1 ( ) 2007 02 16 (2006 5 19 ) 1 1 11 1 12 2 13 Ore 8 14 9 2 (2007 2 16 ) 10 1 11 ( ) ( [T] 131),, s 1 a as 1 [T] 15 (, D ), Lie, (derived category), ( ) [T] Gelfand Manin [GM1] Chapter III, [GM2] Chapter 4 ( [I] XI ) Gelfand Manin [GM1] p147, 6 Definition b), c) ( [GM2] pp89 90 17 Definition
2 1 b, c [I] XI K S FR2, FR3 ) [T] 15, p24 ( ) A S (i), (ii) Lie,,,, 12 [1] (, 40 ) A ( ), S, (a) 1 S S (s, s S ss S) (b) a A, s S a A, s S s a = a s (c) a A, s S as = 0 s S s a = 0 S (a) S A (b), (c) ([T], p24) (c) (c ) (c ) a, a A, s S as = a s s S s a = s a 1 s 1,, s n S a 1,, a n A a 1 s 1 = = a n s n S 2 a i, b i A, s i S a i s i = b i s i (i = 1,, n) s S s a i = s b i (i = 1,, n) 3 S A (s, a) (s, a ) c, c A cs = c s S ca = c a S A S 1 A = S A/, (s, a) S A S 1 A s\a 4 S 1 A (s\a)(s \a ) = (s s)\(a a ), s a = a s, s S, a A, s\a + s \a = s \(ca + c a ), s = cs = c s S, c, c A S 1 A ( )
12 3 5 i S A S 1 A i S (a) = 1\a (a A) i s, s S i S (a) S 1 A 6 Ker i S = { a A s S sa = 0 } 7 S i S, a A a/1 S 1 R, A S 1 A a A, s S s\a = s 1 a [1], [2] ( ) 1 n n = 1 a 1 = 1 n, s 1,, s n+1 S a 1,, a n A a = a 1s 1 = = a ns n S a s = s n+1 (b) a A, s S s = s a 1s 1 = = s a ns n = a s n+1 (a) s S a 1 = s a 1,, a n = s a n, a n+1 = a n + 1 2 n n = 1 (c ) n, a i, b i A, s i S, a i s i = b i s i (i = 1,, n + 1) s S s a i = s b i (i = 1,, n) (c ) s S s a n+1 = s b n+1 1 n = 2, c, c A c s = c s S s = c s = c s S s a i = s b i (i = 1,, n, n+1) 3 a, a, a A, s, s, s S (s, a) (s, a ), (s, a ) (s, a ) c, c, d, d A ca = c a, s 1 = cs = c s S, d a = d a, s 2 = d s = d s S s 1, s 2 1, a 1, a 2 A a 1 cs = a 1 c s = a 2 d s = a 2 d s S a 1 cs = a 2 d s (a 1 c a 2 d )s = 0 (c) t S ta 1 c = ta 2 d (a) ta 1 cs = ta 1 c s = ta 2 d s = ta 2 d s S ta 1 ca = ta 1 c a = ta 2 d a = ta 2 d a (s, a) (s, a ) 3, 1 s 1 \a 1,, s n \a n 4 well-defined well-definedness 41 well-defined 411 (b) a, a A, s, s S s S, a A s a = a s 412 (s s)\(a a ) s, a s i a = a i s, s i S, a i A 1 b 1, b 2 A b 1 s 1 = b 2 s 2 S b 1 a 1s = b 1 s 1a = b 2 s 2a = b 2 a 2 a (c ) t S tb 1 a 1 = tb 2 a 2 tb 1 s 1s = tb 2 s 2 s S, tb 1 a 1a = tb 2 a 2a (s 1s)\(a 1a ) = (s 2s)\(a 2a )
4 1 413 c A, cs S (s, a) (cs, ca) (s s)\(a a ) s a = a s, tca = bs, s, t S, a, b A (tcs)\(ba ) (s s)\(a a ) c, s (b), t S, b A t c = b s t, t (b), t S, b A t t = b t t bs = t tca = b t ca = b b s a = b b a s (c ), t S t t b = t b b a S t t tcs = t b b s s, t t ba = t b b a a (tcs)\(ba ) (s s)\(a a ) 414 (s s)\(a a ) s\a (s 1, a 1 ) (s 2, a 2 ), s i S, a i A c 1, c 2 A t = c 1 s 1 = c 2 s 2 S, b = c 1 a 1 = c 2 a 2 b, s (b), t S, b A t b = b s 413 (s 1s 1, a 1s ) (t t, b a ) (s 2s 2, a 2s ) 415 c A, c s S (s, a ) (c s, c a ) (s s)\(a a ) s a = a s, ta = bc s, s, t S, a, b A (ts, bc a ) (s s, a a ) s, t (b) t S, b A t s = b t b bc s = b ta = t s a = t a s (c ), t S t b bc = t t a t b ts = t t s s S, t b bc a = t t a a (ts, bc a ) (s s, a a ) 416 (s s)\(a a ) s \a (s 1, a 1) (s 2, a 2), s i S, a i A c 1, c 2 A t = c 1s 1 = c 2s 2 S, b = c 1a 1 = c 2a 2 s i a = a i s i, s i S, a i A, (s 1s, a 1a 1) (s 2s, a 2a 2) a, t (b), t S, b A t a = b t 415 (s 1s, a 1a 1) (t s, b b) (s 2s, a 2a 2) 417 (s s)\(a a ) s\a s \a (s 1, a 1 ) (s 2, a 2 ), (s 1, a 1) (s 2, a 2), s ija i = a ijs j, s i, s j, s ij S, a i, a j, a ij A 414, 416 (s 11s 1, a 11a 1) (s 21s 2, a 21a 1) (s 22s 2, a 22a 2) well-defined 42 well-defined 421 2 a, a A, s, s S c, c s = cs = c s S 422 s \(ca + c a ) c, c s i = c i s = c is S, c i, c i A s 1, s 2 2, b 1, b 2 A b 1 s 1 = b 2 s 2 S b 1 c 1 s = b 1 c 1s = b 2 c 2 s = b 2 c 2s 2 n = 2, t S tb 1 c 1 = tb 2 c 2, tb 1 c 1 = tb 2 c 2 tb 1 s 1 = tb 2 s 2 S, tb 1 (c 1 a+c 1a ) = tb 1 c 1 a+tb 1 c 1a = tb 2 c 2 a + tb 2 c 2a = tb 2 (c 2 a + c 2a ) (s 1, c 1 a + c 1a ) (s 2, c 2 a + c 2a ) 423 s \(ca+c a ) s\a (s 1, a 1 ) (s 2, a 2 ), s i S, a i A d 1, d 2 A t = d 1 s 1 = d 2 s 2 S, d 1 a 1 = d 2 a 2 s i = c i s i = c is, c i, c i A t, s 1, s 2 2, b, b 1, b 2 A bt = b 1 s 1 = b 2 s 2 S bd 1 s 1 = td 2 s 2 = b 1 c 1 s 1 = b 1 c 1s = b 2 c 2 s 2 = b 2 c 2s 2 n = 3, t S t bd 1 = t b 1 c 1, t td 2 = t b 2 c 2, t b 1 c 1 = t b 2 c 2 t b 1 s 1 = t b 2 s 2 S, t b 1 c 1 a 1 = t bd 1 a 1 = t bd 2 a 2 = t b 2 c 2 a 2, t b 1 (c 1 a 1 + c 1a ) = t b 1 c 1 a 1 + t b 1 c 1a = t b 2 c 2 a 2 + t b 2 c 2a = t b 2 (c 2 a 2 + c 2a ) (s 1, c 1 a 1 + c 1a ) (s 2, c 2 a 2 + c 2a ) 424 s \(ca + c a ) s \a 423 ( 423 ) 425 s \(ca+c a ) s\a s \a (s 1, a 1 ) (s 2, a 2 ), (s 1, a 1) (s 2, a 2), s ij = c ij s i = c ijs j S, s i, s j S, a i, a j, c ij, c ij A 423,
12 5 424 (s 11, c 11 a 1 + c 11a 1) (s 21, c 21 a 2 + c 21a 1) (s 22, c 22 a 2 + c 22a 2) well-defined 43 s, s, s S, a, a, a A 431 (b) a, s t S, b A ta = bs (b) a, ts t S, b A t a = b ts (s\a)((s \a )(s \a )) = (s\a)((ts )\(ba )) = (t s)\(b ba ) ((s\a)(s \a ))(s \a ) = ((t s)\(b ta ))(s \a ) = ((t s)\(b bs ))(s \a ) = (t s)\(b ba ), 1 t a = b ts, 2 ta = bs, 3 1(b bs ) = (b b)s ((s\a)(s \a ))(s \a ) = (s\a)((s \a )(s \a )) 432 1\1 s1 = 1s, s S, 1 A (1\1)(s\a) = (s1)\(1a) = s\a 1a = a1, 1 S, a A (s\a)(1\1) = (1s)\(a1) = s\a 433 s, s, s 2 n = 3, c, c c A s = cs = c s = c s S ((s\a)+(s \a ))+(s \a ) = (s, ca+c a +c a ) = (s\a) + ((s \a ) + (s \a )) 434 s = cs = c s S, c, c A, (s\a) + (s \a ) = s \(ca + c a ) = s \(c a + ca) = (s \a ) + (s\a) 435 1\0 s = s1 = 1s (1\0)+(s\a) = s\(s0+1a) = s\a 436 s\( a) s\a s = s1, 0 = s0 s\0 = 1\0 s\( a) + s\a = s\(a a) = s\0 = 1\0 437 t = cs = c s S, ua = a t, u S, c, c, a A (s \a )((s\a) + (s \a )) = (s \a )(t\(ca + c a )) = (us )\(a (ca + c a )) = (us )\(a ca + a c a ) = (us )\(a ca) + (us )\(a c a ) = (s \a )(t\(ca)) + (s \a )(t\(c a )) = (s \a )((cs)\(ca)) + (s \a )((c s )\(c a )) = (s \a )(s\a) + (s \a )(s \a ) (b) v, v S, b, b A, vca = bs, v c a = b s 2 d, d A w = dv = d v S wca = dvca = dbs, wc a = d v c a = d b s, w(ca + c a ) = (db + d b )s ((s\a) + (s \a ))(s \a ) = (t\(ca + c a ))(s \a ) = (wt)\((db + d b )a ) = (wt)\(dba + d b a ) = (wt)\(dba ) + (wt)\(d b a ) = (t\(ca))(s \a ) + (t\(c a ))(s \a ) = ((cs)\(ca))(s \a ) + ((c s )\(c a ))(s \a ) = (s\a)(s \a ) + (s \a )(s \a ) S 1 A 5, 6, 7 ( )
6 1 [2] (S 1 A, 20 ) A ( ), S A ( [1] (a) ) ( ) Ã i A Ã (i) s S i(s) Ã ( ) (ii) Ã = { i(s) 1 i(a) s S, a A } (iii) Ker i = { a A s S sa = 0 } ( [1] i S A S 1 A ) 1 f A B A ( ) B, s S f(s) B ( ) s S, a, c A, cs S f(cs) 1 f(ca) = f(s) 1 f(a) 2 ( [1] (b), (c) ) 3 s, s S, a, a A i(s) 1 i(a) = i(s ) 1 i(a ) c, c A cs = c s S ca = c a 4 Φ Ã S 1 A Φ(i(s) 1 i(a)) = s\a (s S, a A), Φ i = i S Φ Ã S 1 A 5 f A B A ( ) B, s S f(s) B ( ) φ Ã B φ i = f i A Ã i S A S 1 A (S 1 A ) 1 f(cs) 1 f(ca) = f(cs) 1 f(c)f(a) = f(cs) 1 f(cs)f(s) 1 f(a) = f(s) 1 f(a) 2 (b) s S, a A (i), (ii) t S, b A i(a)i(s) 1 = i(t) 1 i(b) i(ta bs) = 0, (iii) u S u(ta bs) = 0 s = ut S, a = ub A s a = a s (c) s S, a A, as = 0 i(a)i(s) = i(as) = 0, (i) i(a) = 0 (iii) s S s a = 0 3 s, s S, a, a, c, c A, cs = c s S, ca = c a (i) 1 i(s) 1 i(a) = i(cs) 1 i(ca) = i(c s ) 1 i(c a ) = i(s ) 1 i(a ) s, s S, a, a A, i(s) 1 i(a) = i(s ) 1 i(a ) (b) b A, s S ts = bs (i) i(ta) = i(ts)i(s) 1 i(a) = i(bs )i(s ) 1 i(a ) = i(ba ) (c) u S uta = uba S uts = ubs c = ut, c = ub cs = c s S ca = c a
12 7 4 3 5 4 φ φ Ã B φ i = f (i), (ii), Ã α = i(s) 1 i(a) (s S, a A) f(a) = φ(i(a)) = φ(i(s)i(s) 1 i(a)) = φ(i(s))φ(i(s) 1 i(a)) = f(s)φ(α) φ(α) = f(s) 1 f(a) φ Ã B φ i = f, φ (α) = f(s) 1 f(a), φ = φ φ φ Ã B φ(i(s) 1 i(a)) = f(s) 1 f(a) (s S, a A) 3 s, s S, a, a A, i(s) 1 i(a) = i(s ) 1 i(a ) 1 c, c A cs = c s S, ca = c a, 1 f(s) 1 f(a) = f(cs) 1 f(ca) = f(c s ) 1 f(c a ) = f(s ) 1 f(a ) φ well-defined φ φ i = f φ s, s S, a, a A, α = i(s) 1 i(a), α = i(s ) 1 i(a ) s, s (b) c S, c S s = cs = c s S 1 α = i(s ) 1 i(ca), α = i(s ) 1 i(c a ) α+α = i(s ) 1 i(ca+ c a ) f(s ) 1 f(ca) = f(s) 1 f(a) = φ(α), f(s ) 1 f(c a ) = f(s ) 1 f(a ) = φ(α ) φ(α + α ) = f(s ) 1 f(ca + c a ) = φ(α) + φ(α ) a, s (b) s S, a A s a = a s i(a)i(s ) 1 = i(s ) 1 i(a ) f(a)f(s ) 1 = f(s ) 1 f(a ) φ(αα ) = φ(i(s) 1 i(a)i(s ) 1 i(a )) = φ(i(s) 1 i(s ) 1 i(a )i(a )) = φ(i(s s) 1 i(a a )) = f(s s) 1 f(a a ) = f(s) 1 f(s ) 1 f(a )f(a ) = f(s) 1 f(a)f(s ) 1 f(a ) = φ(α)φ(α ) φ(1) = φ(i(1)) = f(1) = 1 11, [1], [2] (, Lie,, ) [1] [3] ( ) A, S S A ( ) (a) (s, a) (s, a ) c, c A cs = c s S ca = c a (b) (s, a) (s, a ) t S t(s a sa ) = 0 A [1] S 1 A (b) S 1 A (s, a) (s, a ) (s, a) (s, a ) (s, a) (s, a ), c, c A cs = c s S, ca = c a t = csc s S
8 1 A t(s a sa ) = csc s s a csc s sa = ss (c s ca csc a ) = 0 (s, a) (s, a ) (s, a) (s, a ) (s, a) (s, a ) (s, a) (s, a ), t S t(s a sa ) = 0 c = ts, c = ts c, c S, cs = ts s = tss = c s S, ca = ts a = tsa = c a (s, a) (s, a ) 13 Ore 12 (Ore ) ( ) A Ore (Ore domain), A 0, 0 a, b A Aa Ab 0, aa ba 0 [4] (Ore ) A Ore, S = A {0}, S, D = S 1 A D 1 D, A 2 D = { s 1 a a, s A, s 0 } = { bt 1 b, t A, t 0 } D S = A {0} (b), (c) (b) a A, s S a A, s S s a = a s a = 0 s = 1, a = 0 a 0 A Ore Aa As 0 a, s S s a = a s (c) a A, s S as = 0, A a = 0 1 S 1a = 0 D = S 1 A 1, 2 1 A 0 S, A D = S 1 A, D = { a 1 b a, b A, a 0 } a, b A, a 1 b 0 b 0 a 1 b b 1 a D 2 D = S 1 A D s 1 a (a, s A, s 0) a = 0 s 1 a = 0 = 01 1 a 0 A Ore aa sa 0 b, t S at = sb s 1 a = bt 1 D [2] 5 [5] (Ore ) A ( ), K, K A (A K ) A K K A (a) A 0 (b) A K A 0 A 1 A 2 A i A j A i+j A = i=0 A i (A i A )
14 9 (c) n C dim A i Ci n (i ) ( ) f(i) g(i) (i ) lim i f(i)/g(i) = 1) A Ore A Ore A Ore 0 a, b A Aa Ab = 0 aa ba = 0 A = i=0 A i a, b A i0 i 0 A i A j A i+j A i a, A i b A i+i0 Aa Ab = 0 A i a + A i b A dim K A i a = dim K A i b = dim K A i Ci n (i ) dim K A i+i0 C(i + i 0 ) n Ci n (i ), dim K A i+i0 dim K (A i a + A i b) = dim K A i a + dim A i b 2Ci n (i ) aa ba = 0 13 K Ore Weyl Lie Ore 14 [6] ( ) A ( ), K, K A (A K ) (i) S F A (F ), a A, f F n Z >0, α 0,, α n K n α i f n i af i = α 0 f n a + α 1 f n 1 af + α 2 f n 2 af 2 + + α n af n = 0, α 0 0 i=0 (ii) S S 14 α 0 0 α n 0 [1] (b), (c) (c) (ii) (b) f F, a A, (i) t = f n S, b = α 1 0 (α 1 f n 1 a + α 2 f n 2 af + + α n af n 1 ) ta = bf S f 1,, f m S s = f m f m 1 f 1 m a A t S, b A ta = bs m = 1 m 1 t S, b S t a = b f m 1 f 1 m = 1 t S, b A t b = bf m t t a = t b f m 1 f 1 = bf m f m 1 f 1 = bs m
10 15 symmetrizable Kac-Moody Lie g U q (g) U q (n ) Chevalley generators {f 1,, f l } S 2 (2007 2 16 ) [1], [2] Goodearl and Warfield [GW], Chapter 10 McConnel and Robson [MR], Chapter 2 [GM1] Gelfand, S I and Manin, Yu I, Methods of Homological Algebra, Springer, 1996 [GM2] Gelfand, S I and Manin, Yu I, Homological Algebra, Algebra V, Encyclopaedia of Mathematical Sciences, Volume 38, Springer-Verlag, 1994 [GW] Goodearl, K R and Warfield, R B, Jr, An introduction to noncommutative Noetherian rings Second edition, London Mathematical Society Student Texts, 61, Cambridge University Press, Cambridge, 2004, xxiv+344 pp [MR] McConnell, J C and Robson, J C, Noncommutative Noetherian rings, With the cooperation of L W Small, Revised edition, Graduate Studies in Mathematics, 30, American Mathematical Society, Providence, RI, 2001, xx+636 pp [I] [T], B,,,, 1986, 1997, 3, 17,, 1998