March 4, R R R- R R

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1 March 4, R R R- R R Hom Hom Hom Hom M M M

2 2 1. R R-. G G + R (1) (2) 1 (3) a(b + c) = ab + ac, (b + c)a = ba + ca a, b, c R 0 1 (endomorphism) 1.1. G End(G), g End(G) + g g ( + g)(x) = (x) + g(x) (g)(x) = (g(x)) End(G) G R M : R M M R M (1) r (a + b) = r a + r b (2) (r + s) a = r a + s a (3) (rs) a = r (s a) (4) 1 a = a (r, a) r a : R M M (ϕr)(a) = r a ϕ : R End(G) ϕ : R End(G) : R M M R R- (let R-module) R M 1.1. (1) R 0 R M 0 M 0 R x = 0 M x M (2) R 1 R 1 R x M M ( 1 R )x 1. R- 0 r0 = 0 0 1

3 3 2. ( ) M Z- x + + x (n ) n > 0 nx = 0 n = 0 (( n)x) n < 0 x M, n Z 3. R R- R 4. R R R R- R-

4 R- R-. ( ) M N M (1) 0 N (2) x, y N x + y N (3) x N x N R- M N R x N, r R rx N N M R- N M (3) 0.1(2) (2) M R- 5. M R- M M R- M {0 M } M R R R- R (1) Z ( ) (2) n n M n (K) 1 0 n M n (K) M R N M ( ) M/N R r(m + N) = (rm) + N r R, m M R- N R- R- M R- N 1, N 2 N 1, N 2 N 1 + N 2 = {x 1 + x 2 x 1 N 1, x 2 N 2 } 1.2. N 1 + N 2 M R- R- M R- N i (i I) N i = x i x i N i i I N i (i I) i I, 1.3. i I N i M R- R- M R- N i (i I) M R i I N i

5 5 R- M S S R- S S = Rs = N s S S N M S R R-. R- M R- N : M N R- (1) (x + y) = (x) + (y) ( ) (2) (rx) = r(x) 1.1. (2) (x) (x) (rx) = r(x) 7. N R- M R- (1) N M (2) M M/N R- : M N Ker = {x M (x) = 0} M Im = {(x) x M} N (kernel) (image) 1.5. Ker Im R- R- : M N (1 1 ) Ker = 0 ( ) Im = N 1.2. R- : M N 1 : N M R- R- R- (isomorphism) R- M, N M N M N ( R-) M = N 8. id M : M M, x x 1.3 ( ). : M N R- R- M/ Ker Im x + Ker (x) 1.6. (1) K, H M (H + K)/K = H/(H K) (2) K L M M/L = (M/K)/(L/K) 1.7. A B g C (1) g (2) g g

6 {a, b} (x, y) (x, y) = {{x}, {x, y}} 2-tuple S 1, S 2 {(x, y) x S 1, y S 2 } S 1, S 2 S 1 S 2 n 3 n-tuple π 1 :S 1 S 2 S 1, (x, y) x π 2 :S 1 S 2 S 2, (x, y) y (x 1,..., x n 1, x n ) = ((x 1,..., x n 1 ), x n ) () R- P π i : P M i (i I) R- N i : N M i i = π i ϕ ϕ : N P 1 (P, π i ) R- M i (direct product) R- S ι i : M i S (i I) R- N i : M i N i = ϕι i ϕ : S N 1 (S, ι i ) R- M i (direct sum) M i (i I) R- i I M i π j : i I M i M j (j I) x, y i I M i x + y π j (x + y) = π j (x) + π j (y) j I π j (rx) = rπ j (x) j I R i I M i R- π j R- x j π j (x) = 0 i I M i R- R- i I M i R- M j x j x 0 ι j : M i i I M i R- j R- i I M i ι j ( i I M i, ι j ) M i

7 (P, π i ) (P, π i ) R- M i (i I) : P P π i = π i 2.2. (S, ι i ) (S, ι i ) R- M i (i I) : S S ι i = ι i 2.1. R- N i (i I) M : N i M, (x i ) x i (1) Im = N i (2) Ker = 0 N j i j N i = 0 j I j I N j i j N i = 0 N i M N i N i ( ) g : M M/N i, x (x + N i ) (1) Ker g = N i (2) I Im g = M/N i N j + i j N i = M j I

8 C n+1 C n C n (chain complex) 2 R- L M g N g = 0 Im Ker g Im = Ker g M (exact) 9. (1) 0 M 0 M M = 0 (2) 0 M g N M g (3) L M 0 M R-... M n 1 M n M n+1... R- (exact sequence) 0 M 1 M 2 M 3 0 (short exact sequence) 10. R- M N N 0 N M M/N R- ( R- ) (diagram) (commutative) 11. C g = qp A B p g q D 2

9 9 L 7 h g = h 12. M g N 1 2 M 1 M2 M3 h 1 h 2 h 3 g 1 g 2 N 1 N2 N3 ( ) g 1 h 1 = h 2 1, g 2 h 2 = h 3 2 ( ) g 2 g 1 h 1 = g 2 h 2 1 = h (5 ) M 1 M2 M3 M4 M5 h 1 h 2 h 3 h 4 h 5 g 1 g 2 g 3 g 4 N 1 N2 N3 N4 N5 (1) h 1 h 2, h 4 h 3 (2) h 5 h 2, h 4 h 3 (3) h 1, h 2, h 4, h 5 h 3.(1) Ker h 3 = 0 Ker h 3 x h 4 3 (x) = g 3 h 3 (x) = 0 h 4 3 (x) = 0, x Ker 3 = Im 2 x = 2 (y) y M 2 g 2 h 2 (y) = h 3 2 (y) = h 3 (x) = 0 h 2 (y) Ker g 2 = Im g 1 h 2 (y) = g 1 (z) z N 1 h 1 z = h 1 (u) u M 1 h 2 1 (u) = g 1 h 1 (u) = g 1 (z) = h 2 (y) h 2 y = 1 (u) x = 2 (y) = 2 1 (u) = 0 (2) x N 3 h 4 g 3 (x) = h 4 (y) y M 4 h 5 4 (y) = g 4 h 4 (y) = g 4 g 3 (x) = 0 h 5 4 (y) = 0, y Ker 4 = Im 3 y = 3 (z) z M 3 g 3 h 3 (z) = h 4 3 (z) = h 4 (y) = g 3 (x), x h 3 (z) Ker g 3 = Im g 2 x h 3 (z) = g 2 (u) u N 2 h 2 u = h 2 (v) v M 2 h 3 2 (v) = g 2 h 2 (v) = g 2 (u) = x h 3 (z) x = h 3 (z) + h 3 2 (v) = h 3 (z + 2 (v)) N 3 = Im h 3 (3) (1),(2)

10 R- : M N Ker Im R- Coker Ker = {x M (x) = 0} Im = {(x) x M} Coker = N/ Im ϕ : X M ϕ = 0 ϕ = 0 φ : N X φ = 0 φ = k : K M (1) k = 0 (2) h = 0 h : X M h = kϕ ϕ : X K Ker M c : N C (1) c = 0 (2) h = 0 h : N X h = ϕc ϕ : C X N Coker 4.1. R- L M g N (1) g Im = Ker g (2) g (3) g g R- : L M h = 1 L h : M L h h End(M) Coker M R- g : M N gk = 1 N k : N M

11 11 k kg End(M) Ker g M 4.2. R- 0 L M g N 0 g

12 (category theory 3 ) (category) (object) (morphism,arrow) (source,domain) (target,codomain) A B A B A B A (identity) A id A 1 A g D B = C A g D A B C A B id A = id B = A B g C h D h(g) = (hg) C A B C(A, B) 13. R- R- A B g = id A, g = id B B g A (isomorphism) 5.1. ( x, y x = y x = y) (monomorphism) ( x, y x = y x = y) (epimorphism) 5.2. A B g C (1) g g (2) g g (3) g (4) g g R- 3 category theory

13 Z Q Q R ϕ, ψ n Z ϕ(n) = ψ(n) n 0 ϕ( 1 n ) = ϕ(n) 1 = ψ(n) 1 = ψ( 1 n ) ϕ( 1 n ) = ψ( 1 n ) ϕ(m) = ψ(m) (m Z) ϕ( m n ) = ψ( m n ) m n

14 C D F F 4 C A B D F (A) F () F (B) F (id A ) = id F (A) F (g) = F (g)f () F C D (unctor) (covariant unctor) C A B D F (A) F () F (B) F (id A ) = id F (A) F (g) = F ()F (g) F C D (contravariant unctor) 15. A B ( ) A B (orgetul unctor) R- R R- 6. Hom 6.1. Hom. C A C X Y C(A, X) C(A, Y ) g g C(A, ) C C X C X Y (X Y ) X C(A, X) C(A, ) C X ( ) C(A, X) C(A,) C(A, Y ) C(Y, A) C(X, A) g g C(, A) C C X C X Y (X Y ) X C(X, A) Y ( ) C(Y, A) C(,A) C(X, A) 4

15 15 C(, A) 6.2. Hom. C R- C(A, ) C(, A) Hom(A, ) Hom(, A) M, N R-, g Hom(M, N) + g ( + g)(x) = (x) + g(x) ( + g)(x + y) = (x + y) + g(x + y) = ((x) + (y)) + (g(x) + g(y)) = ((x) + g(x)) + ((y) + g(y)) = ( + g)(x) + ( + g)(y) ( + g)(rx) = (rx) + g(rx) = r(x) + rg(x) = r((x) + g(x)) = r( + g)(x) + g Hom(M, N) (1) 0(x) = 0 (2) (( +g)+h)(x) = ((x)+g(x))+h(x) = (x)+(g(x)+h(x)) = ( +(g+h))(x) ( + g) + h = + (g + h) () (3) ( )(x) = (x) (4) ( + g)(x) = (x) + g(x) = g(x) + (x) = (g + )(x) + g = g + () Hom(M, N) 16. m, n Hom(Z/Zm, Z/Zn) = Z(n/(m, n))/zn = Z/Z(m, n) (m, n) m n 17. m 6.1. R- 0 L (1) R- A 0 Hom(A, L) Hom(Z/Zm, Z) = 0 M (2) R- A 0 Hom(N, A) Hom(A,) Hom(A, M) Hom(g,A) Hom(M, A) g N Hom(A, g) Hom(, A) Hom(A,g) Hom(A, N) Hom(,A) Hom(L, A)

16 Hom. N i ( i ) Hom(M, N i ) i = π i i Hom(M, N i ) F (( i )) 6.2. F : Hom(M, Ni ) Hom(M, N i ). ( i ), (g i ) Hom(M, N i ) = F (( i )), g = F ((g i )) π i i + g i = π i + π i g = π i ( + g) i F (( i ) + (g i )) = F (( i + g i )) = + g = F (( i )) + F ((g i )) F Hom(M, N i ) i = π i F (( i )) = F = 0 i = 0 i F M i ( i ) Hom(M i, N) i = ι i i Hom( M i, N) G(( i )) 6.3. G : Hom(Mi, N) Hom( M i, N)

17 M R- N R- M N F (x 1 + x 2, y) (x 1, y) (x 2, y) x 1, x 2 M, y N (x, y 1 + y 2 ) (x, y 1 ) (x, y 2 ) (xr, y) (x, ry) x M, y 1, y 2 N x M, y N, r R F G F/G M, N (tensor product) M N M R N (x, y) M N x y M N n i x i y i (, n i Z) {x y x M, y N} M N M N (x 1 + x 2 ) y = x 1 y + x 2 y x (y 1 + y 2 ) = x y 1 + x y 2 xr y = x ry 7.1. M N 0 y = 0 x 0 = R- M R- N M N A ϕ ϕ(x 1 + x 2, y) = ϕ(x 1, y) + ϕ(x 2, y) ϕ(x, y 1 + y 2 ) = ϕ(x, y 1 ) + ϕ(x, y 2 ) ϕ(xr, y) = ϕ(x, ry) ϕ (balanced mapping) M N M N ϕ : (x, y) x y 7.1. ϕ : M N M N, (x, y) x y ψ : M N A ψ = ϕ : M N A 8. M 8.1. M. M R- M R- R- N M N : N 1 N 2 R- M N 1 1 M ϕ 2 M N 2 M N2 ϕ 1 M N 1 M N1 ψ 1 M ϕ 2 M N 2 M N2

18 18 ψ ϕ 1, ϕ 2 ψ : x y x (y) ψ 1 M M N 1 N 2 g N 3 ϕ 1 M N 1 M N1 1 M 1 M ϕ 2 M N 2 M N2 1 M g 1 M g ϕ 3 M N 3 M N3 (?) 1 M (g) = (1 M g)(1 M ) 1 N : N N 1 M 1 N = 1 M N M R M M R- R- M N 1 N 1 N 2 1 M M N 2 g N M g M N M (M N 2 )/ Im(1 M ) M N 3 I = Im(1 M ) (1 M g)(1 M ) = 1 M (g) = 1 M 0 = 0 α(m x + I) = m g(x), m M, x N 2 α : (M N 2 )/I M N 3 β 0 : M N 3 (M N 2 )/I y N 3 g(x) = y x N 2 (g ) m y m x + I g(x) = g(x 1 ) = y (z, w N 2 ) m x m x 1 = m (x x 1 ) I x β 0 β 0 β(m y) = m x + I β : M N 3 (M N 2 )/I {m y m M, y N 3 } αβ M N 3 βα (M N 2 )/I M N 3 = (M N2 )/I Ker(1 M g) = Im(1 M ) 1 M g

19 R M R- M R M, (m, r) mr 5 : M R M, m r mr. m M (m 1) = m ( m i r i ) = mi r i ( m i r i ) = 0 m i r i = (m i r i 1) = ( m i r i ) 1 = a, b Z- Z a Z Z/Za 0 Z/bZ Z/bZ Z/bZ ( a) Z/bZ Z/bZ Z/Za 0 Z/bZ ( a) (az + bz)/bz = (a, b)z/bz Z/bZ Z/Za = Z/(a, b)z a b Z/bZ Z/Za = ( ) M N i = (M N i ) i I i I 8.3. M R- N i R- N i N i ι i 1 M N M ι i i M ( i I N ) i M N i. 1 M ι i i : M N i X ( ) ϕ : M N i X i I ϕ(x, (u i ) i I ) = i (x u i ) i I ϕ ϕ : M ( N i ) X (x (u i ) i I ) = i I i (x u i ) i = (1 M ι i ) i I x y (x M, y Im ι i ) M ( N i ) 5 R-

20 R- M R- 0 Hom(M, A) 0 A B Hom(M,) Hom(M, B) g C 0 Hom(M,g) Hom(M, C) 0 R- (projective) ( 6.1(1)) R- g B C Hom(M,g) Hom(M, B) Hom(M, C) M ψ ϕ g B C g ϕ ψ 9.1. i I M i i I M i. g : B C R- ϕ : i I M i C R- i I M i gψ i = ϕι i ψ i : M i B ψ i = ι i ψ ψ ϕ gψ gψι i = gψ i = ϕι i gψ = ϕ P = M N ι M, π M M M g : B C ϕ : M C P gψ = ϕπ M ψ : P B gψι M = ϕπ M ι M = ϕ1 M = ϕ Hom(M, g) 9.3. R- R. M = R ϕ(1 R ) g b ψ(r) = rb ψ ψ ϕ = gψ R Z- Z/nZ (n Z, n > 1) 9.4. (1) M (2) M. (1) (2). M : F M F B = F, C = M, g = ϕ = 1 M ψ = 1 M ψ : M F F = M Ker (2) (1)

21 R- M R- 0 Hom(C, M) 0 A B Hom(g,M) Hom(B, M) g C 0 Hom(,M) Hom(A, M) 0 R- (,,injective) ( 6.1(2)) R- A B Hom(,M) Hom(B, M) Hom(A, M) A B ϕ ψ M ϕ ψ i I M i i I M i 9.6. R- (Baer) 9.7. R M R- (1) M (2) R I M R M. (1) (2) (2) (3). R- A B : A M B i A i (A i, i ) A i A A i B (A i, i ) (A j, j ) A i A j i = j Ai (A i, i ) (A j, j ) ( 0, A 0 ) A 0 = B B \ A 0 b {r R rb A 0 } M ϕ(r) = 0 (rb) ϕ ϕ(r) = rm m M g : A 0 + Rb M, g(a + rb) = 0 (a) + rm (g, A 0 + Rb) ( 0, A 0 ) Q Z-

D 24 D D D

D 24 D D D 5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6