine algebraic varieties
over a field of characteristic 0, R be embedded as a subring in S, and F : S
--> S be an endomorphism such that F(R) subset R. Suppose that every ideal of
height 1 in R generates a proper ideal in S, and the spectrum of R has no
selfintersection points. We show that if F is an automorphism so is F|_R : R
--> R. When R and S have the same transcendence degree then the fact that F|_R
is an automorphisms implies that F is an automorphism.Comment: 16 page
Equivariant virtual Betti numbers
We define a generalised Euler characteristic for arc-symmetric sets endowed
with a group action. It coincides with equivariant homology for compact
nonsingular sets, but is different in general. We lay emphasis on the
particular case of , and give an application to the study of the
singularities of Nash function germs via an analog of the motivic zeta function
of Denef & Loeser.Comment: 20 pages, to appear in Ann. Inst. Fourie
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