17,713 research outputs found
On the behavior of test ideals under finite morphisms
We derive transformation rules for test ideals and -singularities under an
arbitrary finite surjective morphism of normal varieties in
prime characteristic . The main technique is to relate homomorphisms
, such as Frobenius splittings, to homomorphisms . In the simplest cases, these rules mirror transformation
rules for multiplier ideals in characteristic zero. As a corollary, we deduce
sufficient conditions which imply that trace is surjective, i.e.
.Comment: 33 pages. The appendix has been removed (it will appear in a
different work). Minor changes and typos corrected throughout. To appear in
the Journal of Algebraic Geometr
Additive unit representations in global fields - A survey
We give an overview on recent results concerning additive unit
representations. Furthermore the solutions of some open questions are included.
The central problem is whether and how certain rings are (additively) generated
by their units. This has been investigated for several types of rings related
to global fields, most importantly rings of algebraic integers. We also state
some open problems and conjectures which we consider to be important in this
field.Comment: 13 page
Projective equivalence of ideals in Noetherian integral domains
Let I be a nonzero proper ideal in a Noetherian integral domain R. In this
paper we establish the existence of a finite separable integral extension
domain A of R and a positive integer m such that all the Rees integers of IA
are equal to m. Moreover, if R has altitude one, then all the Rees integers of
J = Rad(IA) are equal to one and the ideals J^m and IA have the same integral
closure. Thus Rad(IA) = J is a projectively full radical ideal that is
projectively equivalent to IA. In particular, if R is Dedekind, then there
exists a Dedekind domain A having the following properties: (i) A is a finite
separable integral extension of R; and (ii) there exists a radical ideal J of A
and a positive integer m such that IA = J^m.Comment: 20 page
Transcendental extensions of a valuation domain of rank one
Let be a valuation domain of rank one and quotient field . Let
be a fixed algebraic closure of the -adic completion
of and let be the integral closure of in . We describe a relevant class of valuation domains
of the field of rational functions which lie over , which are
indexed by the elements , namely,
.
If is discrete and is a uniformizer, then a valuation domain
of is of this form if and only if the residue field degree
is finite and , for some , where is the maximal ideal
of . In general, for we have
if and only if and are conjugated over
. Finally, we show that the set of
irreducible polynomials over endowed with an ultrametric distance
introduced by Krasner is homeomorphic to the space endowed with the Zariski topology.Comment: accepted for publication in the Proceedings of the AMS (2016);
comments are welcome
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