Index — Tome I

A combined index of notation and terminology for Tome I. Page numbers refer to the 2011 SMF re-edition.

Entry	Pages
Abelian (variety)	428
Adjoint (representation)	70, 90
Admissible (composition law)	129
Affine (groups)	20, 403
$A l f_{/} k$ (category of $k$ -algebras of finite length)	515
$α_{p, k}$	99
Anti-affine ( $k$ -groups)	429
$a (P)$ (sheaf associated to the presheaf $P$ )	209
$Aut (X)$	9, 12
Bialgebra (cocommutative)	544
Bigèbre (cf. also bialgebra)	20
Good `O_S`-module	81
Good group functor	83
Boolean (locally — space)	363
Cantor (set / space of —)	364
Characteristic (subgroup)	377
Category of commutative $k$ -groups (affine)	555
Category of commutative $k$ -groups (algebraic)	323
Category of commutative $k$ -groups (algebraic affine)	323
Category of commutative $k$ -groups (quasi-compact)	327
`Ĉ`	1
$\hat{C}_{/} S$	3
$\hat{C}_{/} S^{d}$ , $\hat{C}_{S}^{b}$	3
Central (subgroup)	17
Centralizer	15, 366
$C e n t r_{G} (X)$	15
$C e n t r (G)$	17
Chevalley (theorems of)	420, 422, 428
Coalgebra $H (X)$ of a formal variety $X$	530
Coalgebras (cocommutative)	455
Coalgebras in groups	457, 477
Cogèbre (cf. also coalgebra)	409
Cohen-Macaulay	294
Commutators (sheaf of)	388
Commutators (subgroup of)	386
Comodules	28, 409
$A lp / k$ (category of profinite $k$ -algebras)	514
Connected component $G^{0}$	299, 308, 345
Local components $A_{m}$ , $M_{m}$	502, 511
Condition (E)	59
Connected (geometrically)	297
Conormal (sheaf)	46, 147
Cokernel (of a pair of arrows)	178, 186, 250, 310, 492, 519
Constant (objects)	10
Constant in `(Sch)`	20
Equivalence couple	254
Covering (morphism, family)	200
Covariant (bialgebra, algebra)	545, 547
Sieves	196
Bracket on $L i e (G / S)$	85
$c (X, Y, f)$	152
$Δ_{X / S}^{(1)}$	105
$S$ -Derivation (of an $S$ -morphism $Y \to X$ )	443
`Cb` (= `Ĉ`)	1
Invariant derivations	89
Descent (datum of)	182
Descent (morphism of)	183
Descent of projectivity	413
Effective descent (morphism of)	183
$S$ -Deviations (of order $⩽ n$ )	441
Fiber dimension	350
$D_{O_{S}} (M)$	52
Cartier duality	461, 547
$d (Y_{0}, Y)$	148
Affine envelope $G_{a f}$	405, 427
Effective epimorphisms	178
Universal effective epimorphisms	178
Universal epimorphisms	177
Equivariant (objects and modules)	38
Ringed quotient space	250
Tangent space at the point $u$	56
Homogeneous spaces of formal groups	582, 590
Essentially free	367
Étale (formally — algebra)	539
Étale (formal group)	556
Étale (formal variety)	539
Extension by zero	416
Sheaf associated to a presheaf	208
Relative sheaf	222
Quotient sheaves	217, 259, 272, 277, 280, 283, 286, 287, 393, 492, 493, 537, 552
Vector fibration $V (F)$	25
Principal homogeneous bundle	231
Tangent bundle	55
Faithful (operation)	17
Functor $Aut (X)$	9, 12
Ring functor $O$ on `(Sch)`	22
Functor $Hom (X, Y)$	7
Functor $Hom_{Z / S} (X, Y)$	50
Functor $\prod_{Z / S} Y$	51, 368, 373
Functor $X \mapsto X_{e}$	542
Module functors $V$ , $W$	24
Formal (scheme)	503, 517
Formal (variety)	518
Formally étale (algebra)	539
Formally homogeneous (space)	246
Formally principal homogeneous	102, 230
Frobenius morphism (absolute) $f r (S)$	461, 574
Frobenius morphism (relative) $F r (X / S)$	461, 573
$Γ * (M)$	522
$G_{a}$	21
$G_{m}$	22
$G$ - $O$ -modules	19
$G$ -`O_S`-modules	27, 43
$G_{re d}$ over a perfect field	291
$G_{re d}$ over a non-perfect field	297
Group (in a category)	12, 13
Group of operators (object with —)	15
Étale group $G / G^{0}$	323
Smooth group over a field	296
Groups with connected fibers	353
Affine groups over $S$ Dedekind	430, 431
Diagonalizable groups	23
Diagonalizable groups (cohomology of)	37
Quasi-compact groups over a field $k$	326
Groupoids	251
$H$ -set	62
$h_{A}$ (functor $Hom_{A lp / k} (A, -)$ )	515
Height $⩽ 1$ ( $S$ -groups, formal groups)	489, 492, 579
Height $⩽ n$ ( $S$ -group, formal group)	464, 575, 590
$H_{0}^{\lor} (S, P)$	205
$h_{M}^{c}$ (functor $Hom_{c} (M, -)$ )	506
Homogeneous (space)	246
Crossed homomorphisms	74
$Hom (X, Y)$	7
$Hom_{O} (F, F^{'})$	19
$Hom_{Z / S} (X, Y)$	50
$Hom_{(Z / S) - g r .} (X, Y)$	74
$H (X)$	530
$h_{X}$	1
Regular immersion	149, 161, 597
Infinitesimal ( $S$ -group, formal group)	484, 557, 569
Infinitesimal automorphisms	98
Infinitesimal endomorphisms	68, 69
Complete intersection	597
Complete intersection (locally)	149, 160
Invariant (subgroup)	17
Invariants (subobject of —)	15, 28
$I_{S} (M)$	52
Jacobson (formulas of)	471, 472
$Ker (f)$	17
Koszul (complex of)	149
$L F (A)$ (category of $A$ -modules of finite length)	506
$L i e (X / S)$	63, 70
$L i e (X / S, M)$	63, 70
$L i e (f)$ derived morphism of $f$	70
$L i e_{0} (X / S, M)$	77
$L i e (G / S)$	89, 90
$L i e (G_{m, S} / S)$ , $L i e (D_{S} (M) / S)$	91
$L i e (_{n G})$ derived morphism of $g \mapsto g^{n}$	64
Inverse limits of group schemes	396
Linearity of affine algebraic groups flat over $S$ regular of dim $⩽ 2$	433
Linearity of affine algebraic groups over a field	417
Smoothness of $G / F r^{n} G$	493
Smoothness of $G$ over $k$ of characteristic zero	336, 571
`LP`	205
$L_{u, X / S}$	56
$L_{X / S} (M)$	56
`L_X`	104
`L₀_X`, $L_{X}^{'}$	114
`(M)`-effectivity	193
`M†`, $N *$	507
Monomorphisms and closed immersions	301, 327, 335
Morphism $g \mapsto g^{n}$ is étale	495
Morphism $g \mapsto g^{n}$ is zero	496
$μ_{n}$ ( $n$ -th roots of unity)	23
Noether (isomorphism theorems of)	235
Dual numbers over $S$ (scheme of —)	52
Normalizer	15, 366
Norm of an invertible sheaf	267
Norm of a finite locally free $A$ -algebra	261
$N or m_{G} (X)$	15
Kernel	17
$N_{Y / X}$ conormal sheaf of $Y$ in $X$	46, 147
$Ω_{X / S}^{1}$	54, 89, 105
$ω_{X / S}^{1}$	89
$O$ -modules	18
Flat $O_{k}$ -modules	520
Differential operators	441
Invariant differential operators on $G$	449
$p$ -Lie algebra	470
$p$ -Lie algebra of an $S$ -group	480
$p$ -Lie algebra of a formal group	575
$PC (A)$ (category of pseudocompact $A$ -modules)	503
$\prod_{T / S} F$ (cohomology of)	124
$\prod_{Z / S} Y$	51
Pointed (cogebra, bigebra)	556, 565
Pointwise irreducible (geometrically)	306
Presheaves (category of)	1
Separated presheaves	202
Pre-equivalence relation	249, 253
Pretopology	201
Primitive (elements)	459, 561, 566
Completed tensor product $\hat{\otimes}$	508, 509, 511, 516, 519
Profinite ( $k$ -algebra)	514
Pseudobasis	507
Pseudocompact (ring)	501
Pseudocompact (module)	503
Squarable (morphism)	179
Quasi-sections	268
Quasi-separated (scheme)	307
Quasi-separated over $S$ (schemes)	363, 403, 409
Quotient by a groupoid (finite and flat)	259
Quotient by a groupoid (flat, not necessarily proper)	277
Quotient by a groupoid (proper and flat)	272
Quotients $G / H$ over $A$ local Artinian	311, 315
Quotients in $Va f_{/} k$	534, 537
Quotients by a group scheme	282
$r (A)$ (Jacobson radical of $A$ )	502
Refinements	199
Equivalence relation	186
Equivalence relation (effective)	191
Equivalence relation (universally effective)	191
Representability of $\prod_{Z / S} Y$	368, 373
Representability of centralizers	371, 376
Representability of normalizers	371, 376
Representable (functor)	2
Solvable or nilpotent (groups)	390
Restriction of scalars à la Weil	51
Retrocompact	307
$S$ - $H$ -functor	62
`(Sch)`, $(S c h) / S$ , $(S c h_{/} S)$	20
Schematically dense	374
Schematically dominant	325, 326, 378, 427
Semidirect (product)	16, 17, 552
Separated (every group over a field is —)	292
$\hat{S}_{k} (E)$ (completed symmetric algebra)	523
Subgroup generated by $f : X \to G$	378
$Spec * (U)$	456
$Sp f * (C)$	530
$St a b_{G} (x)$	15
Stabilizer	15, 44
Strictly rational (point)	293
Topology (étfg)	244
Topology (fpqc), (fppf), (ét), (étf)	241
Topology (canonical)	204
Topology (chaotic / coarse)	211, 245
Topology (Zariski)	236
Topology (flat)	537, 552
Topologies	199
Topologically free	507
Topologically nilpotent	502
Topologically flat (morphism)	527
Topologically flat (formal variety)	529
Torsor	231
Transporter	366
Strict transporter	366
Very good group functor	88, 89, 459
$T_{X / S}$	55
$T_{X / S} (M)$	55
$T_{X / S}^{'} (M)$	77
Multiplicative type ( $k$ -group of —)	558
Unipotent ( $k$ -group)	558, 563
$Υ (A)$ (maximal open ideals of $A$ )	502
$Va f_{/} k$ (category of formal varieties over $k$ )	518
$Va f_{/} S^{f ♭}$	525
$Va f_{/} k^{\overset{e}{ˊ}} t$	539
Abelian variety	428
Formal variety $Sp f * (C)$ of a coalgebra $C$	530
Verschiebung	468
$V_{k}^{f} (N)$	520
$V (Ω_{X / S}^{1})$	55
$V_{k}^{f} (E)$	524
$V_{k}^{f, 0} (E)$	524
$X$ underlying set of a scheme $X$	249
$X^{+} = \prod_{S_{J} / S} X_{J}$	102
$X_{e}$ (étale formal variety associated to $X$ )	542
$\hat{X}_{/ S}$ (formal variety associated to the $S$ -scheme $X$ )	525
Yoneda (lemma of)	1

Séminaire de Géométrie Algébrique du Bois-Marie — English

Index — Tome I