Introduction to Affine Group SchemesSpringer Science & Business Media, 13 nov. 1979 - 164 pages Ah Love! Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre' KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme. |
Table des matières
II | 3 |
V | 4 |
VI | 5 |
VII | 7 |
VIII | 9 |
IX | 11 |
X | 13 |
XI | 14 |
LXVI | 87 |
LXVII | 88 |
LXVIII | 89 |
LXX | 92 |
LXXI | 93 |
LXXII | 95 |
LXXIII | 96 |
LXXIV | 97 |
XII | 16 |
XIV | 21 |
XV | 22 |
XVI | 24 |
XVII | 25 |
XIX | 28 |
XX | 29 |
XXI | 30 |
XXIII | 32 |
XXIV | 33 |
XXV | 37 |
XXVI | 39 |
XXVII | 40 |
XXVIII | 41 |
XXX | 42 |
XXXI | 43 |
XXXII | 46 |
XXXIII | 47 |
XXXIV | 49 |
XXXVI | 50 |
XXXVII | 51 |
XXXVIII | 52 |
XXXIX | 54 |
XL | 55 |
XLII | 56 |
XLIII | 57 |
XLIV | 58 |
XLV | 59 |
XLVI | 62 |
XLVII | 63 |
XLVIII | 65 |
XLIX | 66 |
L | 68 |
LI | 69 |
LIII | 70 |
LV | 73 |
LVI | 74 |
LVII | 75 |
LIX | 76 |
LX | 77 |
LXI | 81 |
LXII | 83 |
LXIII | 84 |
LXIV | 85 |
LXV | 86 |
LXXV | 101 |
LXXVI | 103 |
LXXVII | 104 |
LXXVIII | 105 |
LXXIX | 106 |
LXXX | 107 |
LXXXI | 109 |
LXXXII | 110 |
LXXXIII | 111 |
LXXXIV | 112 |
LXXXV | 114 |
LXXXVI | 115 |
LXXXVIII | 116 |
XC | 117 |
XCI | 118 |
XCII | 121 |
XCIII | 122 |
XCIV | 123 |
XCV | 125 |
XCVI | 129 |
XCVII | 131 |
XCVIII | 132 |
XCIX | 133 |
C | 134 |
CII | 135 |
CIII | 136 |
CIV | 138 |
CV | 140 |
CVI | 141 |
CVII | 142 |
CIX | 144 |
CX | 145 |
CXI | 147 |
CXII | 148 |
CXIII | 151 |
CXIV | 152 |
CXVI | 153 |
CXVIII | 154 |
CXIX | 155 |
CXXI | 156 |
CXXIII | 158 |
161 | |
162 | |
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Expressions et termes fréquents
A-module abelian group affine algebraic group affine group scheme algebra map algebraic affine group algebraic matrix group algebraically closed automorphisms basis bijection character group closed sets closed subgroup closure cocycle cohomology commutative comodule connected components constant group contains Corollary corresponding decomposition defined descent data diagonalizable equations etale faithful flatness faithfully flat field extension finite group scheme finite-dimensional fraction field functions G acts G₁ Galois group functor Hence Hom(G homomorphism Hopf algebra Hopf subalgebra idempotents induces injective invariant invertible irreducible isomorphic k-algebra kernel lemma Let char(k Let G Lie algebra Lie(G linear representation maximal ideal module mult multiplicative type natural maps nilpotent nontrivial nonzero normal subgroup principal homogeneous space PROOF quotient map representable functors representation of G represented scheme G sending separable algebras smooth solvable span Spec structure subset subspace Suppose surjective tensor Theorem topology v₁ vanish y₁ Yoneda lemma zero