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El. knyga: Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field

(University of Michigan, Ann Arbor)
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Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry.

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Recenzijos

'All together, this excellent text fills a long-standing gap in the textbook literature on algebraic groups. It presents the modern theory of group schemes in a very comprehensive, systematic, detailed and lucid manner, with numerous illustrating examples and exercises. It is fair to say that this reader-friendly textbook on algebraic groups is the long-desired modern successor to the old, venerable standard primers ' Werner Kleinert, zbMath 'The author invests quite a lot to make difficult things understandable, and as a result, it is a real pleasure to read the book. All in all, with no doubt, Milne's new book will remain for decades an indispensable source for everybody interested in algebraic groups.' Boris Č. Kunyavski, MathSciNet ' fulfills the dual purpose of providing an updated account of the theory of reductive groups while at the same time serving as an accessible entry point into the general theory of reductive group schemes.' Igor A. Rapinchuk, Bulletin of the American Mathematical Society

Daugiau informacijos

Comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, with few prerequisites.
Preface xv
Introduction 1(2)
Conventions and notation 3(3)
1 Definitions and Basic Properties 6(33)
a Definition
6(6)
b Basic properties of algebraic groups
12(6)
c Algebraic subgroups
18(4)
d Examples
22(1)
e Kernels and exact sequences
23(3)
f Group actions
26(2)
g The homomorphism theorem for smooth groups
28(1)
h Closed subfunctors: definitions and statements
29(1)
i Transporters
30(1)
j Normalizers
31(2)
k Centralizers
33(2)
l Closed subfunctors: proofs
35(3)
Exercises
38(1)
2 Examples and Basic Constructions 39(25)
a Affine algebraic groups
39(5)
b Etale group schemes
44(1)
c Anti-affine algebraic groups
45(1)
d Homomorphisms of algebraic groups
46(3)
e Products
49(1)
f Semidirect products
50(1)
g The group of connected components
51(2)
h The algebraic subgroup generated by a map
53(4)
i Restriction of scalars
57(3)
j Torsors
60(1)
Exercises
61(3)
3 Affine Algebraic Groups and Hopf Algebras 64(19)
a The comultiplication map
64(1)
b Hopf algebras
65(1)
c Hopf algebras and algebraic groups
66(1)
d Hopf subalgebras
67(1)
e Hopf subalgebras of O(G) versus subgroups of G
68(1)
f Subgroups of G(k) versus algebraic subgroups of G
68(2)
g Affine algebraic groups in characteristic zero are smooth
70(2)
h Smoothness in characteristic p not = to 0
72(1)
i Faithful flatness for Hopf algebras
73(1)
j The homomorphism theorem for affine algebraic groups
74(2)
k Forms of algebraic groups
76(5)
Exercises
81(2)
4 Linear Representations of Algebraic Groups 83(15)
a Representations and comodules
83(2)
b Stabilizers
85(1)
c Representations are unions of finite-dimensional representations
86(1)
d Affine algebraic groups are linear
86(2)
e Constructing all finite-dimensional representations
88(2)
f Semisimple representations
90(2)
g Characters and eigenspaces
92(2)
h Chevalley's theorem
94(2)
i The subspace fixed by a group
96(1)
Exercises
97(1)
5 Group Theory; the Isomorphism Theorems 98(26)
a The isomorphism theorems for abstract groups
98(1)
b Quotient maps
99(3)
c Existence of quotients
102(4)
d Monomorphisms of algebraic groups
106(2)
e The homomorphism theorem
108(3)
f The isomorphism theorem
111(1)
g The correspondence theorem
112(2)
h The connected-etale exact sequence
114(1)
i The category of commutative algebraic groups
115(1)
j Sheaves
116(2)
k The isomorphism theorems for functors to groups
118(1)
l The isomorphism theorems for sheaves of groups
118(1)
m The isomorphism theorems for algebraic groups
119(2)
n Some category theory
121(1)
Exercises
122(2)
6 Subnormal Series; Solvable and Nilpotent Algebraic Groups 124(14)
a Subnormal series
124(2)
b Isogenies
126(1)
c Composition series for algebraic groups
127(2)
d The derived groups and commutator groups
129(2)
e Solvable algebraic groups
131(2)
f Nilpotent algebraic groups
133(1)
g Existence of a largest algebraic subgroup with a given property
134(1)
h Semisimple and reductive groups
135(1)
i A standard example
136(2)
7 Algebraic Groups Acting on Schemes 138(10)
a Group actions
138(1)
b The fixed subscheme
138(1)
c Orbits and isotropy groups
139(2)
d The functor defined by projective space
141(1)
e Quotients of affine algebraic groups
141(4)
f Linear actions on schemes
145(1)
g Flag varieties
146(1)
Exercises
146(2)
8 The Structure of General Algebraic Groups 148(15)
a Summary
148(1)
b Normal affine algebraic subgroups
149(1)
c Pseudo-abelian varieties
149(1)
d Local actions
150(1)
e Anti-affine algebraic groups and abelian varieties
151(1)
f Rosenlicht's decomposition theorem
151(2)
g Rosenlicht's dichotomy
153(1)
h The Barsotti-Chevalley theorem
154(2)
i Anti-affine groups
156(3)
j Extensions of abelian varieties by affine algebraic groups: a survey
159(1)
k Homogeneous spaces are quasi-projective
160(2)
Exercises
162(1)
9 Tannaka Duality; Jordan Decompositions 163(23)
a Recovering a group from its representations
163(3)
b Jordan decompositions
166(5)
c Characterizing categories of representations
171(3)
d Categories of comodules over a coalgebra
174(4)
e Proof of Theorem 9.24
178(5)
f Tannakian categories
183(1)
g Properties of G versus those of Rep(G)
184(2)
10 The Lie Algebra of an Algebraic Group 186(23)
a Definition
186(2)
b The Lie algebra of an algebraic group
188(2)
c Basic properties of the Lie algebra
190(1)
d The adjoins representation; definition of the bracket
191(3)
e Description of the Lie algebra in terms of derivations
194(2)
f Stabilizers
196(1)
g Centres
197(1)
h Centralizers
197(1)
i An example of Chevalley
198(1)
j The universal enveloping algebra
199(5)
k The universal enveloping p-algebra
204(3)
l The algebra of distributions (hyperalgebra) of an algebraic group
207(1)
Exercises
208(1)
11 Finite Group Schemes 209(21)
a Generalities
209(2)
b Locally free finite group schemes over a base ring
211(1)
c Cartier duality
212(3)
d Finite group schemes of order p
215(1)
e Derivations of Hopf algebras
216(2)
f Structure of the underlying scheme of a finite group scheme
218(2)
g Finite group schemes of order n are killed by n
220(2)
h Finite group schemes of height at most one
222(2)
i The Verschiebung morphism
224(2)
j The Witt schemes Wn
226(1)
k Commutative group schemes over a perfect field
227(2)
Exercises
229(1)
12 Groups of Multiplicative Type; Linearly Reductive Groups 230(24)
a The characters of an algebraic group
230(1)
b The algebraic group D(M)
230(3)
c Diagonalizable groups
233(1)
d Diagonalizable representations
234(2)
e Tori
236(1)
f Groups of multiplicative type
236(3)
g Classification of groups of multiplicative type
239(2)
h Representations of a group of multiplicative type
241(1)
i Density and rigidity
242(3)
j Central tori as almost-factors
245(1)
k Maps to tori
246(2)
l Linearly reductive groups
248(2)
m Unirationality
250(2)
Exercises
252(2)
13 Tori Acting on Schemes 254(25)
a The smoothness of the fixed subscheme
254(4)
b Limits in schemes
258(2)
c The concentrator scheme in the affine case
260(3)
d Limits in algebraic groups
263(4)
e Luna maps
267(5)
f The Bialynicki-Birula decomposition
272(4)
g Proof of the Bialynicki-Birula decomposition
276(2)
Exercises
278(1)
14 Unipotent Algebraic Groups 279(23)
a Preliminaries from linear algebra
279(1)
b Unipotent algebraic groups
280(6)
c Unipotent elements in algebraic groups
286(2)
d Unipotent algebraic groups in characteristic zero
288(4)
e Unipotent algebraic groups in nonzero characteristic
292(6)
f Algebraic groups isomorphic to Ga
298(1)
g Split and wound unipotent groups
299(2)
Exercises
301(1)
15 Cohomology and Extensions 302(22)
a Crossed homomorphisms
302(2)
b Hochschild cohomology
304(3)
c Hochschild extensions
307(3)
d The cohomology of linear representations
310(1)
e Linearly reductive groups
311(1)
f Applications to homomorphisms
312(1)
g Applications to centralizers
313(1)
h Calculation of some extensions
313(10)
Exercises
323(1)
16 The Structure of Solvable Algebraic Groups 324(28)
a Trigonalizable algebraic groups
324(3)
b Commutative algebraic groups
327(3)
c Structure of trigonalizable algebraic groups
330(4)
d Solvable algebraic groups
334(4)
e Connectedness
338(2)
f Nilpotent algebraic groups
340(3)
g Split solvable groups
343(3)
h Complements on unipotent algebraic groups
346(1)
i Tori acting on algebraic groups
347(4)
Exercises
351(1)
17 Borel Subgroups and Applications 352(35)
a The Borel fixed point theorem
352(1)
b Borel subgroups and maximal tori
353(8)
c The density theorem
361(2)
d Centralizers of tori
363(3)
e The normalizes of a Borel subgroup
366(2)
f The variety of Borel subgroups
368(2)
g Chevalley's description of the unipotent radical
370(3)
h Proof of Chevalley's theorem
373(2)
i Borel and parabolic subgroups over an arbitrary base field
375(1)
j Maximal tori and Cartan subgroups over an arbitrary base field
376(6)
k Algebraic groups over finite fields
382(2)
l Split algebraic groups
384(1)
Exercises
385(2)
18 The Geometry of Algebraic Groups 387(10)
a Central and multiplicative isogenies
387(1)
b The universal covering
388(1)
c Line bundles and characters
389(3)
d Existence of a universal covering
392(1)
e Applications
393(2)
f Proof of theorem 18.15
395(1)
Exercises
396(1)
19 Semisimple and Reductive Groups 397(10)
a Semisimple groups
397(2)
b Reductive groups
399(2)
c The rank of a group variety
401(2)
d Deconstructing reductive groups
403(3)
Exercises
406(1)
20 Algebraic Groups of Semisimple Rank One 407(17)
a Group varieties of semisimple rank 0
407(1)
b Homogeneous curves
408(1)
c The automorphism group of the projective line
409(1)
d A fixed point theorem for actions of tori
410(2)
e Group varieties of semisimple rank 1
412(2)
f Split reductive groups of semisimple rank 1
414(1)
g Properties of SL2
415(3)
h Classification of the split reductive groups of semisimple rank 1
418(1)
i The forms of SL2, GL2, and PGL2
419(2)
j Classification of reductive groups of semisimple rank one
421(1)
k Review of SL2
422(1)
Exercises
423(1)
21 Split Reductive Groups 424(39)
a Split reductive groups and their roots
424(3)
b Centres of reductive groups
427(1)
c The root datum of a split reductive group
428(5)
d Borel subgroups; Weyl groups; Tits systems
433(6)
e Complements on semisimple groups
439(3)
f Complements on reductive groups
442(2)
g Unipotent subgroups normalized by T
444(2)
h The Bruhat decomposition
446(6)
i Parabolic subgroups
452(4)
j The root data of the classical semisimple groups
456(5)
Exercises
461(2)
22 Representations of Reductive Groups 463(20)
a The semisimple representations of a split reductive group
463(10)
b Characters and Grothendieck groups
473(2)
c Semisimplicity in characteristic zero
475(4)
d Weyl's character formula
479(2)
e Relation to the representations of Lie(G)
481(1)
Exercises
482(1)
23 The Isogeny and Existence Theorems 483(29)
a Isogenies of groups and of root data
483(4)
b Proof of the isogeny theorem
487(5)
c Complements
492(3)
d Pinnings
495(2)
e Automorphisms
497(2)
f Quasi-split forms
499(2)
g Statement of the existence theorem; applications
501(2)
h Proof of the existence theorem
503(8)
Exercises
511(1)
24 Construction of the Semisimple Groups 512(32)
a Deconstructing semisimple algebraic groups
512(2)
b Generalities on forms of semisimple groups
514(2)
c The centres of semisimple groups
516(2)
d Semisimple algebras
518(2)
e Algebras with involution
520(3)
f The geometrically almost-simple groups of type A
523(3)
g The geometrically almost-simple groups of type C
526(1)
h Clifford algebras
527(4)
i The spin groups
531(2)
j The geometrically almost-simple group of types B and D
533(1)
k The classical groups in terms of sesquilinear forms
534(4)
l The exceptional groups
538(4)
m The trialitarian groups (groups of subtype 3D4 and 6D4)
542(1)
Exercises
542(2)
25 Additional Topics 544(22)
a Parabolic subgroups of reductive groups
544(4)
b The small root system
548(3)
c The Satake-Tits classification
551(2)
d Representation theory
553(4)
e Pseudo-reductive groups
557(1)
f Nonreductive groups: Levi subgroups
558(1)
g Galois cohomology
559(6)
Exercises
565(1)
Appendix A Review of Algebraic Geometry 566(20)
a Affine algebraic schemes
566(3)
b Algebraic schemes
569(2)
c Subschemes
571(1)
d Algebraic schemes as functors
572(2)
e Fibred products of algebraic schemes
574(1)
f Algebraic varieties
575(1)
g The dimension of an algebraic scheme
576(1)
h Tangent spaces; smooth points; regular points
577(2)
i Etale schemes
579(1)
j Galois descent for closed subschemes
580(1)
k Flat and smooth morphisms
581(1)
l The fibres of regular maps
582(1)
m Complete schemes; proper maps
583(1)
n The Picard group
584(1)
o Flat descent
584(2)
Appendix B Existence of Quotients of Algebraic Groups 586(21)
a Equivalence relations
586(5)
b Existence of quotients in the finite affine case
591(5)
c Existence of quotients in the finite case
596(3)
d Existence of quotients in the presence of quasi-sections
599(3)
e Existence generically of a quotient
602(2)
f Existence of quotients of algebraic groups
604(3)
Appendix C Root Data 607(20)
a Preliminaries
607(1)
b Reflection groups
608(2)
c Root systems
610(3)
d Root data
613(2)
e Duals of root data
615(5)
f Deconstructing root data
620(1)
g Classification of reduced root systems
621(6)
References 627(10)
Index 637
J. S. Milne is Professor Emeritus at the University of Michigan, Ann Arbor. His previous books include Etale Cohomology (1980) and Arithmetic Duality Theorems (2006).
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