Additive Chow groups with higher modulus and the generalized de Rham-Witt complex [Elektronische Ressource] / von Kay Rülling

Description

Additive Chow groups with higher modulusandthe generalized de Rham-Witt complexDissertationzur Erlangung des GradesDoktor der Naturwissenschaften(Dr. rer. nat.)vorgelegt beimFachbereich Mathematik und Informatikder Universit¨at Duisburg-Essenvon Kay Rull¨ ingaus EssenTag der Disputation 11. Mai 2005Vorsitzender der Pruf¨ ungskommission Prof. Dr. Axel KlawonnErster Gutachter Prof. Dr. H´el`ene EsnaultZweiter Gutachter Prof. Dr. Eckart ViehwegAcknowledgmentsI am very grateful to H´el`ene Esnault for giving me the subject of this thesis and forher excellent guidance and support, not only during the work on this problem, butfrom the very beginning of my mathematical studies.IwanttoexpressmygratitudetoLarsHesselholtforexplainingtheWittvectorsand the generalized de Rham-Witt complex to me.I would like to thank Spencer Bloch and Gerd Faltings for several hints, inparticular, to construct the trace for the Witt vectors via the norm map.I want to thank Stefan Kukulies for the many discussions concerning the foun-dations of algebraic geometry, Andre Chatzistamatiou for the careful reading of theintroduction and Wioletta Syzdek for the supply with chocolate.Finally I want to thank my parents for making untroubled studies possible.

Sujets

• Mathematics

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2005
• Nombre de lectures: 34
• Langue: English

Extrait

Additive

the

Chow

gro

generalized

ups with higher modulus and e Rham-Witt complex

d

Dissertation zur Erlangung des Grades Doktor der Naturwissenschaften (Dr. rer. nat.)

vorgelegt beim Fachbereich Mathematik und Informatik der Universit¨t Duisburg-Essen a

Tag der Disputation

vonKayRu¨lling aus Essen

VorsitzenderderPr¨ufungskommission Erster Gutachter Zweiter Gutachter

11. Mai 2005

Prof. Dr. Axel Klawonn Prof.Dr.He´le`neEsnault Prof. Dr. Eckart Viehweg

Acknowledgments

IamverygratefultoHe´le`neEsnaultforgivingmethesubjectofthisthesisandfor her excellent guidance and support, not only during the work on this problem, but from the very beginning of my mathematical studies. I want to express my gratitude to Lars Hesselholt for explaining the Witt vectors and the generalized de Rham-Witt complex to me. I would like to thank Spencer Bloch and Gerd Faltings for several hints, in particular, to construct the trace for the Witt vectors via the norm map. I want to thank Stefan Kukulies for the many discussions concerning the foun- dations of algebraic geometry, Andre Chatzistamatiou for the careful reading of the introduction and Wioletta Syzdek for the supply with chocolate. Finally I want to thank my parents for making untroubled studies possible.

Contents

0

1

2

3

4

A

Introduction

Witt Vectors

The generalized de Rham-Witt Complex

A Residue Theorem

Additive cubical Chow groups with higher modulus

Intersection Theory for Cartier Divisors

References

1

5

14

29

54

77

81

0

Introduction

1

LetXbe an equidimensional scheme of finite type over a fieldkand write Δ n = Speck[t 0  . . .  t n ]/( P n i=0 t i −1). In [Bl86] Bloch introduced higher Chow groups CH p (X n), generalizing the Chow groups ofX(i.e. CH p (X0) = CH p (X)). Roughly speaking, these are defined by considering the quotient ofp-codimensional cycles in X×Δ n in suitable good position modulo the boundary ofp-codimensional cycles inX×Δ n+1 , where the boundary is given by intersecting with the faces (t i = 0) and then take the alternating sum. This construction is usually referred to as the simplicial definition of the higher Chow groups. There is also a cubical one, which mainly differs by taking (P 1 \ {1}) n instead of Δ n (see [To92]) and one can show that these two definitions of Bloch’s higher Chow groups coincide. One of the rare cases, where the higher Chow groups can be computed, which means to give a presentation in terms of generators and relations, is the caseX= Speckandp=n, i.e. formal sums of points in Δ n (resp. (P 1 \{01∞}) n ) modulo the boundary of formal sums of curves in good position in Δ n+1 (resp. (P 1 \ {1}) n+1  this case Nesterenko-Suslin). In and Totaro proved (see [NeSu89], [To92])

CH n (k n) ∼ = K nM (k)

whereK nM (k) are the degreenelements in the Milnor ring ofk. We observe that one could replace Δ n in the definition of the higher Chow groups by Speck[t 0  . . .  t n ]/( P t i −λ), for anyλ∈k × . In [BlEs03a] Bloch and Esnault investigated the degenerated caseλ They obtain a theory of additive higher= 0. Chow groups, SH p (X n), and prove in particular

−1 SH n (k n) = ∼ Ω k/ n Z 

using a presentation of Ω n /k − Z 1 as a quotient of the anti commutative graded ring k⊗ Z V ∗ k × modulo the graded ideal generated bya⊗a+ (1−a)⊗(1−a). In [BlEs03b, Section 6.] Bloch and Esnault construct a cubical version of the additive higher Chow groups (so far only for fields and on the level of zero cycles). Since these groups are our main object of study, we give a more precise definition. Denote by Z d (X) the group ofd-dimensional cycles onXand write

X n =G m ×(P 1 \ {1}) n 

with coordinates (x y 1  . . .  y n ).

Define Z 1 (X n ; 1) to be the subgroup of Z 1 (X n )which is freely generated by 1- dimensional subvarietiesC⊂X n ,C6⊂ S i (y i = 0∞) satisfying the following prop- erties (a) (Good position) (y i =j).[C]∈Z 0 (X n−1 \ S i (y i = 0∞)), fori= 1 . . .  n, j= 0∞.

(b)

(Modulus2condition) Ifν compactification ofC, then

(1)

:

C→P 1 ×  P 1  n e

n 2[ν ∗ (x= 0)]≤ X [ν ∗ (y i i=1

= 1)]

is the normalization

e in Z 0 (C).

of

the

2

There is a map∂= P n i=1 (−1) i (∂ i 0 −∂ i ∞ ) Bloch and Esnault define

0

INTRODUCTION

: Z 1 (X n ; 1)→Z 0 (X n−1 \Y n−1 ) and

TH n (k n) = TH n (k n; 1)  Z 0 (X n−1 \Y n−1 ) = ∂Z 1 (X n ; 1) .

They show by similar methods as before, assuming 1/6∈k,

n−1 TH n (k n  = ; 1) ∼ Ω k/Z .

Now the natural question is, what happens, if we replace the modulus 2 condition in (b), by a modulus (m condition, i.e. replace the 2 in (1) by (+ 1)m+ 1) and TH n (k n; 1) by TH n (k n;m)? The answer to this question, which is given in this thesis, is motivated by the following considerations. Denote by Pic(A k 1 (m+ 1){0}) the equivalence classes of divisors inA 1 , supported onG m  divisors. TwoD,D 0 being equivalent iffD−D 0 = div(f /h), for somef h∈1 +tk[t] withf−h∈t m+1 k[t], this isf /h≡1 mod (m+ 1){0}in the language of [Se88, Chapter III,§1]. Now given such functionsfandhwe can define a curveC⊂G m ×P 1 \ {1}by the equation h(x)y−f(x) = 0 and one easily checks that this curve satisfies the modulus (m+ 1)- condition. Thus we obtain a well defined and surjective map

(2)

Pic(A 1 (m+ 1){0})−→TH 1 (k1;m).

Now it is quiet reasonable to believe this map to be an isomorphism (at least for m= 1, both sides equalk we may identify Pic(, by the above). ButA 1 (m+ 1){0}) with the group  + 1 1t + m t + k[ 1 k t [ ] t]  × , which in turn may be identified with the ring of generalized Witt vectors of lengthmoverk,W m (k we hope to Hence), (see [Bl78]). obtain an isomorphism TH 1 (k1;m) ∼ = W m (k). If this is true, then the groups TH n (k n;m) should in general give something, which generalizestheabsoluteK¨ahlerdifferentialsontheonehandandthebigWittrings of finite length on the other. The natural suspect for this is the generalized de Rham-Witt complex of Hesselholt-Madsen,W m Ω k n , which generalizes thep-typical de Rham-Witt complex of Bloch-Deligne-Illusie. And indeed the main theorem of this thesis is

Theorem.Letkbe a field of characteristic6= 2 we have for all. Thenn m≥1an isomorphism TH n (k n;m) = ∼ W m Ω n k −1 .

The first part of the proof, namely to define the map from the additive Chow groups to the de Rham-Witt complex, is analogous to the proof of the casem= 1 by Bloch and Esnault. To verify that the map TH n (k n;m)→W m Ω k n−1 is well defined, we use a reciprocity law (as it was done in [NeSu89], [To92], [BlEs03a] and [BlEs03b]). Here it is a ”sum-of-residues-equal-zero” theorem, which we prove in section 3, generalizing the well known residue formula for differentials on smooth projective curves. To obtain the inverse map we use the universality of the de Rham-Witt complex, i.e. we equip the additive Chow groups with a structure of