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xContentseDescription of the Lie algebra in terms of derivations . . . . . .fStabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .gCentres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .hCentralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . .iAn example of Chevalley . . . . . . . . . . . . . . . . . . . . .jThe universal enveloping algebra . . . . . . . . . . . . . . . . .kThe universal enveloping p-algebra . . . . . . . . . . . . . . .lThe algebra of distributions (hyperalgebra) of an algebraic groupExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19419619719719819920420720811 Finite Group SchemesaGeneralities . . . . . . . . . . . . . . . . . . . . . . . . . .bLocally free finite group schemes over a base ring . . . . . .cCartier duality . . . . . . . . . . . . . . . . . . . . . . . . .dFinite group schemes of order p . . . . . . . . . . . . . . .eDerivations of Hopf algebras . . . . . . . . . . . . . . . . .fStructure of the underlying scheme of a finite group schemegFinite group schemes of order n are killed by n . . . . . . .hFinite group schemes of height at most one . . . . . . . . .iThe Verschiebung morphism . . . . . . . . . . . . . . . . .jThe Witt schemes Wn . . . . . . . . . . . . . . . . . . . . .kCommutative group schemes over a perfect field . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20920921121221521621822022222422622722912 Groups of Multiplicative Type; Linearly Reductive GroupsaThe characters of an algebraic group . . . . . . . . . . .bThe algebraic group D.M / . . . . . . . . . . . . . . . .cDiagonalizable groups . . . . . . . . . . . . . . . . . .dDiagonalizable representations . . . . . . . . . . . . . .eTori . . . . . . . . . . . . . . . . . . . . . . . . . . . .fGroups of multiplicative type . . . . . . . . . . . . . . .gClassification of groups of multiplicative type . . . . . .hRepresentations of a group of multiplicative type . . . .iDensity and rigidity . . . . . . . . . . . . . . . . . . . .jCentral tori as almost-factors . . . . . . . . . . . . . . .kMaps to tori . . . . . . . . . . . . . . . . . . . . . . . .lLinearly reductive groups . . . . . . . . . . . . . . . . .mUnirationality . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .23023023023323423623623924124224524624825025213 Tori Acting on SchemesaThe smoothness of the fixed subscheme .bLimits in schemes . . . . . . . . . . . . .cThe concentrator scheme in the affine casedLimits in algebraic groups . . . . . . . .254254258260263.

ContentsxieLuna maps . . . . . . . . . . . . . . . . . . .fThe Białynicki-Birula decomposition . . . . .gProof of the Białynicki-Birula decompositionExercises . . . . . . . . . . . . . . . . . . . . . .26727227627814 Unipotent Algebraic GroupsaPreliminaries from linear algebra . . . . . . . . . . .bUnipotent algebraic groups . . . . . . . . . . . . . .cUnipotent elements in algebraic groups . . . . . . .dUnipotent algebraic groups in characteristic zero . .eUnipotent algebraic groups in nonzero characteristicfAlgebraic groups isomorphic to Ga . . . . . . . . . .gSplit and wound unipotent groups . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .27927928028628829229829930115 Cohomology and ExtensionsaCrossed homomorphisms . . . . . . . . .bHochschild cohomology . . . . . . . . .cHochschild extensions . . . . . . . . . .dThe cohomology of linear representationseLinearly reductive groups . . . . . . . . .fApplications to homomorphisms . . . . .gApplications to centralizers . . . . . . . .hCalculation of some extensions . . . . . .Exercises . . . . . . . . . . . . . . . . . . . .30230230430731031131231331332316 The Structure of Solvable Algebraic GroupsaTrigonalizable algebraic groups . . . . . . .bCommutative algebraic groups . . . . . . .cStructure of trigonalizable algebraic groupsdSolvable algebraic groups . . . . . . . . . .eConnectedness . . . . . . . . . . . . . . .fNilpotent algebraic groups . . . . . . . . .gSplit solvable groups . . . . . . . . . . . .hComplements on unipotent algebraic groupsiTori acting on algebraic groups . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . .32432432733033433834034334634735117 Borel Subgroups and ApplicationsaThe Borel fixed point theorem . . .bBorel subgroups and maximal tori .cThe density theorem . . . . . . . . .dCentralizers of tori . . . . . . . . .eThe normalizer of a Borel subgroupfThe variety of Borel subgroups . . .352352353361363366368.

xiiContentsgChevalley’s description of the unipotent radical . . . . . . . .hProof of Chevalley’s theorem . . . . . . . . . . . . . . . . . .iBorel and parabolic subgroups over an arbitrary base field . . .jMaximal tori and Cartan subgroups over an arbitrary base fieldkAlgebraic groups over finite fields . . . . . . . . . . . . . . .lSplit algebraic groups . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37037337537638238438518 The Geometry of Algebraic GroupsaCentral and multiplicative isogeniesbThe universal covering . . . . . . .cLine bundles and characters . . . . .dExistence of a universal covering . .eApplications . . . . . . . . . . . . .fProof of theorem 18.15 . . . . . . .Exercises . . . . . . . . . . . . . . . . .38738738838939239339539619 Semisimple and Reductive GroupsaSemisimple groups . . . . . . .bReductive groups . . . . . . . .cThe rank of a group variety . . .dDeconstructing reductive groupsExercises . . . . . . . . . . . . . . .39739739940140340620 Algebraic Groups of Semisimple Rank OneaGroup varieties of semisimple rank 0 . . . . . . . . . . . . . . .bHomogeneous curves . . . . . . . . . . . . . . . . . . . . . . .cThe automorphism group of the projective line . . . . . . . . . .dA fixed point theorem for actions of tori . . . . . . . . . . . . .eGroup varieties of semisimple rank 1. . . . . . . . . . . . . . .fSplit reductive groups of semisimple rank 1. . . . . . . . . . . .gProperties of SL2 . . . . . . . . . . . . . . . . . . . . . . . . .hClassification of the split reductive groups of semisimple rank 1iThe forms of SL2 , GL2 , and PGL2 . . . . . . . . . . . . . . . .jClassification of reductive groups of semisimple rank one . . . .kReview of SL2 . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40740740840941041241441541841942142242321 Split Reductive GroupsaSplit reductive groups and their roots . . . .bCentres of reductive groups . . . . . . . . .cThe root datum of a split reductive group . .dBorel subgroups; Weyl groups; Tits systemseComplements on semisimple groups . . . .fComplements on reductive groups . . . . .424424427428433439442.

ContentsgUnipotent subgroups normalized by T . . . . .hThe Bruhat decomposition . . . . . . . . . . .iParabolic subgroups . . . . . . . . . . . . . . .jThe root data of the classical semisimple groupsExercises . . . . . . . . . . . . . . . . . . . . . . .xiii.44444645245646122 Representations of Reductive GroupsaThe semisimple representations of a split reductive groupbCharacters and Grothendieck groups . . . . . . . . . . .cSemisimplicity in characteristic zero . . . . . . . . . . .dWeyl’s character formula . . . . . . . . . . . . . . . . .eRelation to the representations of Lie.G/ . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .46346347347547948148223 The Isogeny and Existence TheoremsaIsogenies of groups and of root data . . . . . .bProof of the isogeny theorem . . . . . . . . . .cComplements . . . . . . . . . . . . . . . . . .dPinnings . . . . . . . . . . . . . . . . . . . . .eAutomorphisms . . . . . . . . . . . . . . . . .fQuasi-split forms . . . . . . . . . . . . . . . .gStatement of the existence theorem; applicationshProof of the existence theorem . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . .48348348749249549749950150351124 Construction of the Semisimple GroupsaDeconstructing semisimple algebraic groups . . . . . . .bGeneralities on forms of semisimple groups . . . . . . .cThe centres of semisimple groups . . . . . . . . . . . .dSemisimple algebras . . . . . . . . . . . . . . . . . . .eAlgebras with involution . . . . . . . . . . . . . . . . .fThe geometrically almost-simple groups of type A . . . .gThe geometrically almost-simple groups of type C . . .hClifford algebras . . . . . . . . . . . . . . . . . . . . .iThe spin groups . . . . . . . . . . . . . . . . . . . . . .jThe geometrically almost-simple group of types B and DkThe classical groups in terms of sesquilinear forms . . .lThe exceptional groups . . . . . . . . . . . . . . . . . .m The trialitarian groups (groups of subtype 3D4 and 6D4 )Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .512512514516518520523526527531533534538542542.25 Additional Topics544aParabolic subgroups of reductive groups . . . . . . . . . . . . . 544bThe small root system . . . . . . . . . . . . . . . . . . . . . . . 548cThe Satake–Tits classification . . . . . . . . . . . . . . . . . . . 551

xivContentsdRepresentation theory . . . . . . . . .ePseudo-reductive groups . . . . . . .fNonreductive groups: Levi subgroupsgGalois cohomology . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . .553557558559565Appendix A Review of Algebraic GeometryaAffine algebraic schemes . . . . . . . . . . .bAlgebraic schemes . . . . . . . . . . . . . .cSubschemes . . . . . . . . . . . . . . . . . .dAlgebraic schemes as functors . . . . . . . .eFibred products of algebraic schemes . . . . .fAlgebraic varieties . . . . . . . . . . . . . .gThe dimension of an algebraic scheme . . . .hTangent spaces; smooth points; regular pointsiÉtale schemes . . . . . . . . . . . . . . . . .jGalois descent for closed subschemes . . . .kFlat and smooth morphisms . . . . . . . . . .lThe fibres of regular maps . . . . . . . . . .mComplete schemes; proper maps . . . . . . .nThe Picard group . . . . . . . . . . . . . . .oFlat descent . . . . . . . . . . . . . . . . . ppendix B Existence of Quotients of Algebraic GroupsaEquivalence relations . . . . . . . . . . . . . . . . . .bExistence of quotients in the finite affine case . . . . .cExistence of quotients in the finite case . . . . . . . . .dExistence of quotients in the presence of quasi-sectionseExistence generically of a quotient . . . . . . . . . . .fExistence of quotients of algebraic groups . . . . . . .586586591596599602604Appendix C Root DataaPreliminaries . . . . . . . . . . . . .bReflection groups . . . . . . . . . . .cRoot systems . . . . . . . . . . . . .dRoot data . . . . . . . . . . . . . . .eDuals of root data . . . . . . . . . . .fDeconstructing root data . . . . . . .gClassification of reduced root x637

PrefaceFor one who attempts to unravel the story, the problemsare as perplexing as a mass of hemp with a thousandloose ends.Dream of the Red Chamber, Tsao Hsueh-Chin.This book represents my attempt to write a modern successor to the threestandard works, all titled Linear Algebraic Groups, by Borel, Humphreys, andSpringer. More specifically, it is an exposition of the theory of group schemes offinite type over a field, based on modern algebraic geometry, but with minimalprerequisites.It has been clear for fifty years that such a work has been needed.1 WhenBorel, Chevalley, and others introduced algebraic geometry into the theory ofalgebraic groups, the foundations they used were those of the period (e.g., Weil1946), and most subsequent writers on algebraic groups have followed them.Specifically, nilpotents are not allowed, and the terminology used conflicts withthat of modern algebraic geometry. For example, algebraic groups are usuallyidentified with their points in some large algebraically closed field K, and analgebraic group over a subfield k of K is an algebraic group over K equippedwith a k-structure. The kernel of a k-homomorphism of algebraic k-groups is anobject over K (not k) which need not be defined over k.In the modern approach, nilpotents are allowed,2 an algebraic k-group isintrinsically defined over k, and the kernel of a homomorphism of algebraicgroups over k is (of course) defined over k. Instead of identifying an algebraicgroup with its points in some “universal” field, it is more convenient to identify itwith the functor of k-algebras it defines.The advantages of the modern approach are manifold. For example, theinfinitesimal theory is built into it from the start instead of entering only in an adhoc fashion through the Lie algebra. The Noether isomorphism theorems hold for1 “Another remorse concerns the language adopted for the algebrogeometrical foundation of thetheory . two such languages are briefly introduced . the language of algebraic sets . and theGrothendieck language of schemes. Later on, the preference is given to the language of algebraic sets. If things were to be done again, I would probably rather choose the scheme viewpoint . which isnot only more general but also, in many respects, more satisfactory.” Tits 1968, p. 2.2 To anyone who asked why we need to allow nilpotents, Grothendieck would say that they arealready there in nature; neglecting them obscures our vision.xv

xviPrefacealgebraic group schemes, and so the intuition from abstract group theory applies.The kernels of infinitesimal homomorphisms become visible as algebraic groupschemes.The first systematic exposition of the theory of group schemes was in SGA 3.As was natural for its authors (Demazure, Grothendieck, . . . ), they worked overan arbitrary base scheme and they used the full theory of schemes (EGA andSGA). Most subsequent authors on group schemes have followed them. Theonly books I know of that give an elementary treatment of group schemes areWaterhouse 1979 and Demazure and Gabriel 1970. In writing this book, I haverelied heavily on both, but neither goes very far. For example, neither treats thestructure theory of reductive groups, which is a central part of the theory.As noted, the modern theory is more general than the old theory. The extragenerality gives a richer and more attractive theory, but it does not come for free:some proofs are more difficult (because they prove stronger statements). In thiswork, I have avoided any appeal to advanced scheme theory. Unpleasantly technical arguments that I have not been able to avoid have been placed in separatesections where they can be ignored by all but the most serious students. By considering only schemes algebraic over a field, we avoid many of the technicalitiesthat plague the general theory. Also, the theory over a field has many specialfeatures that do not generalize to arbitrary bases.Acknowledgements: The exposition incorporates simplifications to the generaltheory from Iversen 1976, Luna 1999, Steinberg 1999, Springer 1998, and othersources. In writing this book, the following works have been especially usefulto me: Demazure and Gabriel 1966; Demazure and Gabriel 1970; Waterhouse1979; the expository writings of Springer, especially Springer 1994, 1998; onlinenotes of Casselman, Ngo, Perrin, and Pink, as well as the discussions, oftenanonymous, on https://mathoverflow.net/. Also I wish to thank all thosewho have commented on the various notes posted on my website.

IntroductionThe book can be divided roughly into five parts.A. Basic theory of general algebraic groups (Chapters 1–8)The first eight chapters cover the general theory of algebraic group schemes (notnecessarily affine) over a field. After defining them and giving some examples,we show that most of the basic theory of abstract groups (subgroups, normalsubgroups, normalizers, centralizers, Noether isomorphism theorems, subnormalseries, etc.) carries over with little change to algebraic group schemes. Werelate affine algebraic group schemes to Hopf algebras, and we prove that allalgebraic group schemes in characteristic zero are smooth. We study the linearrepresentations of algebraic group schemes and their actions on algebraic schemes.We show that every algebraic group scheme is an extension of an étale groupscheme by a connected algebraic group scheme, and that every smooth connectedgroup scheme over a perfect field is an extension of an abelian variety by an affinegroup scheme (Barsotti–Chevalley theorem).Beginning with Chapter 9, all group schemes are affine.B. Preliminaries on affine algebraic groups (Chapters 9–11)The next three chapters are preliminary to the more detailed study of affinealgebraic group schemes in the later chapters. They cover basic Tannakiantheory, in which the category of representations of an algebraic group schemeplays the role of the topological dual of a locally compact abelian group, Jordandecompositions, the Lie algebra of an algebraic group, and the structure of finitegroup schemes. Throughout this work we emphasize the Tannakian point of viewin which the group and its category of representations are placed on an equalfooting.C. Solvable affine algebraic groups (Chapters 12–16)The next five chapters study solvable algebraic group schemes. Among these arethe diagonalizable groups, the unipotent groups, and the trigonalizable groups.1

2IntroductionAn algebraic group G is diagonalizable if every linear representation of Gis a direct sum of one-dimensional representations; in other words if, relative tosome basis, the image of G lies in the algebraic subgroup of diagonal matrices inGLn . An algebraic group that becomes diagonalizable over an extension of thebase field is said to be of multiplicative type.An algebraic group G is unipotent if every nonzero representation of Gcontains a nonzero fixed vector. This implies that every representation has abasis for which the image of G lies in the algebraic subgroup of strictly uppertriangular matrices in GLn .An algebraic group G is trigonalizable if every simple representation hasdimension one. This implies that every representation has a basis for whichthe image of G lies in the algebraic subgroup of upper triangular matrices inGLn . The trigonalizable groups are exactly the extensions of diagonalizablegroups by unipotent groups. Trigonalizable groups are solvable, and the Lie–Kolchin theorem says that all smooth connected solvable algebraic groups becometrigonalizable over a finite extension of the base field.D. Reductive algebraic groups (Chapters 17–25)This is the heart of the book, The first seven chapters develop in detail the structuretheory of split reductive groups and their representations in terms of their rootdata. Chapter 24 exhibits all the almost-simple algebraic groups, and Chapter 25explains how the theory of split groups extends to the nonsplit case.E. AppendicesThe first appendix reviews the definitions and

vi The book is a comprehensive introduction to the theory of algebraic group schemes over fie