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Endomorphism from Galois antiautomorphism. (English) Zbl 0841.12001

The present paper may possibly contain some nice ideas, but unfortunately it is written in such a vague style that even the definitions remain unclear.
Author’s introduction: The aim of this paper is to introduce an endomorphism based upon Eisenstein homology. Galois homology was previously well worked out and Eisenstein cohomology was extensively studied, but it seems that the Eisenstein homology was never taken up.
Considering that Galois homology results from a Galois anti-automorphism, it is proved that every endomorphism generated from a Galois anti-automorphism can be decomposed into the direct sum of a Galois homology class and its complementary Galois cohomology class. All developments are initiated from a polynomial ring \(A[x_1,\cdots,x_m]\) in \(m\) indeterminates whose specialization is required to generate a sequential and graded sheaf of rings \(\theta^m\).
Part I gives basic algebraic notions necessary for generating a graded sheaf of rings from a Galois extension, i.e. essentially a specialization, called emergent, from a ring of polynomials \(A[x_1,\cdots,x_m]\) giving rise to a set of compact connected algebraic subgroups which correspond to the different sections of the sheaf of rings \(\theta^m\). Part II refers to the introduction of Eisenstein homology based upon a Galois anti-automorphism. Part III gives the conditions for an endomorphism generated from a Galois anti-automorphism.

MSC:

12F10 Separable extensions, Galois theory
11F75 Cohomology of arithmetic groups
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