208 research outputs found
An application of the almost purity theorem to the homological conjectures
The aim of this article is to establish the existence of big Cohen-Macaulay
algebras in mixed characteristic in some special situation. The main result
follows from the so-called almost purity theorem proved by Davis and Kedlaya.Comment: to appear in Journal of Pure and Applied Algebr
Almost Cohen-Macaulay algebras in mixed characteristic via Fontaine rings
In the present paper, it is proved that any complete local domain of mixed
characteristic has a weakly almost Cohen-Macaulay algebra in the sense that
some system of parameters is a weakly almost regular sequence, which is a
notion defined via a valuation. The central idea of this result originates from
the main statement obtained by Heitmann to prove the Monomial Conjecture in
dimension 3. In fact, A weakly almost Cohen-Macaulay algebra is constructed
over the absolute integral closure of a complete local domain by applying the
methods of Fontaine rings and Witt vectors. A connection of the main theorem
with the Monomial Conjecture is also discussed.Comment: To appear in Illinois J. of Mat
The Frobenius action on local cohomology modules in mixed characteristic
R. Heitmann's proof of the Direct Summand Conjecture has opened a new
approach to the study of homological conjectures in mixed characteristic.
Inspired by his work and by the methods of almost ring theory, we discuss a
normalized length for certain torsion modules, which was introduced by G.
Faltings. Using the normalized length and the Frobenius map, we prove some
results of local cohomology for local rings in mixed characteristic, which has
an immediate implication for the subject of splinters studied by A. Singh.Comment: 15 pages, to appear in Compositio Mat
Integral perfectoid big Cohen-Macaulay algebras via Andr\'e's theorem
The main result of this article is to prove that any Noetherian local domain
of mixed characteristic maps to an integral perfectoid big Cohen-Macaulay
algebra. The proof of this result is based on the construction of almost
Cohen-Macaulay algebras in mixed characteristic due to Yves Andr\'e. Moreover,
we prove that the absolute integral closure of a complete Noetherian local
domain of mixed characteristic maps to an integral perfectoid big
Cohen-Macaulay algebra.Comment: Final version: to appear in Math. Annale
An embedding problem of Noetherian rings into the Witt vectors
The aim of this article is to prove some results on the existence of an
integral extension domain of a complete local Noetherian domain in mixed
characteristic having certain distinguished properties with respect to
the Frobenius map. We prove the main results by constructing required extension
domains via Witt vectors and the method of maximal \'etale extensions. It is
worth remarking that the resulting algebras have deep connections with the
homological conjectures and the rings in -adic Hodge theory.Comment: minor change
F-coherent rings with applications to tight closure theory
The aim of this paper is to introduce a new class of Noetherian rings of
positive characteristic in terms of perfect closures and study their basic
properties. If the perfect closure of a Noetherian ring is coherent, we call it
an -coherent ring. Some interesting applications are given in connection
with tight closure theory. In particular, we discuss relationships between
-coherent rings and -pure, -regular, and -injective rings. The
final section discusses how the coherent property effects the behavior of tight
closure for general perfect rings.Comment: 10 pages, comments are welcom
On the Witt vectors of perfect rings in positive characteristic
The purpose of this article is to prove some results on the Witt vectors of
perfect -algebras. Let be a perfect -algebra
for a prime integer and assume that has the property . Then
does the ring of Witt vectors of also have ? A main theorem
gives an affirmative answer for \mathbf{P}="\mbox{integrally closed}" under a
very mild condition.Comment: to appear in Communications in Algebr
Almost Cohen-Macaulay and almost regular algebras via almost flat extensions
The present paper deals with various aspects of the notion of almost
Cohen-Macaulay property, which was introduced and studied by Roberts, Singh and
Srinivas. We employ the definition of almost zero modules as defined by a value
map, which is different from the version of Gabber-Ramero. We prove that, if
the local cohomology modules of an algebra of certain type over a local
Noetherian ring are almost zero, maps to a big Cohen-Macaulay algebra. Then
we study how the almost Cohen-Macaulay property behaves under almost faithfully
flat extension. As a consequence, we study the structure of -coherent rings
of positive characteristic in terms of almost regularity.Comment: to appear in J. Commutative Algebr
Specialization Method in Krull Dimension two and Euler System Theory over Normal Deformation Rings
The aim of this article is to establish the specialization method on
characteristic ideals for finitely generated torsion modules over a complete
local normal domain R that is module-finite over , where
is the ring of integers of a finite extension of the field of p-adic
integers . The specialization method is a technique that recovers the
information on the characteristic ideal from ,
where I varies in a certain family of nonzero principal ideals of R. As
applications, we prove Euler system bound over Cohen-Macaulay normal domains by
combining the main results in an earlier article of the first named author and
then we prove one of divisibilities of the Iwasawa main conjecture for
two-variable Hida deformations generalizing the main theorem obtained in an
article of the first named author
An elementary proof of Cohen-Gabber theorem in the equal characteristic case
The aim of this article is to give a new proof of Cohen-Gabber theorem in the
equal characteristic case.Comment: 13 pages, to appear in Tohoku Math.
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