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Approximate roots of a valuation and the Pierce-Birkhoff conjecture

F. Lucas, J. Madden, D. Schaub, M. Spivakovsky (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

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 A . 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...

Archimedean frames, revisited

Jorge Martinez (2008)

Commentationes Mathematicae Universitatis Carolinae

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,...

Arithmetic of non-principal orders in algebraic number fields

Andreas Philipp (2010)

Actes des rencontres du CIRM

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R 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.

Artinian cofinite modules over complete Noetherian local rings

Behrouz Sadeghi, Kamal Bahmanpour, Jafar A'zami (2013)

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R -module. In this paper it is shown that if 𝔭 is a prime ideal of R such that dim R / 𝔭 = 1 and ( 0 : M 𝔭 ) is not finitely generated and for each i ≥ 2 the R -module Ext R i ( M , R / 𝔭 ) is of finite length, then the R -module Ext R 1 ( M , R / 𝔭 ) is not of finite length. Using this result, it is shown that for all finitely generated R -modules N with Supp ( N ) ⊆ V ( I ) and for all integers i ≥ 0 , the R -modules Ext R i ( N , M ) are of finite length, if and only if, for all finitely generated R -modules N with Supp ( N ) ⊆ V ( I ) and...

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