Lecture notes on motivic cohomology PDF
Preview Lecture notes on motivic cohomology
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
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