Lecture notes on motivic cohomology

207 Pages·2006·1.266 MB·English

by  Carlo Mazza, Vladimir Voevodsky, Charles Weibel

Lecture Notes on
Motivic Cohomology
Contents
Preface vii
Introduction xi
Part1. PresheaveswithTransfers 1
Lecture1. Thecategoryoffinitecorrespondences 3
Appendix1A. ThecategoryCor 7
S
Lecture2. Presheaveswithtransfers 13
Homotopyinvariantpresheaves 17
Lecture3. Motiviccohomology 21
Lecture4. Weightonemotiviccohomology 25
Lecture5. RelationtoMilnorK-Theory 29
Part2. E´taleMotivicTheory 35
Lecture6. E´talesheaveswithtransfers 37
Lecture7. TherelativePicardgroupandSuslin’sRigidityTheorem 47
Lecture8. Derivedtensorproducts 55
Appendix8A. Tensortriangulatedcategories 63
Lecture9. A1-weakequivalence 67
E´taleA1-localcomplexes 71
Lecture10. E´talemotiviccohomologyandalgebraicsingularhomology 75
Part3. NisnevichSheaveswithTransfers 81
Lecture11. Standardtriples 83
Lecture12. Nisnevichsheaves 89
Thecdhtopology 94
v
vi CONTENTS
Lecture13. Nisnevichsheaveswithtransfers 99
Cdhsheaveswithtransfers 105
Part4. TheTriangulatedCategoryofMotives 107
Lecture14. Thecategoryofmotives 109
NisnevichA1-localcomplexes 111
MotiveswithQ-coefficients 116
Lecture15. ThecomplexZ(n)andPn 119
Lecture16. Equidimensionalcycles 125
Motiveswithcompactsupport 128
Part5. HigherChowGroups 133
Lecture17. HigherChowgroups 135
Appendix17A. Cyclemaps 143
Lecture18. HigherChowgroupsandequidimensionalcycles 149
Appendix18A. Genericequidimensionality 155
Lecture19. MotiviccohomologyandhigherChowgroups 159
Lecture20. Geometricmotives 167
Part6. ZariskiSheaveswithTransfers 173
Lecture21. Coveringmorphismsoftriples 175
Lecture22. Zariskisheaveswithtransfers 183
Lecture23. Contractions 191
Lecture24. Homotopyinvarianceofcohomology 197
Bibliography 203
Glossary 207
Index 211
Preface
ThisbookwaswrittenbyCarloMazzaandCharlesWeibelonthebasisofthe
lectures on motivic cohomology which I gave at the Institute for Advanced Study
inPrincetonin1999/2000.
Fromthepointofviewtakenintheselectures,motiviccohomologywithcoef-
ficientsinanabeliangroupAisafamilyofcontravariantfunctors
Hp,q(−,A):Sm/k→Ab
fromsmoothschemesoveragivenfieldktoabeliangroups,indexedbyintegers p
andq. TheideaofmotiviccohomologygoesbacktoP.Deligne,A.Beilinsonand
S.Lichtenbaum.
Mostoftheknownandexpectedpropertiesofmotiviccohomology(predicted
in[ABS87]and[Lic84])canbedividedintotwofamilies. Thefirstfamilyconcerns
properties of motivic cohomology itself – there are theorems about homotopy in-
variance, Mayer-VietorisandGysinlongexactsequences, projectivebundles, etc.
This family also contains conjectures such as the Beilinson-Soule´ vanishing con-
jecture(Hp,q=0for p<0)andtheBeilinson-Lichtenbaumconjecture,whichcan
beinterpretedasapartiale´taledescentpropertyformotiviccohomology. Thesec-
ond family of properties relates motivic cohomology to other known invariants of
algebraicvarietiesandrings. Thepowerofmotiviccohomologyasatoolforprov-
ing results in algebra and algebraic geometry lies in the interaction of the results
in these two families; applying general theorems of motivic cohomology to the
specificcasesofclassicalinvariants,onegetsnewresultsabouttheseinvariants.
Theideaoftheselectureswastodefinemotiviccohomologyandtogivecare-
ful proofs for the elementary results in the second family. In this sense they are
complementary to the study of [VSF00], where the emphasis is on the properties
ofmotiviccohomologyitself. Thestructureoftheproofsforcesustodealwiththe
mainpropertiesofmotiviccohomologyaswell(suchashomotopyinvariance). As
aresult,theselecturescoveraconsiderableportionofthematerialof[VSF00],but
fromadifferentpointofview.
Onecandistinguishthefollowing“elementary”comparisonresultsformotivic
cohomology. Unless otherwise specified, all schemes below are assumed to be
smoothor(inthecaseoflocalorsemilocalschemes)limitsofsmoothschemes.
vii
viii PREFACE
(1) Hp,q(X,A)=0forq<0,andforaconnectedX onehas
(cid:1)
A for p=0
Hp,0(X,A)=
0 for p(cid:2)=0;
(2) onehas

 O∗(X) for p=1
Hp,1(X,Z)= Pic(X) for p=2

0 for p(cid:2)=1,2;
(3) for a field k, one has Hp,p(Spec(k),A)=KM(k)⊗A where KM(k) is the
p p
p-thMilnorK-groupofk(see[Mil70]);
(4) for a strictly Hensel local scheme S over k and an integer n prime to
char(k),onehas
(cid:1)
µ⊗q(S) for p=0
Hp,q(S,Z/n)= n
0 for p(cid:2)=0
where µ(S)isthegroupsofn-throotsofunityinS;
n
(5) one has Hp,q(X,A)=CHq(X,2q−p;A). HereCHi(X,j;A) denotes the
higherChowgroupsofX introducedbyS.Blochin[Blo86],[Blo94]. In
particular,
H2q,q(X,A)=CHq(X)⊗A,
where CHq(X) is the classical Chow group of cycles of codimension q
modulorationalequivalence.
Theisomorphism betweenmotiviccohomologyandhigherChowgroupsleadsto
connections between motivic cohomology and algebraic K-theory, but we do not
discuss these connections in the present lectures. See [Blo94], [BL94], [FS02],
[Lev98]and[SV00].
Deeper comparison results include the theorem of M. Levine comparing
CHi(X,j;Q)withthegradedpiecesofthegammafiltrationinK∗(X)⊗Q[Lev94],
andtheconstructionofthespectralsequencerelatingmotiviccohomologyandal-
gebraicK-theoryforarbitrarycoefficientsin[BL94]and[FS02].
The lectures in this book may be divided into two parts, corresponding to the
fall and spring terms. The fall term lectures contain the definition of motivic co-
homology and the proofs for all of the comparison results listed above except the
last one. The spring term lectures include more advanced results in the theory of
sheaveswithtransfersandtheproofofthefinalcomparisonresult(5).
ThedefinitionofmotiviccohomologyusedheregoesbacktotheworkofAn-
dreiSuslininabout1985. AsIunderstandit,whenhecameupwiththisdefinition
he wasable toprove the firstthree of the comparison results statedabove. Inpar-
ticular, the proof of comparison (3) between motivic cohomology and Milnor’s
K-groups given in these lectures is exactly Suslin’s original proof. The proofs of
the last two comparison results, (4) and (5), are also based on results of Suslin.
Suslin’sformulationoftheRigidityTheorem([Sus83];seetheorem7.20)isakey
PREFACE ix
result needed for the proof of (4), and Suslin’s moving lemma (theorem 18A.1
below) isakeyresultneededfortheproofof(5).
It took ten years and two main new ideas to finish the proofs of the compar-
isons(4)and(5). Thefirstone,whichoriginatedinthecontextoftheqfh-topology
andwaslatertransferredtosheaveswithtransfers(definition2.1),isthatthesheaf
of finite cycles Z (X) is the free object generated by X. This idea led to a group
tr
of results, the most important of which is lemma 6.23. The second idea, which is
the main result of [CohTh], is represented here by theorem 13.8. Taken together
they allow one to efficiently do homotopy theory in the category of sheaves with
transfers.
Aconsiderablepartofthefirsthalfofthelecturesisoccupiedbytheproof of
(4). Insteadofstatingitintheformusedabove,weproveamoredetailedtheorem.
For a given weight q, the motivic cohomology groups Hp,q(X,A) are defined as
thehypercohomology(intheZariskitopology)ofX withcoefficientsinacomplex
of sheaves A(q)| . This complex is the restriction to the small Zariski site of X
XZar
(i.e.,thecategoryofopensubsetsofX)ofacomplexA(q)definedonthesiteofall
smooth schemes over k with the Zariski and even the e´tale topology. Restricting
A(q) to the small e´tale site of X, we may consider the e´tale version of motivic
cohomology,
Hp,q(X,A):=Hp(X,A(q)| ).
L e´t Xe´t
The subscript L is in honor of Steve Lichtenbaum, who first envisioned this con-
structionin[Lic94].
Theorem10.2assertsthatthee´talemotiviccohomology ofanyX withcoeffi-
cients in Z/n(q) where n is prime to char(k) are isomorphic to Hp(X,µ⊗q). This
e´t n
implies comparison result (4), since the Zariski and the e´tale motivic cohomol-
ogy of a strictly Hensel local scheme X agree. There should also be an analog of
(4) for the case of Z/(cid:1)r coefficients where (cid:1)=char(k), involving the logarithmic
de Rham-Witt sheaves νq[−q], but I do not know much about it. We refer the
r
readerto[GL00]formoreinformation.
VladimirVoevodsky
InstituteforAdvancedStudy
May2001
Introduction
This book isdivided intosixmainparts. The firstpart (lectures1–5) presents
thedefinitions andthefirstthreecomparisonresults. Thesecondpart(lectures6–
10)presentsthee´taleversionofthetheory,focussingoncoefficients,1/m∈k. As
Suslin’sRigidity Theorem7.20 demonstrates, akey role isplayedby locallycon-
stante´talesheavessuchas µ⊗i,whicharequasi-isomorphictothemotivicZ/m(i)
m
bytheorem10.3. ThetensortriangulatedcategoryDM−(k,Z/m)ofe´talemotives
e´t
is constructed in lecture 9 and shown to be equivalent to the derived category of
discreteZ/m-modulesovertheGaloisgroupG=Gal(k /k)intheorem9.35.
sep
Thefirstmaingoalofthelecturenotes,carriedoutinlectures11–16, istoin-
troducethetensortriangulatedcategoryDMeff,−(k,R)ofeffective motives andits
Nis
subcategoryofeffectivegeometricmotivesDMeff. ThemotiveM(X)ofascheme
gm
X is an object of DMeff,−(k,R), and belongs to DMeff if X is smooth. This re-
Nis gm
quiresanunderstandingofthecohomologicalpropertiesofsheavesassociatedwith
homotopy invariant presheaves with transfers for the Zariski, Nisnevich and cdh
topologies. This is addressed in the third part (lectures 11–13). Lecture 11 intro-
duces the technical notion of a standard triple, and uses it to prove that homotopy
invariant presheaves with transfers satisfy a Zariski purity property. Lecture 12
introduces the Nisnevich and cdh topologies, and lecture 13 considers Nisnevich
sheaveswithtransfersandtheirassociatedcdhsheaves.
A crucial role in this development isplayed by theorem 13.8: ifF is a homo-
topy invariant presheaf with transfers, and k is a perfect field, then the associated
NisnevichsheafF ishomotopyinvariant,andsoisitscohomology. Forreasons
Nis
ofexposition,theproofofthisresultispostponedandoccupieslectures21to24.
In the fourth part (lectures 14–16) we introduce the categories DMeff,−(k,R)
Nis
andDMeff. Themainpropertiesofthesecategories—homotopy,Mayer-Vietoris,
gm
projectivebundledecomposition,blow-uptriangles,Gysinsequence,theCancella-
tionTheorem, andtheconnectionwithChowmotives—aresummarizedin14.5.
We also show (in 15.9) that the product on motivic cohomology (defined in 3.12)
isgraded-commutative andinagreement(forcoefficientsQ)withthee´taletheory
presentedinlectures9and10(see14.30).
Lecture 16 introduces equidimensional algebraic cycles. These are used
to construct the Suslin-Friedlander motivic complexes ZSF(i), which are quasi-
isomorphictothemotiviccomplexesZ(i);thisrequiresthefieldtobeperfect(see
16.7). TheyarealsousedtodefinemotiveswithcompactsupportMc(X). Thebasic
xi
xii INTRODUCTION
theorywithcompactsupport complements thetheorypresentedinlecture14; this
requires the field to admit resolution of singularities. This lecture concludes with
theuseofFriedlander-Voevodsky duality(see16.24)toestablishtheCancellation
Theorem16.25;thisletsusembedeffectivemotivesintothetriangulatedcategory
ofallmotives.
Thesecondmaingoalofthisbookistoestablishthefinalcomparison(theorem
19.1)withBloch’shigherChowgroups: foranysmoothseparatedschemeX over
aperfectfieldk,wehave
Hp,q(X,Z)∼=CHq(X,2q−p).
This is carried out in the fifth part (lectures 17–19). In lecture 17, we introduce
Bloch’s higher Chow groups and show (in 17.21) that they are presheaves with
transfers over any field. Suslin’s comparison (18.3) of higher Chow groups with
equidimensional cycle groups over any affine scheme is given in lecture 18, and
the link between equidimensional cycle groups and motivic cohomology is given
inlecture19.
WebrieflyrevisitthetriangulatedcategoryDM ofgeometricmotivesinlec-
gm
ture20. Weworkoveraperfectfieldwhichadmitsresolutionofsingularities. First
weembedGrothendieck’sclassiccategoryofChowmotivesasafullsubcategory.
We then construct the dual of any geometric motive and use it to define internal
Hom objects Hom(X,Y). The lecture culminates in theorem 20.17, which states
thatthisstructuremakesDM arigidtensorcategory.
gm
Thefinalpart(lectures21–24)isdedicatedtotheproofoftheorem13.8. Using
technicalresultsfromlecture21,weprove(in22.3)thatF ishomotopyinvariant.
Nis
Theproofthatitscohomologyishomotopy invariant(24.1) isgiveninlecture24.
WeconcludewithaproofthatthesheafF admitsa“Gersten”resolution.
Nis
During the writing of the book, we received many suggestions and comments
from the mathematical community. One popular suggestion was that we include
some of the more well known and useful properties of motives that were missing
fromtheoriginallectures,inordertomaketheexpositionofthetheorymorecom-
plete. For this reason, a substantialamount of material hasbeenadded tolectures
12–14, 16 and 20. Another suggestion was that we warn the reader that the ex-
ercises vary in difficulty and content, from the concrete to the abstract; some are
learningexercisesandsomeaugmenttheideaspresentedinthetext.
InFigure1wegivearoughbird’seyeviewofthestructureofthebookandhow
thevariouslecturesdependuponeachother. Lectures1and2aremissingfromthe
figurebecausetheyareprerequisitesforallotherlectures. Wesplitlecture13into
twopartstoclarifythattheresultsinthesecondhalfofthelecturecruciallydepend
on theorem 13.8, which is proven in lecture 24. The dependency chart (and this
Introduction)shouldserveasaguidetothereader.
INTRODUCTION xiii
FIGURE 1. Dependencygraphofthelectures
xiv INTRODUCTION
Acknowledgements
The authors are deeplyindebted tothe Institute for Advanced Study, theClay
Mathematics Institute and Rutgers University for the support provided by these
institutionsduringthewritingofthisbook. Inaddition,wearegratefultoThomas
Geisser,whocarefullyproofreadanearlierversionandprovideduswithhislecture
notes on this subject, which were used as an outline for the additions in lectures
12,16and20.
In addition, the authors were supported on numerous grants during the writ-
ing phase (2000–2005) of this book. This includes support from the NSF, NSA,
INDAMandtheinstitutionsnamedabove.
CarloMazza
VladimirVoevodsky
CharlesA.Weibel
December2005