( ) Lemma 2.2. X ultra filter (1) X = X 1 X 2 X 1 X 2 (2) X = X 1 X 2 X 3... X N X 1, X 2,..., X N (3) disjoint union X j Definition 2.3. X ultra filt

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1 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE ( ) 1. Introduction (1) (2) universality C ( ) R (1) (2) ultra filter 0 (1) (1) ( ) (2) (2) (3) 2. ultra filter Definition 2.1. X F filter (1) F A, B = A B F. (2) F A, A A 1 X = A 1 F. filter ultra filter ltra filter

2 ( ) Lemma 2.2. X ultra filter (1) X = X 1 X 2 X 1 X 2 (2) X = X 1 X 2 X 3... X N X 1, X 2,..., X N (3) disjoint union X j Definition 2.3. X ultra filter principal ultra filter x X = { X; x } principal filter ultra filter X x (Bourbaki principal filter trivial filter ) Lemma (3) Lemma 2.4. X ultra filter E principal filter non-principal ultra filter 2.1. :ltra filter. ultra filter Lemma 2.5. X C b (X) C b (X) (C - ) C b (X) Y = Spm(C b (X)) C b (X) 5 (1) Y ( C b (X) ) (2) C b (X) C C- ϕ (3) C b (X) C - (4) X (Stone-Čech ) (5) X ultra filter ultra filter X principal ultra filter X non-principal ultra filter X ( wikipedia ( web RL

3 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE ) ) Proof. (2) ϕ (1) ϕ ( ) C b (X) C - ( Banach ) M (M C b (X) ) C b (X)/M Banach C (Gelfand-Mazur C - ( ) ) (3) (1)-(3) (4) (1) X K φ K C(K) f f ( ) φ φ (f) φ : C(K) C b (X) - C - C(K) K (Stone-Weierstass : C - ) Y = Spm(C b (X)) K Y Y (5) X ultra filter (3) C b (X) f f ( ) f K > 0 f(x) D K = {z C; z K} D K K ɛ > 0 D K ɛ B 1,..., B N X = D K = B 1 B 2... B N N f 1 (B j ) j=1 ultra filter j f 1 (B j ) disjoint union j f 1 (B j ) f() ɛ f() C

4 ( ) f() ( ) c f C f c f p Z/pZ F p p r F p r F p inj lim r F p r F p Definition 2.6. P = Spm(Z) (Z ) non-principal ltra filter Q = p F p /( 0) Q ( ) = p F p /( 0) p F p (a p ) p P (a p F p ) p F p ( 0) p F p I Q I = {(a p ) p P p F p ; such that a p = 0 for all p } Q ( ) Proposition 2.7. (1) Q ( ), Q 0 (2) Q ( ) Proof. (1) Q, Q ( ) Q f = (f p ) Q 0 ( ) E 1 = {p Spec 1 (O); f p 0} E E E 1 ( f = 0 ) ( ) E 1 f

5 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE g = (g p ) f 1 if p E 1 g p = 0 otherwise Q 0 n n = 0 Q E 0 n p E 0 pz E 0 principal ultra filter Q ( ) F p Q ( ) F (X) = X n + a (n 1) X n 1 + a (n 2) X n a (1) X + a (0) (a (j) Q ( ) ) Q( ) p F p a (j) = (a p (j) ) p a p (j) F p F p (X) = X n + a (n 1) p X n 1 + a (n 2) p X n a (1) p X + a(0) p F p F p x (1) p, x (2) p,..., x (n) p ( ) x (j) = (x (j) ) p Q ( ) (j = 1,..., n) x F Q ( ) {x (i) p } n i=1 n! p F (X) p σ p S n x (σ) = (x (σp(1)) ) p Q ( ) F S n τ τ = {p Spm(Z); σ p = τ}

6 ( ) Spm(Z) = τ Sn τ disjoint union ultra filter τ p σ p τ x (σ) ( ) x (τ(1)) ultra filter (S n ) Proposition 2.8. ultra filter Q Q Q ( ) Lemma Lemma 2.9. f Z[X] \ Z f modulo p F p p Proof. f Z[X] \ Z f modulo p F p p p 1, p 2,..., p N f(x) f(x + c) f f x > 0 f(x) > 0 {f(j); j = 1, 2, 3,..., } {p e 1 1 p e p e N N ; e 1, e 2,..., e N N} #({p e 1 1 pe pe N N ; e 1, e 2,..., e N N} N 2 m) #{e 1, e 2,..., e N N m } = m N #({f(j); j = 1, 2, 3,..., } N 2 m) 2 m/d (d f ) Lemma f 1, f 2,..., f n Z[X] \ Z F p p Proof. Q f 1, f 2,..., f n α 1, α 2,..., α N K K L L Q ( 0 ) L Q β β Q Lemma

7 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE (Proposition 2.8 ) Z monic S S = {f 1, f 2,..., f n } S S = {p Spm(Z); f 1, f 2,..., f n F p } S S, T S S T = S T 0 = { Spm(Z); S S such that S } 0 filter ( 0 filter base { S ; S S} ) 0 ultra filter ( ) non principal Z Q Lemma Q ( ) Q R. Proof. Q ( ) F q #Q ( ) #R Q ( ) Q Q Q S 1 = R/Z ι p : F p S 1 ι p : F p = Z/pZ (n mod Z) (n/p mod Z) R/Z. well-defined π π(π ((a p ))) = lim p ι p(a p ) limit filter limit ultra filter (Lemma 2.2 ) S 1 α a p F p ι p (a p ) α π((a p )) = α π Q ( ) Corollary Q ( ) = C. Proof. 0 (transcendense base ) (ultra filter ) F p p C

8 ( ) p f p p ( ) 2.8 F p limit ultra filter 3. non commutative algebraic space of finite arithmetic type abel A. Rosenberg [4] Grothendieck abel C abel abel Rosenberg abel 3.1. R-abel. Definition 3.1. R abel C R-abel C (1) M 1, M 2 Ob(C) Hom C (M 1, M 2 ) R- (2) R- (a.f) g = a.(f g) = f (a.g) a R C ( ) f, g R-abel R-abel R- (additive ) R- R- (R module) R-abel R-abel C augmented (R module) C R- Definition 3.2. C R R-abel R I R/I-abel C/I C Ob(C/I) = {M Ob(C); IM = 0}. ( IM = 0 a I 0 ) M a M

9 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE Definition 3.3. k p 0 little non commutative algebraic space of finite type over k (1) X k finite type (2) X A A O X - O X - (X, A) C qcoh (X, A) A- Ø X - augmented k-abel Definition 3.4. R K O non commutative algebraic space of finite arithmetic type X (over R) (1) R- ( ) C qcoh (X) (2) p Spm(O) little non commutative algebraic space of finite type (X (p), A X(p) ) over R/p (3) p Spm(O) C coh /p = C qcoh (X (p), A X(p) ) non commutative algebraic space of finite arithmetic type NC ( finite arithmetic type ) R NC morphism (little NC ( ) ) Definition 3.5. R K O X, Y R NC X Y morphism (1) C qcoh (Y ) C qcoh (X) ( ). (2) p Spm(O) (X (p), A X(p) ) (Y (p),ay (p) ) little NC f (p). modulo p p NC localization, étale, smooth X smooth p Spm(O) X (p) smooth A X(p) X (p) O X(p) - ( X (p) p O/p ) NC affine O- A C qcoh (X) (A-modules)

10 ( ) A (A modules) NC- (4.2 ) Weyl (4.1 ) Lie (4.3 ) X (p) A/pA Spec Weyl Lie NC 4.1. Weyl algebra. 4. Definition 4.1. k Weyl A n (k) k ( ) A n (k) = k ξ 1, ξ 2,..., ξ n, η 1, η 2,..., η n /(η j ξ i ξ i η j δ ij ; 1 i, j n), (where δ ij is the Kronecker s delta.) (A n (k) (k) ind-scheme k- k k k ξ, η k k ) A n (k) Z n (k) 2n ξ p 1, ξp 2,..., ξp n, ηp 1, ηp 2,..., ηp n k A n (k) Z n (k) matrix bundle section Weyl. k p > 0 Weyl k- Z n (k) k- ( [5] ) Z n (k) p- S n (k) = k[t 1, T 2,..., T n, 1, 2,..., n ] T i = (ξ p i )1/p, j = (η p j )1/p S n (k) n- ( ) A n (k) Proposition 4.2. k p > 0 φ : A n A n k- f : Spec S n Spec S n φ morphism G (4.1) G(f )G 1 = + ω

11 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE (4.2) ω = n i=1 (ω T i dt i + ω i d i ) n ω p T i + ( / T i ) p 1 j (ω Ti T j ) = 0 T j=1 i (i = 1, 2,..., n) n ω p i + ( / i ) p 1 j (ω i T j ) = 0 i j=1 where T i = ˆψ(T i ) = f (T i ), i = ˆψ( i ) = f ( i ) ω Ti ω i ω T i i 1- ω suffix ( ) p = 3, n = 1 ξ 1 ξ 1 η 1 ξ 1, η 1 η 1 ξ 1 η 1 φ φ f étale φ ( ) p f étale Weyl p >> 0 dt 1 d 1 + dt 2 d dt n d n ultra filter 0. ultra filter Q Weyl A n (Q ) symplectic dt 1 d 1 + dt 2 d dt n d n A n (Q ) h 0 symplectic Weyl A n Q - Weyl Weyl localization ultra filter. A n k ξ, η, ξ 1 ξ ξ 2, η 1 2 ξ 1 η

12 ( ) Lemma 4.3. f A n (Z) f p Z n (F p ) ( p ) A n (Z) f modulo p ((A n ) f )(F p ) = A n (F p ) Zn(F p) Z n (F p )[(f p ) 1 ] Z n (F p )[(f p ) 1 ] finitely generated f localize Lemma 4.4. Let p be a prime. Then in k ξ, η /(ηξ ξη 1), we have the following identity. (1) ξ t η t = (ξη)(ξη 1)(ξη 1)... (ξη (t 1)) (2) η t ξ t = (ξη + 1)(ξη + 2)... (ξη + t) (3) (ξη) p ξη = ξ p η p. (4) Let f(w) = (w a 1 )... (w a l ) k[w], a i k. Then we have ( ) (f(ξη)η s ) p = (ξ p η p a p i + a i) η sp. for any positive integer s which is relatively prime with p. (5) For any polynomial f, g k[w], we have i [f(ξη)η t, g(ξη)ξ t ] = F (ξη) F (ξη t) where F (w) = w(w + 1)(w + 2)... (w t + 1)f(w)g(w + t). f, s > 0 f(ξη)η s localization f(ξη) η f(ξη) inverse η Lemma 4.5. (θ 2 2)η (θ = ξη) ((θ 2 2)η) p = ξ 2p η 3p 4(1 ( 2 ξ 2p η 3p (2: ) p ))ηp = ξ 2p η 3p 8η p (2: ). (( 2 ) is the Legendre s symbol). p ultra filter 2 B = A n [ξ 1, η 1 ] (θ 2 2) (mod p ) ultra filter 2 B (θ 2 2) mod p ultra filter inverse algebra algebra mod p limit

13 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE 4.2. Cuntz O 2. Cunts O 2 ( ) O 2 = e 1, e 2, e 1, e 2 ; e 1 e 1 = 1, e 2 e 2 = 1, e 1 e 1 + e 2e 2 = 1, e 2 e 1 = 0, e 1 e 2 = 0, ( O 2 C - C -completion Cuntz C ( ) O 2 O 2 ) O 2 ( ) e 1, e 2 1 = e 1 e 1e 2 e 2 = 2 O 2 ( ) O 2 O 3, O 4,..., O n,... O (O ) Cunts K- [1] Weyl Cuntz C Lie g (g) g Theorem 4.6. (Serre) C Lie g {x i, y i, h i 1 i l} ( Lie [2] 18.1 ) (1) [h i h j ] = 0 (2) [x i y i ] = h i, [x i y j ] = 0 if i j. (3) [h k x j ] = α j, α i x j, [h i y j ] = α j, α i y j, (4) (ad x i ) α j,α i +1 (x j ) = 0 (5) (ad y i ) α j,α i +1 (x j ) = 0 g {x i, y i, h i } g Q = Q {x i, y i, h i } Q p F p ( V Q F p suffix V Q, V Fp ) g h = C {h i }, n + = C {x i }, n = C {y i }

14 ( ) g = h n + n ( ) h {h i } n + ( n ) {x i } ( {y i } ) g Q = h Q n +Q n Q Q- n + n {n 1, n 2..., n s } n i ad-nilpotent p > 0 Q F p p k g k p restricted Lie [3] Lie restricted Lie Lie Theorem 4.7. k Lie g k Killing form A g k ad(a) p (B) = [A [p], B] ( B g k ) A [p] g k [A p, B] = [A [p], B] ( B g k ) p g ( ) h [p] i = h i (i = 1,..., l), n [p] j = 0 (j = 1,..., s) ( ) h p i h i (i = 1,..., l), n p j (j = 1,..., s) casimir 0 g (Cartan sub algebra ) r g Z r ( Z [2] 23.3 Coxeter Z ) Z modulo p g k

15 NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE Poincaré-Birkoff-Witt ( ) k ( ) R k = k[{h p i h i (i = 1,..., l), n p j (j = 1,..., s)}] (g k ) R k g k (g k ) ϕ ϕ(x) = x p x [p] restricted Lie ϕ p-semilinear ϕ(c 1 x 1 + c 2 x 2 ) = c p 1ϕ(x 1 ) + c p 2ϕ(x 2 ) c 1, c 2 k, x 1, x 2 g k R k ϕ Poincaré- Birkoff-Witt ϕ Spec(R k ) g g ( Frobenius ) (g Fp ) O-coherent algebra sheaf O- locally free [3] g u-algebra (g) g algebra sheaf : math.rt/ ( ) [email protected] References [1] J. Cuntz, A new look at kk-theory, K-Theory 1 (1987), [2] J. E. Humphreys, Introduction to lie algebras and representation theory, Springer-Verlag, [3] N. Jacobson, Lie algebras, Interscience publishers, [4] A. L. Rosenberg, Noncommutative schemes, Compositio Math. 112, no. 1 (1998), [5] Y. Tsuchimoto, Preliminaries on Dixmier conjecture, Mem. Fac. Sci. Kochi niv. Ser.A Math.. 24 (2003),