Terminological Index

5 b, No. c.d. by means of A, 5 b, c.d. Similarly, G.C. refers to the general conventions. The symbol Gr introduces the correspding term i n the terminology of Grothendieck, i n case his tenmimlcgy is different frcm OUTS.

W e r e f e r to chapter A,

additive

additive group

I I , § 1, 2.2

adjoint

adjoint representation

11,s 4, 4 . 1

aff ine

affine line affine affine affine affine

I,§ 1, 3.3 and 6.1

functor over another functor mrphisn schene S-schene

algebra

X-algebra , S-algebra %-algebra

algebraic

algebraic curve (Gr. courk alggbrique irreductible) algebraic hull of a sub-Lie-algebra algebraic schene, locally algebraic scheme (Gr. sch&a de prgsentation f i n i e , sc& localenent de prgsentation f i n i e ) algebraic subalgebra of a Lie-algebra

115 5, 3.1

1115 6, 2.4 1,s 3,

2.1

1115 6, 2.4

associative

associative l a w of c a n p s i t i o n

I I , § 1, 1.1

augmentst ion

augmentation of a bialgebra

1115 1, 1.6

biduality

biduality hmcxmrphissn

1115 1115

Boolean

Boolean space Boolean group

1,s 1, 2.12 11,s 1, 2.12

bracket

bracket i n the Lie-algebra of a group

1115 4, 4.2

character

character of a group-functor

11,s 1, 2.9

characteristic

characteristic subgroup

11,s 1, 3.9

Cartier

C a r t i e r algebra, C a r t i e r dual of a f i n i t e

central

central subgroup central extension

bialgebra

cutmutative group

centre

351

1, 1.6 1, 2.10

11,s 1, 2.10 11,s

1115 1115

1, 1.3 3, 3.2

1, 3.9

352

TERMINOLOGICAL INDEX

centralizer

11,s 1, 3.4

clean

clean mrphisn, clean scheme

1,s 4, 3.2

closed

closed enbeaaing closed irrage of a mrphisn of schemes closed mrphisn locally closed subfunctor closed asJbfunctor, subschane closed subschene defined by an ideal universally closed mrphisn

1,s 1,s 1,s 1,s 1,s 1,s 1,s

closure of a subschene (Gr. &race scmtique)

1,s 2, 6.11

closure

2, 2, 1, 2, 2, 2, 5,

6.1 6.11

4.2 7.2 6.8 6.8 1.3

coefficient cmtative

coefficient of a representation cmtative law of canpsition

11,s 2, 2.3

aanodule

d u l e over a bialgebra

11,s 2, 2.1

canplete

canplete k-scheme (Gr. k - s c M propre et de prgsentation finie sur k) law of canpsition gmtrically connected schane schgne of connected cQnponents group of connected canponents constant mnoid constant scheme constant term of a differential operator constmctible subset of a scheme convolution product of distributions coprcduct of a bialgebra counit of a bialgebra

ccmposition connected constant constructible convolution coprcduct counit curve

open m e r i n g of a %functor algebraic cUrve (Gr. courbe alggbrigue irrductible)

derived

derived group of an algebraic group

mering

diagonalizable differential dimension

1,s 5, 2.1 11,s 1, 1.1

1,s 4, 6.8 1,s 4, 6.6 11,s 5, 1.10 11,s 1, 1.5~) 1,s 1, 6.10 11,s 4, 5.3 1,s 3, 3.1

11,s 4, 6.1 11,s 1, 1.6

11,s 1, 1.6 1,s 1, 3.10 1,s 5, 3.1

11,s 5, 4.8 11,s 4, 5.1 and 5 4, 5.7

deviation diagonal

11,s 1, 1.1

diagonal group diagonal mrphisn of a scheme diagonalizable group, monoid diagonalizable representation differential operator module of differentials dimension of a topological space, local dimension at a pint

11,s 1, 3.11 11,s 2, 2.1 11,s 1, 2.11 11,s 2, 1.7 11,s 4, 5.3 1,s 4, 2.1 1,s 3, 5.1

TERMINOLOGICAL INDEX

353

dimension

dimension of a schene, local dimension a t a point Krull dimension of a ring

1,s 3, 6.1 1,s 3, 5.1

direct

d i r e c t image of an X-module

1,s 2, 1.1

distribution

distribution on a schene 11,s 4, 5.2 algebra of distributions on a group-schene 11,s 4, 6 . 1

dchninant

daninant mrphism of schemes

dual

C a r t i e r dual of a f i n i t e carmutative group scheme algebra of dual numbers

gnbeading closed embedding open anbedaing of g m e t r i c spaces o p n enbedding of Z-functors

1,s 1, 2.4 and 3, 3.7

11,s 1, 2.10 11,s 4, 1.1 1,s 1,s 1,s 1,s

2, 7.1 2, 6 . 1 1, 1.4

1, 3.6

equivalent

equivalent H-extensions

etale

etale algebra, mrphism, scheme

exponential

exponential map

11,s 6 , 3

extension

base extension, extension of scalars H-extension

1,s 1, 6.5 11,s 3, 2 . 1

11,s 3, 2.1

1,s 4, 3.2

factor

simple factor of a representation

11,s 2, 1.5

faithful

f a i t h f u l linear representation

11,s 5, 5.2

faithfully

f a i t h f u l l y f l a t mrphisn f a i t h f u l l y f l a t k-schene

1,s 2, 3.1 1,s 2, 3.3

fibre

f i b r e of a mrphisn of functors

1,s 1, 5.8 ard

finite

f i n i t e mrphisn

1,s 5, 1.1

finitely

f i n i t e l y generated quasicoherent sheaf f i n i t e l y generated mrphisn of schemes f i n i t e l y presented morphisn of schemes locally f i n i t e l y generated mrphisn locally f i n i t e l y presented mrphisn

1,s 1,s 1,s 1,s 1,s

flag

schene of f l a y s

1,s 2, 6.5

flat

f l a t mrphism, f a i t h f u l l y f l a t mrphism f l a t schene, f a i t h f u l l y f l a t scheme

1,s 2, 3.1 1,s 2, 3.3

free

locally f r e e mrphism locally f r e e schene

1,s 5, 1.1 1,s 2, 9.5

Frobenius

Frobenius mrphisn

functions

ring of functions over a geometric space ring of functions over a Z-functor ring of functions over a k-functor

1,s 1, 2.5 1,s 1, 3.3 1,s 1, 6 . 1

functor

Z-f unctor k-functor, S-functor

1,s 1, 3.1 I,§ 1, 6.1

1, 6.3 2, 3, 3, 3, 3,

2.6 1.12 1.6 1.12 1.6

11,s 7, 1.1

354

TERMINOLOGICAL INDEX

functor

underlying Z-functor

generated

closed subgroup generated by a rational point

generic

generic point of an irreducible subset

1,s 1, 2.10

geQnetric

geametric space g m t r i c realisation of a Z-functor geametric realisation of a k-functor

1,s 1, 1.1 1,s 1, 4.2 1,s 1, 6.8

germ of a function over a geometric space

1,s 1, 1.1

1,s 1, 6.3 11,s 5 , 4.5

1,s 1, 3.4

k-group-functor, WFOUP height

k-group-schane

height of an infinitesimal group

H-extension

11,s 1, 1.1 11,s 5, 1.7 11,s 7 , 1.4 11,s 3 , 2.1

Hcchhild

Hochschild ccanplex, W h s c h i l d group

11,s 3, 1.1

hull

algebraic h u l l of a sub-Lie-algebra

111s 6, 2.4

image

image of a mrphisn of T-functors image-functor of a mrphisn of ZB-functors closed image of a mrphisn of schar~s

1,s 1, 4.2 1,s 1, 4.2 1,s 2 , 6.11

induced inessential

g m e t r i c space induced on a subset inessential H-extension

1,s 1, 1.3

infinitesimal

infinitesimal group-schane

injective

injective mrphisn of Z-functors

inner

inner autcmrphisn operation

intqral

i n w a l mrphign

invariant

invariant derivation invariant differential operator invariant fllbgroup

inverse

inverse image of a sheaf

irreducible

irreducible linear representation irreducible topological space

11,s 2 , 1.5 1,s 1, 2.10

isotypical

isotypical linear representation

11,s 2, 1.5

Jacobi

Jambi identity

11,s 4 , 4.3

Jacobson

Jacobson formula

11,s 7 , 3.2

kernel

kernel of a double-arrow I,§ 1, 5.1 kernel of a knnmrphisn of group-functors 11,s 1, 1.3 K r u l l dimension of a ring

law

l a w of canposition on a k-functor

1,s 1, 4.2 11,s 1, 3.3d) 1,s 5, 1.1 11,s 4, 4.6 11,s 4, 6.5 11,s 1, 1.3 1,s 2 , 1.3

1,s 3, 5.1 11,s 1, 1.1

TERMINOLOGICAL INDEX

355

Lie

Lie-algebra of a group-schene L i e p-algebra L i e p-algebra of a group-scheme

11,s 4, 4.8 11,s 7, 3.3 11,s 7, 3.4

linear

linear group, special linear group linear representation

11,s 1, 2.4 11,s 2, 1.1

local

local dimension a t local dimension a t space local embdding of local enbedding of local functor

a point of a scheme

a point of a topological

k-schanes schemes

1,s 1,s 1,s

3, 6.1

3, 5.1 3, 4.4 1,s 3, 4.3 1,s 1, 3.11

model

model , k d e l G.C. X-module, S d u l e , module over a Z-functor 2, 4.1 module of an embeading (Gr. faisceau comml) 4, 1.3 k-G-mdule (Gsgroup-f unctor ) 11,s 2, 1.1 0 -module 11,s 1, 2.5 skeaf of modules over a gecmetric space I,§ 2, 1.1

mnoid

monoid law, mmid-functor, mnoid-scheme

multiplicative

standard multiplicative group (Gr g r o u p multiplicatif

1,s 1,s

.

11,s 1, 1.1

11,s 1,s 1,s

1, 2.8

nationality

nationality of a flag

neighburhood

first neighbourbo.3 nth-neigmurbo.3

4, 11,s 4,

neutral

neutral carrponent of an algebraic group

11,s 5, 1.3

nilpotent

n i l p t e n t elanent of a Lie-algebra

11,s 6, 3.7

nomlizer

11,s

open open open open open open

&ding of geanetric spaces Embedding of Z-functors morphisn of Z-functors subfunctor subscheme cavering of a Z-functor

2, 6.5

I’,

1.1 5.5

3.4

1,s 1,s 1,s

1, 1.4 1, 3.6 1, 4.2 1,s 1, 3.6 1,s 1, 3.11 1, 3.10

1,s

operator

differential operator

11,s 4,

operation

operation of a k-group on a k-scheme

11,s 1, 3.1 3.2

opposite

opposite monoid

11,s 1, 1.1

orbit

5.3

11,s 5, 3.3

partition

partition of unity i n a ring

point

point of a %functor

pcrwer

p*-per

i n a Lie-dgeixa

1,s 1, 3.10

I,§ 1, 4.2 11,s 7, 2.1

and

356

TERMINOLOGICAL INDEX

i n the Lie-algebra of p*-pwer a groupfunctor

11,s 7 ,

3.3

prime

prime spectrum of a r i n g prime spectrum

1,s 1, 2.3 1,s 1, 2.9

projective

projective space, p r o j e c t i v e l i n e

1,s 1, 3.4

proper

proper m r p h i s n of schemes

pure

pure suhmd.de

quasi-coherent

quasi-coherent m d u l e quasi-coherent S-algebra quasi-coherent sheaf of modules

quasi-ccanpact

quasi-cchnpact mrphim q u a s i - m p a c t scheme

quasi-separated

quasi-separated m r p h i s n quasi-separated scheme

1,s 2, 2.2 1,s 2, 1.8

ramified

non-ramified m r p h i s n , schane

1,s 4, 3.2

rank

rank of a l o c a l l y free mrphism

1,s 5, 1.1

rational.

rational pint

reduced

reduced schene, part of a schene reduced r i n g

reducible

carrpletely reducible linear representation 11,s 2, 1.5

resular

regular algebraic curve

representation

1,s 5, 2.1 11,s 2, 1.3 1,s 2, 4 . 1 1,s 2, 5.4 1,s 2, 1.7 1,s 2, 2.3 1,s 2, 1.8

1,s 3, 6.7 1,s 2, 6.11 1,s 2, 6.13

regular representation

l i n e a r representation

regular representation

restriction

base restriction, restriction of scalars

root

group of nth_roots of u n i t y

scheme

schene

Weil r e s t r i c t i o n

(Gr.

1,s 1, 6.4 1,s 1, 6.6 11,s 1, 2.8

foncteur sur les anneaux repr6sentable

par un s c h )

1,s 1, 3.11

(Gr. foncteur sur les k-algebres reprssentable par un k - s c h )

1,s 1, 6.1

k-schene seni-simple

sd-s-le

seni-direct

semi-direct product

separated

separated mrphism, separated %functor

set

II-set

simple

simple factor of a representation simple representation

representation

m t h mrphisn, m t h scheme, k - m t h

schene

11,s 2, 1.5

11,s 1, 3.10 I,§ 2, 7.4 1,s 4, 6.4

11,s 2, 1.5

1,s 4,

4.1

T E ~ O L C G I C A LINDFX

space

space of points of a Z-functor

special

special open subset special linear group

Spectral

spectral space

spectrum

prime spectrum of a ring prime spectrum spectrum of a quasi-coherent sheaf of algebras

(Gr. sch6-m)

357 1,s 1, 4.2 1,s 1, 1.3 11,s 1, 2.4 1,s 1, 2.9 1,s 1, 2.3 1,s 1, 2.9 1,s 2, 5.5

stable

stable suhmdule

11,s 2, 1.5

strict

strict triagonal group

11,s 1, 3.11

structural

structural projection of an S-functor

subscheme

1,s 1, 6.3 1,s 2, 5.1

subset

subset of a Z-functor

1,s 1, 4.2

surjective

surjective mrphism of Z-functors

1,s 1, 4.2

tangent

tangent space of Zariski

translation

translation operation

transprter

1,s 4, 4.15 11,s 1, 3.34 1,s 2, 9.4 and 11,s 1, 3.4

triagonal

triagonal group, strict triagonal group

11,s 1, 3.11

unit

u n i t section

11,s 1, 1.1

univ€xsally

universally closed mrphisn

1,s 5, 1.3

value

value of a function a t a p i n t

1,s

Weil

Weil restriction

1,s 1, 6.6

Zariski

tangent space of Zariski

1,s 4, 4.15

1, 1.1