Commutative Algebra Seminar - Giulio Peruginelli

July 10, 2018
Wednesday, July 18, 2018 - 3:00pm to 4:00pm
Math Tower 154
math_sculpture

Title Valuation domains of the field of rational functions associated to pseudo-convergent sequences 

Speaker:  Giulio Peruginelli - University of Padova
 
Abstract:  Let V be a valuation domain of rank one with quotient field K. Recently, Loper and Werner  gave a natural construction of a valuation domain V_E of the field of rational function K(X) lying over V associated to a pseudo-convergent sequence E=\{s_n\}_{n\in\mathbb{N}} of K, when E is either of transcendental type or of algebraic type and zero-breadth ideal. This
construction has been employed by Loper and Werner in the realm of integer-valued polynomials. We show that their construction works for any valuation domain V and for any pseudo-convergent sequence of K.  We also show that, when V has rank one, the valuation domain V_E is strictly related to another extension of V to K(X) as given by Ostrowski in 1935. We give some results about the Zariski-Riemann spaces of such valuation domains. This is a joint work with Dario Spirito.

If time allows, I will also introduce the more general definition of pseudo-monotone sequence as given by Chabert, which has been an essential  tool to establish when the ring of  integer-valued polynomials over a subset S of V, that is \textnormal{Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\},  is a Pr\"ufer domain.

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