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16261 results in Geometry and topology

Page 184 of 814

  • 20 - Algebraic Groups of Semisimple Rank One

  • J. S. Milne, University of Michigan, Ann Arbor
  • Book:
    Algebraic Groups
    Published online:
    25 October 2017
    Print publication:
    21 September 2017, pp 407-423

  • Summary

    This chapter contains preliminaries for the general study of reductive groups in the next chapter. In particular, we show that every split reductive group of semisimple rank 1 is isomorphic to exactly one of the following groups:

    Group varieties of semisimple rank 0

    Recall (19.20) that the rank of a group variety is the dimension of a maximal torus, and that this does not change under extension of the base field. The semisimple rank of G is the rank of the largest semisimple quotient of Gka.

    THEOREM 20.1. Let G be a connected group variety over a field k.

  • (a) G has rank 0 if and only if it is unipotent.

  • (b) G has semisimple rank 0 if and only if it is solvable.

  • (c) G is reductive of semisimple rank 0 if and only if it is a torus.

    • PROOF. We may suppose that k is algebraically closed.

    (a) This is a restatement of Proposition 16.60.

    (b) If G is solvable, then G D R.G/, and so it has semisimple rank 0. Conversely, if G has semisimple rank 0, then G=R.G/ is unipotent (by (a)) and semisimple (19.2), and hence trivial. Thus, and so G is solvable.

    (c) A torus is certainly reductive of semisimple rank 0. Conversely, if G is reductive of semisimple rank 0, then it is solvable. As Ru.G/ D e, this implies that G is a torus (16.33d).

    Homogeneous curves

    Let C be a regular complete algebraic curve over k. The local ring at a point is a discrete valuation ring containing k with field of fractions, and every such discrete valuation ring arises from a unique P. Therefore, we can identify with the set of such discrete valuation rings in endowed with the topology for which the proper closed subsets are the finite sets. For an open subset U, the ring. Thus, we can recover C from its function field. In particular, two regular complete connected curves over k are isomorphic if they have isomorphic function fields.

  • Preface

  • J. S. Milne, University of Michigan, Ann Arbor
  • Book:
    Algebraic Groups
    Published online:
    25 October 2017
    Print publication:
    21 September 2017, pp xv-xvi

  • Summary

    For one who attempts to unravel the story, the problems are as perplexing as a mass of hemp with a thousand loose ends.

    Dream of the Red Chamber, Tsao Hsueh-Chin.

    This book represents my attempt to write a modern successor to the three standard works, all titled Linear Algebraic Groups, by Borel, Humphreys, and Springer. More specifically, it is an exposition of the theory of group schemes of finite type over a field, based on modern algebraic geometry, but with minimal prerequisites.

    It has been clear for fifty years that such a work has been needed. When Borel, Chevalley, and others introduced algebraic geometry into the theory of algebraic groups, the foundations they used were those of the period (e.g., Weil 1946), and most subsequent writers on algebraic groups have followed them. Specifically, nilpotents are not allowed, and the terminology used conflicts with that of modern algebraic geometry. For example, algebraic groups are usually identified with their points in some large algebraically closed field K, and an algebraic group over a subfield k of K is an algebraic group over K equipped with a k-structure. The kernel of a k-homomorphism of algebraic k-groups is an object over K (not k) which need not be defined over k.

    In the modern approach, nilpotents are allowed, an algebraic k-group is intrinsically defined over k, and the kernel of a homomorphism of algebraic groups over k is (of course) defined over k. Instead of identifying an algebraic group with its points in some “universal” field, it is more convenient to identify it with the functor of k-algebras it defines.

    The advantages of the modern approach are manifold. For example, the infinitesimal theory is built into it from the start instead of entering only in an ad hoc fashion through the Lie algebra. The Noether isomorphism theorems hold for algebraic group schemes, and so the intuition from abstract group theory applies. The kernels of infinitesimal homomorphisms become visible as algebraic group schemes.

  • Introduction

  • J. S. Milne, University of Michigan, Ann Arbor
  • Book:
    Algebraic Groups
    Published online:
    25 October 2017
    Print publication:
    21 September 2017, pp 1-5

  • Summary

    The book can be divided roughly into five parts.

    Basic theory of general algebraic groups (Chapters 1–8)

    The first eight chapters cover the general theory of algebraic group schemes (not necessarily affine) over a field. After defining them and giving some examples, we show that most of the basic theory of abstract groups (subgroups, normal subgroups, normalizers, centralizers, Noether isomorphism theorems, subnormal series, etc.) carries over with little change to algebraic group schemes. We relate affine algebraic group schemes to Hopf algebras, and we prove that all algebraic group schemes in characteristic zero are smooth. We study the linear representations of algebraic group schemes and their actions on algebraic schemes. We show that every algebraic group scheme is an extension of an étale group scheme by a connected algebraic group scheme, and that every smooth connected group scheme over a perfect field is an extension of an abelian variety by an affine group scheme (Barsotti–Chevalley theorem).

    Beginning with Chapter 9, all group schemes are affine.

    Preliminaries on affine algebraic groups (Chapters 9–11)

    The next three chapters are preliminary to the more detailed study of affine algebraic group schemes in the later chapters. They cover basic Tannakian theory, in which the category of representations of an algebraic group scheme plays the role of the topological dual of a locally compact abelian group, Jordan decompositions, the Lie algebra of an algebraic group, and the structure of finite group schemes. Throughout this work we emphasize the Tannakian point of view in which the group and its category of representations are placed on an equal footing.

    Solvable affine algebraic groups (Chapters 12–16)

    The next five chapters study solvable algebraic group schemes. Among these are the diagonalizable groups, the unipotent groups, and the trigonalizable groups.

    An algebraic group G is diagonalizable if every linear representation of G is a direct sum of one-dimensional representations; in other words if, relative to some basis, the image of G lies in the algebraic subgroup of diagonal matrices in GLn. An algebraic group that becomes diagonalizable over an extension of the base field is said to be of multiplicative type.

  • 13 - Tori Acting on Schemes

  • J. S. Milne, University of Michigan, Ann Arbor
  • Book:
    Algebraic Groups
    Published online:
    25 October 2017
    Print publication:
    21 September 2017, pp 254-278

  • Summary

    Schemes with an action of a torus arise frequently in the theory of algebraic groups. In this chapter, we prove the basic theorems concerning such actions. In particular, we prove the Białynicki-Birula decomposition (13.47), which will allow us to show that the Bruhat decomposition exists on the level of scheme

    Throughout, all schemes are algebraic over the field k. Recall that all algebraic groups are affine.

    The smoothness of the fixed subscheme

    Recall that tori are linearly reductive.

    THEOREM 13.1. Let G be a linearly reductive group variety acting on a smooth variety X over k. Then the fixed-point scheme XGis smooth.

    We shall need to use some basic results on regular local rings.

    Let A be a local ring with maximal ideal m and residue field. Let d denote the Krull dimension of A. Every set of generators for m has at least d elements. If there exists a set with d elements, then A is said to be regular, and a set of generators with d elements is called a regular system of parameters for A (Matsumura 1986, p. 105).

  • (a)A local ring A is regular if and only if the canonical map

  • is an isomorphism (Matsumura 1986, 14.4).

  • (b) Assume that A is regular. Let be a regular system of parameters for A, and let for some. Then is local of dimension its maximal ideal is generated by, and so is regular. Every regular quotient of A is of this form (Matsumura 1986, 14.2).

    • We require several lemmas.

    LEMMA 13.3. Let A be a regular local ring of dimension d and m the maximal ideal in A. Let a be an ideal in A, and let. If, for every, there exists a regular system of parameters for A such that

    then is regular (of dimension).

    PROOF. Let, and let n denote the maximal ideal of B.

Algebraic Groups
  • The Theory of Group Schemes of Finite Type over a Field
  • J. S. Milne
  • Published online:
    25 October 2017
    Print publication:
    21 September 2017
  • 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.

Page 184 of 814