The relative Chern character and regulators [Elektronische Ressource] / vorgelegt von Georg Tamme

Sujets

• Mathematics

Informations

• Publié par: universitat_regensburg
• Publié le: 01 janvier 2010
• Nombre de lectures: 11
• Langue: English

Extrait

The Relative Chern Character and Regulators
Dissertation zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
an der Naturwissenschaftlichen Fakultat I { Mathematik der
Universitat Regensburg
vorgelegt von
Georg Tamme
aus Sinzing
2010Promotionsgesuch eingereicht am: 4. Februar 2010
Die Arbeit wurde angeleitet von: Prof. Dr. Guido Kings
Prufungsaussc huss:
Prof. Dr. Helmut Abels (Vorsitzender)
Prof. Dr. Guido Kings (1. Gutachter)
Prof. Amnon Besser, Ben Gurion University (2. Gutachter)
Prof. Dr. Klaus Kunnemann
Prof. Dr. Uwe Jannsen (Ersatzprufer)CONTENTS
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Notations and Conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Part I. The complex theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1. Simplicial Chern-Weil theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1. De Rham cohomology of simplicial complex manifolds. . . . . . . . . . . . 15
1.2. Bundles on simplicial manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3. Connections, curvature and characteristic classes. . . . . . . . . . . . . . . . . 29
1.4. Secondary classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2. Characteristic classes of algebraic bundles. . . . . . . . . . . . . . . . . . . . . . 41
2.1. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2. Chern classes of algebraic bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3. Relative Chern character classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4. Chern classes in Deligne-Beilinson cohomology. . . . . . . . . . . . . . . . . . . . 53
2.5. Comparison of relative and Deligne-Beilinson Chern character
classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3. Relative K-theory and regulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1. Topological K-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2. Relative K-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 CONTENTS
3.3. The relative Chern character. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4. Comparison with the Chern character in Deligne-Beilinson
cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5. Non a ne varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6. The case X = Spec(C): The regulators of Borel and Beilinson. . . . 78
Part II. The p-adic theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1. A noid algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2. Dagger spaces, weak formal schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5. Chern-Weil theory for simplicial dagger spaces. . . . . . . . . . . . . . . . 101
5.1. De Rham cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2. Simplicial bundles and connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3. Secondary classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4. Chern character classes for algebraic bundles. . . . . . . . . . . . . . . . . . . . . 113
6. Re ned and secondary classes for algebraic bundles . . . . . . . . . . . 117
6.1. Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2. Comparison with the secondary classes of section 5.3. . . . . . . . . . . . . 124
6.3. Variant for R-schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7. Relative K-theory and regulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.1. Topological K-theory of a noid and dagger algebras. . . . . . . . . . . . . 129
7.2. Relative K-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.3. The relative Chern character. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.4. The caseX = Spec(R): Comparison with thep-adic Borel regulator140
A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.1. Some homological algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.2. Cohomology on strict simplicial (dagger) spaces. . . . . . . . . . . . . . . . . . 157
A.3. Simplicial groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159CONTENTS 3
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163INTRODUCTION
The starting point for the study of regulators is Dirichlet’s regulator for a
number eld F . Ifr (resp. 2r ) is the number of real (resp. complex) embed-1 2
r +r1 2dings of F , one has the regulator map r :O !H R from the groupF
r +r1 2of units in the ring of integersO ofF to a hyperplane in R . Its kernel isF
nite and its image is a lattice, whose covolume is Dirichlet’s regulator R . InF
ththe late 19 century, Dedekind related this regulator to the residue at s = 1
of the zeta function (s) of the number eld. Using the meromorphic contin-F
uation and the functional equation of proved by Hecke one can formulateF
this relation in the class number formula
hRF(r +r 1)1 2lim (s)s = ;F
s!0 w
where h is the class number of F , w is the number of roots of unity and the
left hand side is the leading coe cient of the Taylor expansion of ats = 0.F
In the 1970’s Quillen introduced higher algebraic K-groups K (O ), i 0,i F

generalizingK (O ) =O and showed, that they are nitely generated. Borel1 F F
r r +r2 1 2constructed higher regulators r :K (O )! R (resp. R ), if n 2n 2n 1 F
is even (resp. odd). He was able to prove, that the kernel of r is nite andn
its image is a lattice, whose covolume is a rational multiple of the leading
coe cient of the Taylor expansion of at the point 1 n.F
In the following, the construction of regulators was extended to the case of
K of a curve by Bloch, and then to all smooth projective varieties over Q by26 INTRODUCTION
Beilinson. In this context the regulator maps for the variety X,
2n iK (X)!H (X ; R(n));i R
D
have values in the Deligne-Beilinson cohomology of X and are obtained by
composing the natural map K (X)!K (X ) with the Chern character mapi i C
D 2n i (1)Ch : K (X )! H (X ; R(n)). Beilinson establishes a whole systemi C Cn;i
D
of conjectures relating these regulators to the leading coe cients of the Taylor
expansions of the L-functions of X at the integers [Be 84].
He also sketches a proof of the fact, that in the case of a number eld, his
regulator maps coincide with Borel’s regulator maps. Then Borel’s theorem
implies Beilinson’s conjectures in this case. Many details of this proof were
given by Rapoport in [Rap88]. With a completely di erent strategy, based on
the comparison of Cheeger-Simons Chern classes with Deligne-Beilinson Chern
classes, Dupont, Hain and Zucker [DHZ00] tried to compare both regulators
and gave good evidence for their conjecture, that Borel’s regulator is in fact
twice Beilinson’s regulator. Later on Burgos [BG02] worked out Beilinson’s
original argument and proved, that the factor is indeed 2.
Nowadays there exists also ap-adic analogue of the above conjectures. Thanks
to Perrin-Riou [PR95] one has a conjectural picture about the existence and
properties of p-adic L-functions, so that one can formulate a p-adic Beilin-
son conjecture for smooth projective varieties over a p-adic eld. There the
Deligne-Beilinson cohomology is replaced by (rigid) syntomic cohomology and
the regulator maps by the corresponding rigid syntomic Chern character.
In [HK06] Huber and Kings show, that one can also construct a p-adic Borel
regulator parallel to the classical Borel regulator, and relate it to the syntomic by an analogue of Beilinson’s comparison argument.
In a di erent direction, Karoubi [Kar87] constructed Chern character maps
(resp. relative Chern character maps) on the algebraic (resp. relative)K-theory
of any real, complex or even ultrametric Banach algebra with values in con-
tinuous cyclic homology, where relative K-theory is the homotopy bre of the
(1) 2n i
There is a natural action of complex conj