64 research outputs found
Towards a classification of stable semistar operations on a Pr\"ufer domain
We study stable semistar operations defined over a Pr\"ufer domain, showing
that, if every ideal of a Pr\"ufer domain has only finitely many minimal
primes, every such closure can be described through semistar operations defined
on valuation overrings of .Comment: to appear in Communications in Algebr
Non-compact subsets of the Zariski space of an integral domain
Let be a minimal valuation overring of an integral domain and let
be the Zariski space of the valuation overrings of .
Starting from a result in the theory of semistar operations, we prove a
criterion under which the set is not compact.
We then use it to prove that, in many cases, is not a
Noetherian space, and apply it to the study of the spaces of Kronecker function
rings and of Noetherian overrings.Comment: To appear in the Illinois Journal of Mathematic
Calculating the density of solutions of equations related to the P\'olya-Ostrowski group through Markov chains
Motivated by a problem in the theory of integer-valued polynomials, we
investigate the natural density of the solutions of equations of the form
, where are fixed integers,
are parameters and and
are functions related to the -adic valuations of the numbers between 1
and . We show that the number of solutions of this equation in
satisfies a recurrence relation, with which we can associate to any pair
a stochastic matrix and a Markov chain. Using this interpretation, we
calculate the density for the case and for the case
, and either or and
are coprime.Comment: to appear in Acta Arithmetic
Extending valuations to the field of rational functions using pseudo-monotone sequences
Let be a valuation domain with quotient field . We show how to
describe all extensions of to when the -adic completion
is algebraically closed, generalizing a similar result obtained
by Ostrowski in the case of one-dimensional valuation domains. This is
accomplished by realizing such extensions by means of pseudo-monotone
sequences, a generalization of pseudo-convergent sequences introduced by
Chabert. We also show that the valuation rings associated to pseudo-convergent
and pseudo-divergent sequences (two classes of pseudo-monotone sequences)
roughly correspond, respectively, to the closed and the open balls of in
the topology induced by .Comment: all comments are welcome!
The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences
Let be a valuation domain with quotient field . Given a
pseudo-convergent sequence in , we study two constructions associating
to a valuation domain of lying over , especially when has
rank one. The first one has been introduced by Ostrowski, the second one more
recently by Loper and Werner. We describe the main properties of these
valuation domains, and we give a notion of equivalence on the set of
pseudo-convergent sequences of characterizing when the associated valuation
domains are equal. Then, we analyze the topological properties of the
Zariski-Riemann spaces formed by these valuation domains.Comment: any comment is welcome! Trans. Amer. Math. Soc. 373 (2020), no. 11,
7959-799
Free groups of ideals
We study the freeness of the group of invertible ideals of
an integral domain , and the freeness of some related groups of (fractional)
ideals. We study the relation between and
, in particular in the locally finite case, and we analyze
in more detail the case where is Noetherian (obtaining a characterization
of when is free for one-dimensional analytically unramified
Noetherian domains) and where is Pr\"ufer
Asymptotic for the number of star operations on one-dimensional Noetherian domains
We study the set of star operations on local Noetherian domains of
dimension such that the conductor (where is the integral
closure of ) is equal to the maximal ideal of . We reduce this problem to
the study of a class of closure operations (more precisely, multiplicative
operations) in a finite extension , where is a field, and
then we study how the cardinality of this set of closures vary as the size of
varies while the structure of remains fixed
The local Picard group of a ring extension
Given an integral domain and a -algebra , we introduce the local
Picard group as the quotient between the Picard group
and the canonical image of in
, and its subgroup generated by the the
integral ideals of that are unitary with respect to . We show that, when
is a ring extension that satisfies certain properties (for
example, when is the ring of polynomial or the ring of
integer-valued polynomials ), it is possible to decompose
as the direct sum , where
ranges in a Jaffard family of . We also study under what hypothesis this
isomorphism holds for pre-Jaffard families of
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