Aspherical neighborhoods on arithmetic surfaces

96 Pages·2016·0.8 MB·English

by  Katharina Hübner

DISSERTATION
submitted
to the
Combined Faculty of the Natural Sciences and Mathematics
of
Heidelberg University, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Dipl.-Phys. Dipl.-Math. Katharina Hübner
Born in: Nürnberg
Oral examination:

Aspherical neighborhoods on
arithmetic surfaces
Advisors:
Prof. Dr. Alexander Schmidt
Prof. Dr. Kay Wingberg

Abstract
Aspherical neighborhoods on arithmetic surfaces
On arithmetic surfaces over local or global rings of integers this thesis exam-
ines whether a geometric point has a basis of étale neighborhoods which are
Kpπ,1q with respect to a full class of finite groups c. These neighborhoods
are also called aspherical neighborhoods. In this thesis we will consider only
classes of finite groups c such that the order of all groups in c is prime to
the residue characteristics of the arithmetic surface in question. In the local
case we construct a basis of Kpπ,1q-neighborhoods for any geometric point
of a normal (but not necessarily regular) arithmetic surface. In the global
case the existence of such bases of neighborhoods is proven under additional
regularity assumptions and a condition on the l-division points of the Jaco-
bian of the generic fibre. Moreover, we assume in the global case case that c
is the class of finite l-groups for a prime number l that is invertible on the
arithmetic surface.
Asphärische Umgebungen auf arithmetischen Flächen
Die vorliegende Arbeit beschäftigt sich mit der Existenz asphärischer étaler
UmgebungsbasenaufarithmetischenFlächen,auchKpπ,1q-Umgebungsbasen
genannt. Genauer wird die Kpπ,1q-Eigenschaft bezüglich einer vollen Klasse
endlicher Gruppen c untersucht, wobei die Ordnung aller Gruppen in c teiler-
fremd zu den Restklassencharakteristiken der jeweiligen arithmetischen Flä-
che ist. Das Basisschema der hier behandelten arithmetischen Flächen ist
dabei stets ein lokaler oder globaler Zahlring. Im lokalen Fall wird für alle
normalen (aber nicht notwendigerweise regulären) arithmetischen Flächen
eine Kpπ,1q-Umgebungsbasis konstruiert. Für den globalen Fall sind zusät-
zlicheReguläritätsannahmenundeineBedingungandiel-Teilungspunkteder
Jacobischen der generischen Faser notwendig. Außerdem beschränkt sich die
Untersuchung auf die Klasse cplq der endlichen l-Gruppen für eine auf der
arithmetischen Fläche invertierbare Primzahl l.

1
Acknowledgements
I would like to thank my advisor Alexander Schmidt, firstly for arousing my interest in Kpπ,1q-
schemes and secondly for many fruitful discussions about this topic. I would like to thank Kay
Wingbergformakingtheefforttobemysecondadvisor. FurtherthanksgotoArminHolschbach,
Johannes Schmidt, Johannes Anschütz, and Malte Witte for answering my various questions. I
wanttothankDavidRohrforproof-readingmythesisalthoughheworksinacompletelydifferent
field. Finally, my thanks go to the members of the mathematical institute for accompanying me
throughmytimeasadoctoralstudent,forinterestingseminars,andformanylivelyconversations
during coffee breaks.
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Contents
Acknowledgements 1
Contents 2
1 Introduction 4
2 Setup and Notation 8
2.1 The Kpπ,1q-property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Arithmetic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Tame coverings of arithmetic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Non-existence of good Artin neighborhoods 17
4 Exceptional fibres 21
4.1 The local structure of tame coverings . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Rational Singularities and the dual graph . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Explicit desingularizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 The generalized exceptional fibre of a desingularized tame covering. . . . . . . . . 33
4.5 Multiplicities of the exceptional divisors . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Cohomology with support 40
5.1 Absolute cohomological purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Cohomology and dual graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3 Killing cohomology with support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Higher direct Images 54
6.1 The Leray Spectral Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Killing the Cohomology of higher direct images . . . . . . . . . . . . . . . . . . . . 56
6.3 The intersection matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.4 The first higher direct image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.5 The second higher direct image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
CONTENTS 3
7 Construction of Kpπ,1q neighborhoods 76
7.1 Stability of certain properties under desingularized tame coverings . . . . . . . . . 76
7.2 Neighborhoods with enough tame coverings . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Construction of suitable étale neighborhoods . . . . . . . . . . . . . . . . . . . . . 83
8 The main results 86
Bibliography 89
4
Chapter 1
Introduction
A classical result in topology states that every CW-complex is locally contractible. In other
words, for every point x in a CW-complex X we find an open neighborhood U of x such that the
inclusiontxuãÑU isahomotopyequivalence. InthecategoryofCW-complexesthisisequivalent
to saying that the induced maps of homotopy groups with base point x are isomorphisms, i.e.,
that txuãÑU is a weak homotopy equivalence. Since the homotopy groups of a point are trivial,
this amounts to saying that every point of X has a neighborhood with trivial homotopy groups.
As a consequence, we can cover X by contractible open subsets. The topological properties of X
are then completely encoded in the way these open subsets are patched together to form X.
We would like to have a similar result in étale homotopy theory. Let us sketch shortly what
étale homotopy theory is about. In [Fri II] it is explained how to functorially assign to a locally
noetherian scheme X with geometric point x¯ a pointed pro-CW-complex, the étale topological
type. Strictly speaking the étale topological type is not a pro-CW-complex but a pro-simplicial
set. After geometric realization, however, we can view it as pro-CW-complex. Its image in the
pro-homotopy category of CW-complexes is denoted X and is called étale homotopy type of X.
ét
It was already constructed in [AM]. To the pro-CW-complex X we can associate its homotopy
ét
pro-groups π pX q. In [SGA3], Exp. X, §6 there is already defined a first homotopy pro-group
n ét
for X, the "pro-groupe fondamentale enlargi" π pX,x¯q. In order for étale homotopy theory to be
1
useful, the first homotopy group of the étale homotopy type X should be related to π pX,x¯q.
ét 1
And indeed, by [AM], Corollary 10.7 we have π pX q–π pX,x¯q. It thus makes sense to define
1 ét 1
the étale homotopy pro-groups of X as
π pX,x¯q:“π pX q.
n n ét
If X is connected, geometrically unibranch, and noetherian, the étale homotopy pro-groups are
profinite (see [AM], Theorem 11.1) and thus can be interpreted as topological groups. In this
case the first homotopy group coincides with the fundamental group defined in [SGA1]. Further-
more, étale homotopy theory is compatible with étale cohomology in the following sense: Via
the isomorphism π pX q – π pX,x¯q the locally constant étale sheaves on X are in one-to-one
1 ét 1
correspondence with the local systems on X . If (cid:65) is a locally constant sheaf on X and A its
ét
corresponding local system on X , we have
ét
HnpX,(cid:65)q–HnpX ,Aq
ét
by [AM], Corollary 10.8.
Let us return to the problem of local contractibility. The étale homotopy type X is in general
ét
not a CW-complex but a pro-CW-complex. In the category of pro-CW-complexes there is a
priori no canonical notion of homotopy equivalence. Yet, we can still speak of a weak homotopy
equivalence by saying that it induces isomorphisms on homotopy pro-groups. Following [AM],
we will write 7-isomorphism instead of weak homotopy equivalence. In general, we cannot expect
to find étale neighborhoods of a geometric point x¯ in X such that all homotopy pro-groups