Unramified Extensions Of Normal Domains
Date of Award
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.
Drost, John Lewin, "Unramified Extensions Of Normal Domains" (1983). Dissertations from ProQuest. 1339.