This is a comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, but with minimal prerequisites. The final manuscript was sent to CUP on February 28, 2017 and it became available in October 2017 (644 pages). The hardback (9781107167483) is attractively printed, and so it is worth buying the printed version. A corrected paperback reprint (9781009018586) will be available sometime. Here is the page at CUP.
Because we allow nilpotents, a statement in this work may be stronger than the statement in Borel 1991 or Springer 1998, even when the two are word for word the same.
Pierre Menard, Jorges Luis Borges's fictional early-twentieth-century scholar was so obsessed with Cervantes that he set out to reproduce word for word "the ninth and thirty-eighth chapters of part I of Don Quixote and a fragment of the twenty-second chapter." Borges, in truly Borgesian fashion, delivered a capricious judgment: "The text of Cervantes and that of Menard are verbally identical, but the second is almost infinitely richer."
Erratum 2017 For 2017 book.
Erratum 2022 For corrected reprint 2022.
Please send corrections and comments to me at jmilne at umich.edu.
Review (Werner Kleinert, zbMath) "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 mentioned above."
Review (Boris Kunyavskiĭ, MR3729270) 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.
From Bjorn Poonen: Two years ago when I taught this class, my thinking was that your book would be too long to use for a semester-long course, so I tried to use another one, but I ended up being frustrated with the old language and ended up borrowing a lot of material from your book anyway. This time around I used your book exclusively. Of course I had to skip many things, but nevertheless things went much better.
A rough preliminary draft of the book is available iAG200. Caution: the final version is substantially rewritten, and the numbering has changed.
The picture illustrates Grothendieck's vision of a pinned reductive group: the body is a maximal torus T, the wings
are the opposite Borel subgroups B, and the pins rigidify the situation.
"Demazure nous indique que, derrière cette terminologie [épinglage], il y a l'image du papillon (que lui a fournie Grothendieck): le corps est un tore maximal T, les ailes sont deux sous-groupes de Borel opposées par rapport à T, on déploie le papillon en étalant les ailes, puis on fixe des éléments dans les groupes additifs (des épingles) pour rigidifier la situation (c.-à-d., pour éliminer les automorphismes)." (SGA 3, XXIII, p.177, new edition.)
Blurb
Algebraic groups play much the same role for algebraists that Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields, including the structure theory of semisimple algebraic groups, written in the language of modern algebraic geometry.* When Borel, Chevalley, and others introduced algebraic geometry into the theory of algebraic groups, they used the algebraic geometry of the day, whose terminology conflicts with that of modern (post 1960) algebraic geometry. As Tits wrote in 1960, "the scheme viewpoint ... is not only more general but also, in many respects, more satisfactory." Indeed, allowing nilpotents gives a much richer and more natural theory.The first eight chapters of the book study general algebraic group schemes over a field. They culminate in a proof of the Barsotti-Chevalley theorem stating that every algebraic group is an extension of an abelian variety by an affine algebraic group. The remaining chapters treat only affine algebraic groups. After a review of the Tannakian philosophy, there are short accounts of Lie algebras and finite group schemes. Solvable algebraic groups are studied in detail in Chapters 12-16. The final eight chapters treat the Borel-Chevalley structure theory of reductive algebraic groups over arbitrary fields. Three appendices review the algebraic geometry needed, the construction of very general quotients of algebraic groups, and the theory of root data.
The exposition incorporates simplifications to the theory by Springer, Steinberg, and others. Although the theory of algebraic groups can be considered a branch of algebraic geometry, most of those using it are not algebraic geometers. In the present work, prequisites have been kept to a minimum. The only requirement is a first course in algebraic geometry including basic commutative algebra.
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*The only previous attempt to write such a book that I know of was
that of Demazure and Gabriel. As they wrote in the foreword of "Groupes
Algébriques (1971)" (my translation):
A. Grothendieck introduced two particularly useful tools into algebraic geometry: the functorial calculus and varieties with nilpotent elements. These tools allow us in particular to better understand phenomena arising from inseparability, by restoring differential calculus in nonzero characteristic, and they considerably simplify the general theory of algebraic groups. Hence we originally intended to develop the now classical theory of semisimple algebraic groups over algebraically closed fields (Borel-Chevalley) within the framework of schemes; it was simply a question of updating the 1956-58 seminar of Chevalley. But we soon realized that there did not exist a suitable book to serve as a reference for the general theory of algebraic groups, and moreover that it was impossible to refer a nonspecialist reader to the Grothendieck's Eléments de Géométrie Algébrique (EGA). Thus we were led to considerably modify and develop our original project, and we now present to the "mathematical community" a first volume devoted to the general theory of algebraic groups, where the case of semisimple groups is not entered into.Alas, no second volume was published.
Table of Contents (hover to see the sections)
-
Definitions and basic properties.
a. Definition.
b. Basic properties of algebraic groups.
c. Algebraic subgroups.
d. Examples.
e. Kernels and exact sequences.
f. Group actions.
g. The homomorphism theorem for smooth groups.
h. Closed subfunctors: definitions and statements.
i. Transporters.
j. Normalizers.
k. Centralizers.
l. Closed subfunctors: proofs.
Exercises. -
Examples and some basic constructions.
a. Affine algebraic groups.
b. Étale group schemes.
c. Anti-affine algebraic groups.
d. Homomorphisms of algebraic groups.
e. The Frobenius endomorphism.
f. Products.
g. Semidirect products.
h. The group of connected components.
i. The algebraic subgroup generated by a map.
j. Forms of algebraic groups.
k. Torsors and extensions
i. Restriction of scalars.
Exercises. -
Affine algebraic groups and Hopf algebras.
a. The comultiplication map.
b. Hopf algebras.
c. Hopf algebras and algebraic groups.
d. Hopf subalgebras.
e. Hopf subalgebras of O(G) versus algebraic subgroups of G.
f. Subgroups of G(k) versus algebraic subgroups of G.
g. Affine algebraic groups in characteristic zero are smooth.
h. Smoothness in characteristic p.
i. Faithful flatness for Hopf algebras.
j. The homomorphism theorem for affine algebraic groups.
Exercises -
Linear representations of algebraic groups.
a. Representations and comodules.
b. Stabilizers.
c. Representations are unions of finite-dimensional representations.
d. Affine algebraic groups are linear.
e. Constructing all finite-dimensional representations.
f. Semisimple representations.
g. Characters and eigenspaces.
h. Chevalley's theorem.
i. The subspace fixed by a group.
Exercises. -
Group theory; the isomorphism theorems.
a. The isomorphism theorems for abstract groups.
b. Quotient maps.
c. Existence of quotients.
d. Monomorphisms of algebraic groups.
e. The homomorphism theorem.
f. The isomorphism theorem.
g. The correspondence theorem.
h. The connected-étale exact sequence.
i. The category of commutative algebraic groups.
j. Sheaves.
k. The isomorphism theorems for group functors.
l. The isomorphism theorems for sheaves of groups.
m. The isomorphism theorems for algebraic groups.
n. Some category theory.
Exercises -
Subnormal series; solvable and nilpotent algebraic groups.
a. Subnormal series.
b. Isogenies.
c. Composition series for algebraic groups.
d. Solvable and nilpotent algebraic groups.
e. The derived group of an algebraic group.
f. Nilpotent algebraic groups.
g. Existence of a largest algebraic subgroup with a given property.
h. Semisimple and reductive groups.
i. A standard example. -
Algebraic groups acting on schemes.
a. Group actions.
b. Linear actions: affine case.
c. The fixed subscheme.
d. Orbits and isotropy groups.
e. The functor defined by projective space.
f. Quotients: definition and properties.
g. Quotients: construction in the affine case.
h. Linear actions: projective case.
i. Complements.
j. Flag varieties.
k. Homogeneous spaces are quasi-projective.
Exercises. -
The structure of general algebraic groups.
a. Summary.
b. Generalities.
c. Local actions.
d. Anti-affine algebraic groups and abelian varieties.
e. Rosenlicht's decomposition theorem.
f. Rosenlicht's dichotomy.
g. The Barsotti-Chevalley theorem.
h. Anti-affine groups.
i. Extensions of abelian varieties by affine algebraic groups: a survey.
Exercises -
Tannaka duality; Jordan decomposition.
a. Recovering a group from its representations.
b. Application to Jordan decompositions.
c. Characterizations of categories of representations.
d. Proof of Theorem 10.24.
e. Tannakian categories
f. Rosenlicht's dichotomy.
g. Properties of G versus those of Rep(G): a summary. -
The Lie algebra of an algebraic group.
a. Definition.
b. The Lie algebra of an algebraic group.
c. Basic properties of Lie algebras.
d. The adjoint representation; definition of the bracket.
e. Description of the Lie algebra in terms of derivations.
f. Stabilizers.
g. Centres.
h. Centralizers.
i. An example of Chevalley.
j. The universal covering algebra.
k. The universal enveloping p-algebra.
Exercises. -
Finite group schemes.
a. Generalities.
b. Locally free finite group schemes over a base ring.
c. Cartier duality.
d. Finite group schemes of order p.
e. Derivations of Hopf algebras.
f. Structure of the underlying scheme of a finite group scheme.
g. Finite group schemes of order n are killed by n.
h. Finite group schemes of height at most one.
i. The Verschiebung morphism.
j. The Witt schemes Wn.
k. Commutative group schemes over a perfect field. -
Groups of multiplicative type; linearly reductive groups..
a. The characters of an algebraic group.
b. The algebraic group D(M).
c. Diagonalizable groups.
d. Diagonalizable representations.
e. Tori.
f. Groups of multiplicative type.
g. Classification of groups of multiplicative type.
h. Representations of groups of multiplicative type.
i. Density and rigidity.
j. Central tori as almost factors.
k. Maps to tori.
l. Linearly reductive groups.
m. Unirationality.
Exercises -
Tori acting on schemes.
a. The smoothness of the fixed subscheme.
b. Limits in affine and projective space.
c. Limits in affine algebraic varieties.
d. Limits in algebraic groups.
e. Tori acting on smooth affine varieties.
f. Luna maps.
g. Decomposition of a variety under the action of a torus.
h. Proof of the Bialynicki-Birula decomposition.
Exercises -
Unipotent algebraic groups.
a. Preliminaries from linear algebra.
b. Unipotent algebraic groups.
c. Unipotent algebraic groups in characteristic zero.
d. Unipotent algebraic groups in nonzero characteristic.
e. Split and wound unipotent groups.
Exercises. -
Cohomology and extensions.
a. Crossed homomorphisms.
b. Hochschild cohomology.
c. Hochschild extensions.
d. The cohomology of linear representations.
e. Linearly reductive groups.
f. Applications to homomorphisms.
g. Applications to centralizers.
h. Calculation of some extensions.
Exercises -
The structure of solvable algebraic groups.
a. Trigonalizable algebraic groups
b. Commutative algebraic groups.
c. Structure of trigonalizable algebraic groups.
d. Solvable algebraic groups.
e. Connectedness.
f. Nilpotent algebraic groups.
g. Split solvable groups.
h. Complements on unipotent algebraic groups.
i. Tori acting on algebraic groups.
j. Tori acting on solvable algebraic groups.
k. Summary.
Exercises. -
Borel subgroups; Cartan subgroups.
a. The Borel fixed point theorem.
b. Borel subgroups and maximal tori.
c. The density theorem.
d. Centralizers of tori.
e. The normalizer of a Borel subgroup.
f. The variety of Borel subgroups.
g. Chevalley's description of the unipotent radical.
h. Proof of Chevalley's theorem.
i. Borel and parabolic subgroups over an arbitrary field.
j. Maximal tori and Cartan subgroups over an arbitrary field.
k. Algebraic groups over finite fields.
i. Split algebraic groups.
Exercises. -
The geometry of algebraic groups.
a. Definitions.
b. The universal covering.
c. Line bundles and characters.
d. Existence of a universal covering.
e. Applications.
f. Proof of Theorem 19.14.
Exercises. -
Semisimple and reductive groups.
a. Semisimple groups
b. Reductive groups.
c. The rank of a group variety.
d. Deconstructing reductive algebraic groups.
Exercises. -
Algebraic groups of semisimple rank one.
a. Group varieties of semisimple rank 0.
b. Homogeneous curves.
c. The automorphism group of the projective line.
d. A fixed point theorem for actions of tori.
e. Group varieties of semisimple rank 1.
f. Split reductive groups of semisimple rank 1.
g. Properties of SL2.
h. Classification of the split reductive groups of semisimple rank 1.
i. The forms of SL2, GL2, PGL2.
j. Classification of reductive groups of semisimple rank one.
k. Review of SL2.
Exercises. -
Split reductive groups.
a. Split reductive groups and their roots.
b. The centre of G in terms of its roots.
c. Review of SL2.
d. The root datum of a split reductive group.
e. Borel subgroups; Weyl groups; Tits systems.
f. Complements on semisimple groups.
g. Complements on reductive groups.
h. Subgroups normalized by T.
j. The big cell.
k. Parabolic subgroups.
l. The root data of the classical semisimple groups.
Exercises. -
Representations of reductive groups.
a. The semisimple representations of a split reductive group.
b. Characters and Grothendieck groups.
c. Semisimplicity in characteristic zero.
d. The Chern class of a simple representation.
e. Construction of the simple representations of G.
f. Weyl's character formula.
g. Relation to the representations of Lie(G).
h. Hyperalgebras.
Exercises. -
The isogeny and existence theorems.
a. Isogenies of groups and root data.
b. Proof of the isogeny theorem.
c. Complements.
d. Pinnings.
e. Automorphisms.
f. Quasi-split forms
g. Statement of the existence theorem; applications
f. Proof of the existence theorem.
Exercises. -
Construction of the semisimple groups.
a. Deconstructing semisimple algebraic groups.
b. Generalities on forms of reductive groups
c. The centres of semisimple groups
d. Semisimple algebras
e. Algebras with involution.
f. The geometrically almost-simple groups of type A.
i. The spin groups.
j. The geometrically almost-simple group of type B and D.
k. The classical groups in terms of sesquilinear forms.
l. The exceptional groups.
m. The trialitarian groups (groups of subtypes 3D4 and 6D4)
Exercises. -
Additional topics.
a. Parabolic subgroups of reductive groups.
b. The small root system.
c. The Satake-Tits classification.
d. Representations theory.
e. Pseudo-reductive groups.
f. Nonreductive groups: Levi subgroups.
g. Galois cohomology.
Exercises.
- Review of algebraic geometry
-
Existence of quotients of algebraic groups.
a. Equivalence relations.
b. Existence of quotients in the finite affine case.
c. Existence of quotients in the finite case.
d. Existence of quotients in the presence of quasi-sections.
e. Existence generically of a quotient.
f. Existence of quotients of algebraic groups. -
Root data.
a. Preliminaries.
b. Reflection groups.
c. Root systems.
d. Root data.
e. Duals of root data.
f. Deconstructing root data.
g. Classification of reduced root systems.