Mathematical Sciences: Problems Related to Lech's Conjectureand Some Questions on Divisor Class Groups

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There are two areas which will be addressed in this research. In the first, problems will be considered which are related to a conjecture by Lech concerning the behavior of the Hilbert function under faithfully flat morphisms of local noetherian rings. In the second, the principal investigator will study stability properties of the divisor class group CL(R) where R is a local normal excellent henselian ring. This research is concerned with the geometric properties of commutative rings which arise in algebraic geometry. One of the problems to be considered is a conjecture by Lech which has been unsolved for twenty-five years.