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In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring . We give two versions of the construction: the first, much simpler, in a special case (roughly speaking, that of rank 1 valuations), the second – in the case...
Nous montrons ici un théorème d’approximation diophantienne entre le corps des séries formelles en plusieurs variables et son complété pour la topologie de Krull.
This paper extends the notion of an archimedean frame to frames which are not necessarily algebraic. The new notion is called joinfitness and is Choice-free. Assuming the Axiom of Choice and for compact normal algebraic frames, the new and the old coincide. There is a subfunctor from the category of compact normal frames with skeletal maps with joinfit values, which is almost a coreflection. Conditions making it so are briefly discussed. The concept of an infinitesimal element arises naturally,...
Let be an order in an algebraic number field. If is a principal order, then many explicit results on its arithmetic are available. Among others, is half-factorial if and only if the class group of has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.
Let be a complete Noetherian local ring, an ideal of and a nonzero Artinian -module. In this paper it is shown that if is a prime ideal of such that and is not finitely generated and for each the -module is of finite length, then the -module is not of finite length. Using this result, it is shown that for all finitely generated -modules with and for all integers , the -modules are of finite length, if and only if, for all finitely generated -modules with and...
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