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Affine Group Scheme의 이해 (1)
이 인 석
제 23차 대수캠프
2010년 2월
1 / 28
차례
Equivalent Categories
Classical Language
Scheme Language
꿈FRepresentable Functors
Yoneda’s Lemma
Affine Schemes; 꿍꿍이
Affine Group Schemes
Hopf Algebras
Epilogue; (Group) Schemes
2 / 28
서론
Scheme Language ; 왜 어떤 꿈F을 갖고 태어났는가 ?
Scheme Language의 (한때의) 경쟁자들
대수기하에서 제일 싫은 (무서운) 말
“k가 algebraically closed일 때만 설명하면 충분하다”
(다음 기회에)
예를 들어, SLn : k-Alg → Group을 functor로 이해해야
SLn : Ring → Group ??
3 / 28
전문용어해설
“다음 기회에” 혹은 “기회가 있으면”
= “(다행히) 다음 기회는 없다”
“오늘의 주제는 아니므로 생략”
= “좀 알지만 잘은 모른다”
“시간이 없어 생략”
= “잘 모르니 더 캐물으면 미워할 것이다”
오늘 모든 ring과 algebra는 commutative with 1,
homomorphism : 1 7→ 1
4 / 28
Equivalent Categories
Category Theory
Natural Transformation
Isomorphic Categories
Category of k-algebras
Equivalent Categories
Axiom of Choice
Small Categories
5 / 28
Category Theory
categories C,D, objects A, B ∈ Cfull subcategory
small category
morphism f : A → B or f ∈ MorC(A,B)
isomorphism
(covariant) functors F , E : C → D, contravariant functor
functor들의 합성
full and faithful functor
6 / 28
Natural Transformation
natural transformation α : F → E
F (B) E (B)αB
//
F (A)
F (B)
F (f )
²²
F (A) E (A)αA // E (A)
E (B)
E(f )
²²© (f : A → B)
α : F∼−→E ; natural equivalence if αA ; isomorphism
functor category
objects are functors
morphisms are natural transformations
7 / 28
Isomorphic Categories
C ≈ D if there exist F : C → D, E : D → C such that
F ◦ E = idD, E ◦ F = idC
Examples
Ab ≈ Z-Mod , Ring = Z-Alg
k-Alg ≈ Ak (다음 슬라이드)
꿈F들≈ representable functors (다음 시간)
(대개) 너무 자명 (재미 없다)
8 / 28
k-Alg ≈ Ak
k는 fixed ring (base ring)
Ak의 object는 화살표 (ring homomorphism) kα→A
ϕ : (kα→A) → (k
β→B) is an Ak -morphism
if ϕ is a ring homomorphism such that
A Bϕ//
k
A
α
ÄÄÄÄÄÄ
ÄÄÄÄ
ÄÄÄÄ
k
B
β
ÂÂ???
????
????
?
©
Lang은 항상 k-algebra를 이렇게 삼각형으로 정의
왜 ? (내일)
9 / 28
Equivalent Categories
C is equivalent to D if there exist F : C → D, E : D → Csuch that F ◦ E
∼−→idD, E ◦ F∼−→idC
표기법 ; C ≈−→D ( C ≈−−−−→anti
D )
examples
CyclicGr≈−→{Z/nZ | 0 ≤ n ∈ Z}
오늘과 내일 여러 non-trivial (anti-)equivalence들 구경
(다음 슬라이드에 예고편)
이렇게 정의하면,≈−→는 당연히 equivalence relation
10 / 28
예고편
category of
representable
functors over k
≈←−−−−anti
k-Alg≈−−−−→
anti
category of
affine schemes
over k
꿈F‖
Homk-Alg (A,−)
←→ (k → A) ←→ 꿍꿍이 over k
(Spec A → Spec k)
11 / 28
Equivalent Categories
명제 : C ≈−→D if and only if there exist F : C → D such that
(1) F is full and faithful
(2) for each D ∈ D, there exists C ∈ C such that D ≈ F (C )
이 명제 포함하는 책 많지 않음
Proof (⇐) : Want E : D → C. For D ∈ D, choose C ∈ Csuch that D ≈ F (C ) · · · · · ·
Axiom of Choice의 적용 범위 ??
12 / 28
Small Categories
Category Theory에서 제일 골치 아픈 말 ;
“골치 아프면 (전부) small category로 생각해도 좋다”
foundation (axiom system) of category theory
set theoretic axiom system과는 좀 (많이) 다름
그냥 아는 척하면 됨 !
오늘의 주제가 아니므로 · · · · · ·시간도 없고
13 / 28
Classical Language (‘Ancient Language’ ??)
Zariski Topology on An
Affine Coordinate Ring
Category of Algebraic Sets ??
Sheaves
Sheaf of Regular Functions
Category of Ringed Spaces, Category of Affine Varieties
Main Theorem : an Anti-equivalence
Stalks (and Germs)
Products
Linear Algebraic Groups14 / 28
Zariski Topology on An
k ; fixed algebraically closed field, write kn = An
want only polynomial functions are continuous
i.e., want only zero sets(loci) of polynomials are closed
Euclidean topology ; exp, log, sin, cosh, Γ, ζ, · · · · · ·
정의 ; [closed set in An] = V(S), where S ⊆ k[T ]
V(S) = {x ∈ An | f (x) = 0 for all f ∈ S}k[T ] = k[T1, . . . , Tn]
call V(S) an algebraic set in An
An and its subspaces are usually not Housdorff
15 / 28
Affine Coordinate Ring
enough to consider V(I ) for ideals I of k[T ]
V(S) = V(〈S〉)Hilbert’s Finite Basis Theorem ; I is finitely generated
for Y ⊆ An, put I(Y ) = {f ∈ k[T ] | f (x) = 0 for all x ∈ Y }Hilbert’s Nullstellensatz ; I(V(I )) =
√I = · · · · · ·
for an algebraic set X ⊆ An, define
k[X ] = k[T ]/I(X )
= the affine coordinate ring of X
= the k-algebra of polynomial functions on X
ring= algebra ?? basis = generator ?? 16 / 28
Category of Algebraic Sets ??
k − {0} is not an algebraic set (open in A1)
but, want to ‘identify’ k − {0} ≈ X = {(a, b) ∈ A2 | ab = 1}polynomial functions on X ↔ rational functions on k −{0}
C∞-manifold에서의 경험을 생각 · · · · · ·sheaf, germ, stalk, · · · · · ·
17 / 28
Sheaf F on a Topological Space X
an abelian group F(U), for each open subset U in X
sheaf of abelian groups, sheaf of k-algebras, · · · · · ·F(∅) = 0
a homomorphism ρUV : F(U) → F(V ), for each inclusion
V ⊆ U of open subsets in X
ρUU = idF(U), ρV
W ◦ ρUV = ρU
W , if W ⊆ V ⊆ U
ρUV is called a restriction map, write ρU
V (s) = s|Vif U ; open, {Vi} ; open covering of U, si ∈ F(Vi ),
si |Vi∩Vj= sj |Vi∩Vj
for all i , j ,
then there exists a unique s ∈ F(U) s.t. s|Vi= si for all i
i.e., every s ∈ F(U) is “locally defined”
18 / 28
Category of Sheaves on X
category of sheaves on X
sheaf morphisms ϕ : F → Gdirect image sheaf f∗Fdirect limit lim→
오늘 밤 예습 복습
19 / 28
Sheaf of Regular Functions
sheaf of continuous (differentiable, C∞, analytic) functions on
a manifold
정의 : f : Y → k is regular at y , (where y ∈ Y ⊆ An)
iff f is “locally” a rational function
i.e., iff there exist an open nhd U ⊆ Y of y and g , h ∈ k[Y ]
such that f = g/h on U
정의 : sheaf of regular functions OX on X ⊆ An
OX (U) = {f : U → k | f is regular on U}OX is a sheaf of k-algebras
20 / 28
Category of Ringed Spaces
정의 : (X ,FX ) is a ringed space iff
(1) X is a topological space
(2) FX is a sheaf of k-algebras on X
(3) FX (U) consists of k-valued functions on U
정의 : ϕ : (X ,FX ) → (Y ,FY ) ; morphism of ringed spaces iff
(1) ϕ : X → Y is continuous
(2) for each open V ⊆ Y , ϕ induces
ϕ∗V : FY (V ) → FX (ϕ−1(V )), ϕ∗V (f ) = f ◦ ϕ|ϕ−1(V )
내일 등장할 morphism of locally ringed spaces와는 좀 다름
21 / 28
Category of Affine Varieties
정의 : X ⊆ An is an affine variety iff
(X ,OX ) ≈ (Y ,OY ) as ringed spaces for some Y ⊆ Am,
where O is the sheaf of regular functions
affine var is nothing but a new(fancy) name for algebraic set
정의 : Aff Var is a full subcategory of RingedSp
22 / 28
Affine Coordinate Ring
Let X ⊆ An be an algebraic set (affine variety)
recall : k[X ] is only defined for algebraic set X
for an ideal I ≤ k[X ] and P ∈ X , define
VX (I ) = {x ∈ X | f (x) = 0 for all f ∈ I}MP = {f ∈ k[X ] | f (P) = 0}
정리 : k[X ] ≈ OX (X )
X is not necessarily irreducible
see Springer (Linear Algebraic Group) or Hartshorne (p. 72)
23 / 28
Dictionary
affine var (X ,OX ) k-algebra k[X ]
irreducible ⇔ integral domain
closed subsets VX (I ) ↔ radical ideals I
DCC on closed subsets ⇔ ACC on ideals
point P ∈ X ↔ maximal ideal MP
P ∈ VX (I ) ⇔ I ≤ MP
OX (X ) = k[X ]
stalk OP = localization k[X ]MP
OX (D(f )) = k[X ]f...
......
24 / 28
Main Theorem : an Anti-equivalence
k[X ] is reduced (has no non-zero nilpotent element)
if f r ∈ I(X ), then f ∈ I(X )
정리 : Aff Var≈−−−−→
anti[category of f.g. reduced k-algebras]
X 7→ k[X ]
Mor(X ,Y )bij−→ Homk-alg(k[Y ], k[X ]), ϕ 7→ ϕ∗
note : OX (X ) ≈ k[X ]
every algebra is a quotient of a free algebra (=poly algebra)
if A/I is reduced, then I =√
I = IV(I )
25 / 28
Stalks (and Germs)
F ; sheaf on X , P ∈ X
(U, f ) ∼ (V , g) if f |W = g |W for some W ⊆ U ∩ V
P ∈ W ⊆ U ∩ V , U, V , W ; open, f ∈ F(U), g ∈ F(V )
an equivalence class [U, f ] is called a germ at P
the set of all germs is called the stalk at P
i.e., FP ={[U, f ] | P ∈ U, U is open, f ∈ F(U)
}
better to understand FP = lim→ F(U)
direct limit of abelian groups (k-algebras · · · · · · )
OX ,P is a local ring, if (X ,OX ) is an affine variety
26 / 28
Products
identify Am × An = Am+n
if X = V(R), Y = V(S)
where R ⊆ k[S1, . . . , Sm], S ⊆ k[T1, . . . , Tn]
consider R ∪ S ⊆ k[S1, . . . , Sm,T1, . . . , Tn]
define X × Y = V(R ∪ S) ⊆ Am+n
this is indeed the categorical product of affine varieties
the usual product topology is finer than ours
k[X × Y ] ≈ k[X ]⊗k k[Y ]
k[X ]⊗k k[Y ] is the categorical coproduct of k-algebras
27 / 28
Linear Algebraic Groups
(약식)정의 : G is an linear algebraic group iff
(G ,OG ) is an affine variety
G is a group
mult : G × G → G , inv : G → G are morphisms of affine vars
a few ‘separation axioms’
linear algebraic group은 topological group ? NO ! (why ?)
k, k× × k×, GLn(k), SLn(k), On(k), Spn(k), · · · · · ·
정리 : every linear algebraic group is isomorphic to
a closed subgroup of GLn(k) for some n
28 / 28
Affine (Group) Scheme의 이해 (2)
이 인 석
제 23차 대수캠프
2010년 2월
1 / 19
Scheme Language
꿈F
Representable Functors
Yoneda’s Lemma
Affine Schemes; 꿍꿍이
Spectrum
Locally Ringed Spaces
Affine Schemes over k
Affine Group Schemes over k
2 / 19
서론
예를 들어, 다음을 생각
SLn(R) = {(rij) ∈ Rn2 | det(rij)− 1 = 0}On(R) = {(rij) ∈ Rn2 | t(rij) · (rij)− I = 0}µn(R) = {r ∈ R | rn − 1 = 0}E(R) = {(r , s) ∈ R2 | s2 − 5r3 +
√2r − 1 = 0}
우리의 꿈F은SLn, On, µn(R), E 등을 functor로 이해
이 functor 각각에 ‘어떤’ geometric structure를 정의
5,√
2가 사는 곳 = 다항식들이 정의된 곳 = base ring = k
R은 결국 k-algebra일 수밖에
functor들 (즉, 꿈F들) : k-Alg → Set
classical language와의 차이 · · · · · · !3 / 19
꿈F들의 Category
하나의 꿈 F : k-Alg → Set는 다음 data
I ; an index set
Xi ; an indeterminate, (단, i ∈ I )
T ; a subset of the polynomial algebra k[{Xi}i∈I ]
for a k-algebra R, define
F(R) =
{(ri ) ∈
∏i∈I R
∣∣∣∣ f ((ri )) = 0 for all f ∈ T
}
for ξ ∈ Homk-Alg (R, S), define ftn F(ξ) : F(R) → F(S) by
F(ξ) : (ri ) 7→ (ξ(ri )), ( (ri ) ∈ F(R) )
꿈F들의 category는 functor category
즉, 꿈F들 간의 morphism은 natural transformation
4 / 19
Representable Functors over k
정의 : F : k-Alg → Set is a representable functor (over k)
iff there exists a k-algebra A s.t. F ≈ Homk-Alg (A,−)
say “A represents F”
주의 : functor category이므로, ‘≈’와 ‘∼−→’는 같은 의미
정리 :
[꿈F들의category
]=
category of
representable
functors over k
꿈F F ←→ Homk-Alg (A,−)
A = k[{Xi}i∈I ] / 〈T 〉5 / 19
Yoneda’s Lemma and k-algebras
[꿈F들의category
]=
category of
representable
functors over k
≈←−−−−anti
k-Alg
꿈F F ←→ Homk-Alg (A,−) ←→ A
Yoneda’s Lemma는 (정말 妙한) abstract non-sense
No geometric structure yet ! 꿍꿍이가등장할차례
6 / 19
(다시) 서론
지금까지는 전부 abstract non-sense (철학 ?)
affine variety 제외하고
이제, 구름 위로 · · · · · ·
지금까지의 story는사실 神話 시대에는널리알려진내용
요즘은 · · · · · · ?
神話 시대의 관심사는 꿍꿍이 (즉, geometric structure)
scheme = (1) 계획, (2) 음모, 꿍꿍이
scheme의 경쟁자들 · · · · · ·7 / 19
Spec A
여기서는 A는 ring (즉, Z-algebra)
Spec A = {p | p is a prime ideal of A}
topology on SpecA
closed sets ; V(a) = {p | a ≤ p}, (a ; ideal of A)
8 / 19
Structure Sheaf on Spec A
notations
p ∈ Spec A, V , U ; open in Spec A
Ap = localization of A at p, (Ap is a local ring)
OSpec A(U) is the set of functions s : U → ∐p∈U Ap s.t.
s(p) ∈ Ap for each p ∈ U
for p ∈ U, there exist V with p ∈ V ⊆ U, and a, f ∈ A
s.t. f /∈ q and s(q) = a/f ∈ Aq for each q ∈ V
i.e., s is locally a quotient of elements of A
(Spec A,OSpec A) is called the spectrum of A
OSpec A is a sheaf of rings
9 / 19
Spectrum
정의 : for f ∈ A, define D(f ) = SpecA− V(〈f 〉)D(f ) is called a principal open set
principal open sets form a base for the topology of Spec A
명제 : write O = OSpec A
(a) Op ≈ Ap for any p ∈ Spec A
(b) O(D(f )) ≈ Af for any f ∈ A
(c) O(Spec A) ≈ A
10 / 19
Direct Image Sheaf
F ; sheaf on X , f : X → Y ; continuous
direct image sheaf f∗F on Y is defined by
(f∗F)(V ) = F(f −1(V )), (V is open inY )
inverse image sheaf f −1Gneed sheafification · · · · · ·
11 / 19
Category of Sheaves on X
X is a fixed topological space
objects are sheaves (of abelian groups) F ,G on X
morphism ϕ : F → G is a set
{ϕ(U) ∈ HomAb(F(U),G(U)) | U is open in X} s.t.
F(V ) G(V )ϕ(V )
//
F(U)
F(V )
ρUV
²²
F(U) G(U)ϕ(U) // G(U)
G(V )
ρ′UV²²
© (V ⊆ U)
12 / 19
Category of Locally Ringed Spaces
정의 : (X ,OX ) is a locally ringed space iff
X is a topological space with a sheaf OX of rings
stalk OX ,P is a local ring if P ∈ X
spectrum (SpecA,OSpec A) is a locally ringed space
정의 : (f , f #) : (X ,OX ) → (Y ,OY ) is a morphism of
locally ringed spaces iff
f : X → Y is a continuous map
f # : OY → f∗OX is a morphism of sheaves on Y
f #은 f 와 무관
induced map f #P : OY ,f (P) → OX ,P is a local homomorphism
13 / 19
Morphism of Locally Ringed Spaces (LRS)
왜 이렇게 (골치 아프게) 정의 ?
첫째 (유일한 ?) 이유
want : MorRing (A, B)one-to-one−−−−−−→ MorLRS(Spec B, Spec A)
ϕ : A → B induces f : Spec B → Spec A, f (p) = ϕ−1(p)
즉, 우리의 꿈F을 이루기 위해서 !
다른 이유들은 (만약 다른 이유가 있다면) 다음 기회에
14 / 19
비교
classical scheme
ringed space locally ringed space
f : (X ,OX ) → (Y ,OY ) (f , f #) : (X ,OX ) → (Y ,OY )
f induces f ∗V f #은 f 와무관
f # : OY → f∗OX
f ∗V : OY (V ) → OX (f −1(V )) f #(V ) : OY (V ) → OX (f −1(V ))
f ∗V (s) = s ◦ f |f −1(V )
f # induces f #P : OY ,f (P) → OX ,P
15 / 19
Category of Affine Schemes
정의 : X ,OX is an affine scheme (꿍꿍이) iff
(X ,OX ) ≈ (SpecA,OSpecA) as LRS for some ring A
정확히는 [affine scheme] = [affine scheme over Z]
정의 : Aff Sch is a full subcategory of LRS
정리 : Ring≈−−−−→
antiAff Sch
16 / 19
Category of Affine Schemes over k
정의 : ((X , OX ), (f , f ])) is an affine Scheme over k iff
(X , OX ) ; 꿍꿍이, (f , f ]) : X → Spec k ; morphism of LRS
간단히 화살표 (Xf−→Spec k) 로 표기
정의 : (h, h]) ∈ MorAff Sch/k((Yg−→Spec k), (X
f−→Spec k))
iff (h, h]) : (Y , OY ) → (X , OX ) is a morphism of LRS s.t.
Y
Spec k
(g ,g])
ÂÂ???
????
????
?Y X(h,h]) // X
Spec k
(f ,f ])
ÄÄÄÄÄÄ
ÄÄÄÄ
ÄÄÄÄ
©
17 / 19
k-algebras and Affine Schemes over k
A Bϕ//
k
A
α
ÄÄÄÄÄÄ
ÄÄÄÄ
ÄÄÄÄ
k
B
β
ÂÂ???
????
????
?
©
Spec B
Spec k
(g ,g])ÂÂ?
????
????
???Spec B Spec A
(h,h]) // Spec A
Spec k
(f ,f ])ÄÄÄÄ
ÄÄÄÄ
ÄÄÄÄ
ÄÄ
©
18 / 19
Conclusion
category of
representable
functors over k
≈←−−−−anti
k-Alg≈−−−−→
anti
category of
affine schemes
over k
꿈F‖
Homk-Alg (A,−)
←→ (k → A) ←→ 꿍꿍이 over k
(Spec A → Spec k)
19 / 19
Affine Group Scheme의 이해 (3)
이 인 석
제 23차 대수캠프
2010년 2월
1 / 18
Affine Group Schemes
Affine Group Schemes over k
Hopf Algebras
Epilogue ; (Group) Scheme (over k)
2 / 18
Affine Group Schemes
affine scheme/k는 여러 가지 개념의 mixture
꿈F, representable functor, k-algebra, spectrum
affine group scheme도 마찬가지
functor G : k-Alg → Group is an affine group scheme over k
iff G is representable functor over k such that
mult : G× G → G, unit : e → G, inv : G → G ; morphisms
affine scheme = “set functor”,
affine group scheme = “group functor”
3 / 18
Affine Group Schemes
k-alg A가 G를 represent하면, Homk-Alg (A, R)는 group
Homk-Alg (A,R)의 natural abel group structure와는 무관
if ξ : R → S is an k-algebra homomorphism,
then G(ξ) : G(R) → G(S) is a group homomorphism
if Φ : G → H is a morphism of affine group scheme over k,
then ΦR : G(R) → H(R) is a group homomorphism
4 / 18
Product of Affine Schemes over k
명제 : E,F가 A, B에 의해 represent될 때,
categorical product E× F는 다음과 같이 주어진다
(E× F)(R) = E(R)× F(R), (R은 k-algebra)
E(R)× F(R)은 (집합들의) cartesian product
꿈F의 입장에서는 변수 개수 늘린 것E× F는 당연히 A⊗k B에 의해 represent
5 / 18
Affine Group Scheme G over k
G(S)× G(S) G(S)multS
//
G(R)× G(R)
G(S)× G(S)
G(ξ)×G(ξ)
²²
G(R)× G(R) G(R)multR // G(R)
G(S)
G(ξ)
²²© (ξ ∈ Homk-Alg (R,S))
mult : G× G → G는 natural transformation
6 / 18
Group Laws
associativity
G× G Gmult
//
G× G× G
G× G
mult×id
²²
G× G× G G× Gid×mult // G× G
G
mult
²²©
7 / 18
Trivial Group Functor e
e(R) = {e} = trivial group
note : |f(R)| = 1 for all R, then e ≈ f
e(S) f(S)≈//
e(R)
e(S)
e(ξ)
²²
e(R) f(R)≈ // f(R)
f(S)
f(ξ)
²²© (ξ : R → S)
8 / 18
Group Laws
마찬가지로
G× G Gmult
//
e× G
G× G
unit×id
²²
e× G G≈ // G
G
||²²
©
e Gunit
//
G
e²²
G G× Ginv×id // G× G
G
mult
²²©
역으로, 이다섯 (둘은생략) diagram이 G를결정
9 / 18
Examples
k[x ] represents the additive group scheme Ga
Ga(R) = (R, +) = {r ∈ R | no condition}
k[x , y ]/ 〈xy − 1〉 ↔ the multiplicative group scheme Gm
Gm(R) = R×
k[xij , y ]/ 〈y · det(xij)− 1〉 ↔ GLn
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화살표뒤집기
A represents G
A⊗k A A⊗k A⊗k Aid⊗∆
//
A
A⊗k A
∆
²²
A A⊗k A∆ // A⊗k A
A⊗k A⊗k A
∆⊗id
²²©
A k ⊗k A≈//
A
A
||²²
A A⊗k A∆ // A⊗k A
k ⊗k A
ε⊗id
²²©
A⊗k A AS⊗id
//
A
A⊗k A
∆
²²
A kε // k
A²²
©
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Hopf Algebras
앞 다섯 diagram 만족하는 k-algebra는 Hopf algebra
∆ : A → A⊗k A ; comultiplication
ε : A → k ; augmentation
S : A → A ; antipode
정리 : Aff GrSchk≈−−−−→
antiHopfAlgk
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Example
the additive group scheme Ga
Ga(R) = (R, +)
Ga ≈ Homk-Alg (k[x ],−)
∆(x) = x ⊗ 1 + 1⊗ x
ε(x) = 0
S(x) = −x
Yoneda’s Lemma !
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용어남용
大家들의 용어 남용 ;
[ group ] = [ affine group scheme ] = [ Hopf algebra ]
[ group ] = [ group algebra ] = [ Hopf algebra ]
그래서, Hopf algebra이지만, 애당초 group일 수 없는
quantum group을 ‘group’으로 명명
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Cocommutative Hopf algebras
정의 : G is a commutative affine group scheme iff
G(R) are commutative for all R, 즉
G G=//
G× G
G
mult
²²
G× G G× Gtwist // G× G
G
mult
²²©
정의 : A is a cocommutative Hopf algebra iff
A⊗ A A⊗ Atwist
//
A
A⊗ A
∆
²²
A A= // A
A⊗ A
∆
²²©
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Cocommutative Hopf algebras
명제 ; A가 G를 represent하면,
G is commutative ⇔ A is cocommutative
대개
affine group scheme ; commutative but non-cocommutative
group algebra ; non-commutative but cocommutative
최초로 발견된 non-commutative and non-cocommutative
Hopf algebra 가 바로 quantum group
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Affine Group Schemes
명제 ; G의 linear representation은 (즉, G-module은)
Hopf algebra A의 comodule에 대응
· · · · · ·
정리 : 만약 k가 field이면, 모든 affine group scheme은
어떤 GLn의 closed subgroup과 isomorphic
geometry 필요 · · · · · ·
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Epilogue
projective variety, projective scheme · · · · · ·
scheme = LRS which “locally looks like an affine scheme”
schemes over k · · · · · ·
group scheme ? “locally group” ??
다음을 만나도 쫄지 않기를 !
“elliptic curve is a group scheme”
“Soit G un schema en groupe (affine)”
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