16,027 research outputs found
On the section conjecture of Grothendieck
For a given arithmetic scheme, in this paper we will introduce and discuss
the monodromy action on a universal cover of the \'etale fundamental group and
the monodromy action on an \emph{sp}-completion constructed by the graph
functor, respectively; then by these results we will give a proof of the
section conjecture of Grothendieck for arithmetic schemes.Comment: 22 pages. Made Changes in Page 4, Def 2.3; Page 5, "essential equal".
Deleted Page 6, footnote. Removed Typos: "integral scheme" changed into
"integral variety (-ies)" in Remark 5.8, Theorem 5.9, Lemma 5.10, and Page
15, section
On the etale fundamental groups of arithmetic schemes
In this paper we will give the computation of the etale fundamental group of
an arithmetic scheme.Comment: 11 pages. Made changes in: Section 1.1, Page 2; Section 2.2, Page 3;
Section 2.3, Page
The Specializations in a Scheme
In this paper we will obtain some further properties for specializations in a
scheme. Using these results, we will take a picture for a scheme and a picture
for a morphism of schemes. In particular, we will prove that every morphism of
schemes is specialization-preserving and of norm not greater than one (under
some condition); a necessary and sufficient condition will be given for an
injective morphism of irreducible schemes.Comment: Revised. 14 page
Affine Structures on a Ringed Space and Schemes
In this paper we will first introduce the notion of affine structures on a
ringed space and then obtain several properties. Affine structures on a ringed
space, arising mainly from complex analytical spaces of algebraic schemes over
number fields, behave like differential structures on a smooth manifold.
As one does for differential manifolds, we will use pseudogroups of affine
transformations to define affine atlases on a ringed space. An atlas on a space
is said to be an affine structure if it is maximal. An affine structure is
admissible if there is a sheaf on the underlying space such that they are
coincide on all affine charts, which are in deed affine open sets of a scheme.
In a rigour manner, a scheme is defined to be a ringed space with a specified
affine structure if the affine structures are in action in some special cases
such as analytical spaces of algebraic schemes. Particularly, by the whole of
affine structures on a space, we will obtain respectively necessary and
sufficient conditions that two spaces are homeomorphic and that two schemes are
isomorphic, which are the two main theorems of the paper. It follows that the
whole of affine structures on a space and a scheme, as local data, encode and
reflect the global properties of the space and the scheme, respectively.Comment: Final version. 22 pages. to appear in Chinese Ann of Math, Series
On the existence of geometric models for function fields in several variables
In this paper we will give an explicit construction of the geometric model
for a prescribed extension of a function field in several variables over a
number field.
As a by-product, we will also prove the existence of quasi-galois closed
covers of arithmetic schemes (in eprint arXiv:0907.0842).Comment: 9 pages. Made changes in: Section 2.1, Page 3; Section 3.1, Page
Notes on the section conjecture of Grothendieck
In this short note, we will give the key point of the section conjecture of
Grothendieck, that is reformulated by monodromy actions. Here, we will also
give the result of the section conjecture for algebraic schemes over a number
field.Comment: 5 page
On the unramified extension of an arithmetic function field in several variables
In this paper we will give a scheme-theoretic discussion on the unramified
extensions of an arithmetic function field in several variables. The notion of
unramified discussed here is parallel to that in algebraic number theory and
for the case of classical varieties, coincides with that in Lang's theory of
unramified class fields of a function field in several variables. It is twofold
for us to introduce the notion of unramified. One is for the computation of the
\'{e}tale fundamental group of an arithmetic scheme; the other is for an
ideal-theoretic theory of unramified class fields over an arithmetic function
field in several variables. Fortunately, in the paper we will also have
operations on unramified extensions such as base changes, composites,
subfields, transitivity, etc. It will be proved that a purely transcendental
extension over the rational field has a trivial unramified extension. As an
application, it will be seen that the affine scheme of a ring over the ring of
integers in several variables has a trivial \'{e}tale fundamental group.Comment: 13 page
On the arithmetic fundamental groups
In this paper we will define a qc fundamental group for an arithmetic scheme
by quasi-galois closed covers. Then we will give a computation for such a group
and will prove that the etale fundamental group of an arithmetic scheme is a
normal subgroup in our qc fundamental group, which make up the main theorem of
the paper. Hence, our group gives us a prior estimate of the etale fundamental
group. The quotient group reflects the topological properties of the scheme.Comment: 14 pages. Made changes in: convention 1.1; section 1.2; Definition
1.1; Remark 1.
On the transcendental Galois extensions
In this paper the transcendental Galois extensions of a field will be
introduced as counterparts to algebraic Galois ones. There exist several types
of transcendental Galois extensions of a given field, from the weakest one to
the strongest one, such as Galois, tame Galois, strong Galois, and absolute
Galois. The four Galois extensions are distinct from each other in general but
coincide with each other for cases of algebraic extensions. The transcendental
Galois extensions arise from higher relative dimensional Galois covers of
arithmetic schemes. In the paper we will obtain several properties of Galois
extensions in virtue of conjugation and quasi-galois and will draw a comparison
between the Galois extensions. Strong Galois is more accessible. It will be
proved that a purely transcendental extension is strong Galois.Comment: 13 pages. Revised versio
On the \'etale fundamental groups of arithmetic schemes, revised
In this paper we will give a computation of the \'{e}tale fundamental group
of an integral arithmetic scheme. For such a scheme, we will prove that the
\'{e}tale fundamental group is naturally isomorphic to the Galois group of the
maximal formally unramified extension over the function field. It consists of
the main theorem of the paper. Here, formally unramified will be proved to be
arithmetically unramified which is defined in an evident manner and coincides
with that in algebraic number theory. Hence, formally unramified has an
arithmetic sense. At the same time, such a computation coincides with the known
result for a normal noetherian scheme.Comment: Revised. 23 page
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