Unramified Extensions Of Normal Domains

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Degree Name

Doctor of Philosophy (Ph.D.)


Properties of inclusions of normal domains where the larger domain is unramified or separable over the smaller are examined. If these domains are C and A respectively then there is an ideal, the finiteness ideal, of A equal to the set of elements where localizing makes C finite over A. Similarly if B is finite over A then there is an ideal, the discriminant ideal, equal to the set of elements where localizing makes B unramified over A. In the situation that C is an unramified extensions of A normal and B is the integral closure of A in C then both ideals exist. In addition the map from Spec to Spec B is an open immersion defining an ideal of B, the Zariski complement ideal, which is the intersection of those primes of B that do not extend. Finally the branch locus ideal of B is the intersection of all the ramified primes of B. The Zariski complement ideal restricts to the finiteness ideal, and the branch locus ideal to the discriminant ideal.If C above is extended to its Galois closure over A then the above mentioned ideals of A are preserved. In fact, the rings arising from the decomposition of C(CRTIMES)(,A)C into a product of normal domains share the same finiteness ideals and the discriminant ideals of the integral closures of A are the same. The domains also indicate whether the field extension C over A is Galois or not.In the case of C and A are affine over the complex numbers more topological results can be given, involving the fundamental groups of Max C and Max A the spaces of maximal ideals. For example, if the map if finite and unramified it is a topological covering map. Finally, restriction to affine maps of irreducible schemes allows generalizations of the results to the case of schemes to be made.



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