2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =

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2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I A x I a A ax I. A (prime ideal) I, I A, a, b A ab I a I b I.. A Spec(A). Spec(A) := {p A }. 1.1.1. k, k 1 k[x]. k k[x] r, k f k[x] f = a 0 i=1 (x a i) er 1. Spec(k[x]) = {(0)} {(x a) a k}. = (0) Spec(k[x]) = { } k.. A. Spec(A), f A D(f) := {p Spec(A) f / p} A (spectrum), Spec(A). Spec(A) Zariski. Spec(A), Λ l Λ D(f l ). 1.1.2. A I D(I) Spec(A). D(I) := {p Spec(A) I p}. (2) D(I) Zariski. *1 2018/11/08, ver. 0.5.

2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) = D(IJ). IJ := {a 1 b 1 + + a n b n a k I, b k J} A, I J ( ). (4) A {I l l Λ} D( l Λ I l) = l Λ D(I l ). l Λ I l := { l a l a l I l, }. 1.1.3. k 1 k[x]. 1.1.1 Spec(k[x]) = { } k. f k[x] D(f). (i) 0 k[x] D(0) =, (ii) a k \ {0} D(a) = Spec(k[x]) = { } k. (iii) a k D(x a). (0) (x b) (x b) D(x a) (x b) / x a b a. D(x a) = { } (k \ {a}). r (iv) f = a 0 i=1 (x a i) er D(f) = r i=r D((x a i) ei ) = r i=1 D(x a i). (iii) D(f) = { } (k \ {a 1,..., a r }). Spec(k[x]),,, 3., Spec(k[x]) Hausdorff ( T 2 ). = (0) Spec(k[x]). 1.1 ( ). k.. (1) k[x] (0) := {g(x)/f(x) f(x), g(x) k[x], f(0) 0} Spec(k[x] (0) ) = {(0), (x)}. Spec(k[x] (0) ) = {, 0},, { }, {, 0} 3. (2) k 2 k[x, y] 3. (0). f (f). (a, b) k 2 (x a, y b). 1 A Spec(A), φ : A B. 1.1.4. φ : A B, a φ : Spec(B) Spec(A), p φ 1 (p) well-defined, f A ( a φ) 1 (D(f)) = D(φ(f)), Zariski. a φ φ.

2018/10/04 IV/ IV 3/12 1.2.. A (multiplicatively closed subset) *2 S A. (i) 1 A S. (ii) f 1, f 2 S f 1 f 2 S. 1.2.1. S A, M A. (1) S M (s, m) (s, m ) t S, f(s m sm ) = 0 S 1 M := (S M)/ A. (s, m) S M m/s. + : S 1 M S 1 M S 1 M m/s + m /s := (s m + sm )/(ss ), well-defined, S 1 M.. : A S 1 M S 1 M a.(m/s) := (a.m)/s *3, well-defined, S 1 M A. A S 1 M, M S (localization) S 1 M. (2) ψ M : M S 1 M ψ M (m) := m/1 A, well-defined, A. ψ M. 1.2.2. S A. (1) A A S 1 A, : S 1 A S 1 A S 1 A a/s a /s := (aa )/(ss ), well-defined, S 1 A. S 1 A A S. (2) 1.2.1 (2) A ψ A : A S 1 A (1). ψ A. 1.2.3. S A, M A.. : S 1 A S 1 M S 1 M a/s.(m/s ) := (a.m)/(ss ), well-defined, S 1 M S 1 A.. *2. *3 A M A M M (a, m) a.m..

2018/10/04 IV/ IV 4/12 1.2.4. S 1 A : φ : A B φ(s) B, φ : S 1 A B φ = φ ψ A. φ. 1.2 (tensor product) (short exact sequence). 1.2 ( ). A, S A. M A. ( 1.2.4). (1) S 1 A A M S 1 M. (2) A 0 M 1 M 2 M 3 0 S 1 A. 0 S 1 M 1 S 1 M 2 S 1 M 3 0. (3) S 1 S 2 A. 1.2.2 (2) A S2 1 A S 1 S 1, S 1 S 1 2 A, (S 1S 2 ) 1 M (S 1) 1 (S2 1 M). 1.2.5. 1.2 (1) (2), M S 1 A A M. A B, M B A M, (flat). ψ A : A S 1 A. D(f).,. 1.2.6. S 1 A (1) S 1 A, A I I(S 1 A). (2) Spec(S 1 A) = {p(s 1 A) p Spec(A), p S = }. 1.2.7. A. f A, S := {1, f, f 2,...} A. S S 1 A A f. A f := A[x]/(fx 1)., 1 A[x] fx 1 A[x] (fx 1). 1.2.8. A, f A, A f {1 A, f, f 2,...} ( 1.2.7 ). ψ : A A f., ψ a ψ : Spec(A f ) Spec(A) ( 1.1.4 ), a ψ(spec(a f )) = D(f), a ψ : Spec(A f ) D(f). 1.3 Hilbert. A, I. 1.1.2 D(I) V (I). V (I) := Spec(A) \ D(I) = {p Spec(A) p I}.

2018/10/04 IV/ IV 5/12 1.3 ( ). φ : A B. A/ Ker φ Im φ B. (1) φ, Spec B V (Ker φ) Spec(A), a φ V (Ker φ). a φ ( )1 1. (2) a φ V (Ker φ) = Spec A. Ker φ = p Spec(A) p. Hilbert,. 1.3.1 ( ). A, I. f A f V (I) = 0 f I. I.. A I, I := {f A f m I, m Z >0 } I (radical). f V (I) = 0. 1.3.2. A p, A \ p A p := (A \ p) 1 A, A p, Spec(A) p (local ring). 1.2.6.. Spec(A p ) = {qa p q Spec(A), q p}. 1.4 (2) A p (maximal ideal). B, B. m (quotient ring) *4 B/m. 1.4 ( ). A.. (1) A S, A \ S,. (2) Spec(A) p A p pa p. 1.3.3. 1. 1.4 (2) Spec(A) p A p,.. A, p Spec(A). (1) k(p) Spec(A) p (residue field) *5. k(p) := A p /pa p (2) f A p, f f. A A p k(p), f f/1 A f := (f/1 A mod pa p ). *4. *5 (field of fractions) (quotient field).,.

2018/10/04 IV/ IV 6/12 f A T Spec(A), p T f 0, f T = 0. f V (I) = 0. 1.3.1,. 1.3.4. A I p V (I) p = I. 1.3.4. 1.3.1 1.3.4. 1.3.5. A I, J (1) Spec(A) = A = 0. (2) V (I) V (J) I J. (3) V (I) = V (J) I = J. 1.4 Zariski 1.1.3, A Spec(A) Hausdorff.,.,, D(f) Spec(A). Zariski.. A f A, D(f) (quasi-compact) *6. D(f) = i I D(f i ), J I i J D(f i ).. {f i A i I} a := i I (f i) V (f) = V (a). Hilbert ( 1.3.1) f m a m Z 1. f m = i J f ig i J I g i A. f m i J (f i), V (f) = V (f m ) i J V (f i ). D(f) = i J D(f i ). 1. Hausdorff 1, 1.4.1. A, p Spec(A). 1 {p} Spec(A), {p} = V (p)... {p} = F : p F = I p V (I) = V (p).. X x X, {x}, (closed point). 1.4.1. p Spec(A) p.. *6 Hausdorff.

2018/10/04 IV/ IV 7/12. X (irreducible), X = F 1 F 2 X = F 1 X = F 2. 1.4.2. I A, V (I) I. 1.5 ( ). (1) Hilbert 1.3.5 1.4.2. (2) {p} Spec(A),.. 1.4.3. X x, y X, (1) y {x}, x y (generalization), y x (specialization). (2) {x} = X, x X (generic point). 1.6 ( ). 1.4.3 X = Spec(A), x = p, y = q, *7. {x} y p q. 1.6 {x} = {y} p = q.. Spec(A) 1. 1.7 ( ). Spec(A) 1, 2. (1) Spec(A) T 0, 2 x, y Spec(A) U x U y / U. (2) T 0 1. 1.5..,.. X. (1) X Open(X) (Sets) F, X (presheaf of sets)., F (I) (II). (I) X U, F(U). (II) U 1 U 2 r U1,U 2 : F(U 1 ) F(U 2 ), 2. (a) id U : U = U r U,U = id F(U) : F(U) = F(U). (b) U 1 U 2 U 3 r U2,U 3 r U1,U 2 = r U1,U 3 : F(U 1 ) F(U 3 ). r U1,U 2 (restriction map). *7 ver. 0.5 typo.

2018/10/04 IV/ IV 8/12 (2) X F (sheaf of sets). (III) U = i I U i s, t F(U) i I r U,Ui (s) = r U,Ui (t) s = t. {s i F(U i ) i I} U i U j i, j I r Ui,U i U j (s i ) = r Uj,U i U j (s j ), s F(U) i I s i = r U,Ui (s). (locality), (gluing). (3), (Sets),, (Grp), (Mod), (Ring),, ((pre)sheaf of groups, modules, rings).., s V := r U,V (s). F r F U,V. stalk germ *8. (inductive limit) (direct limit) lim *9.. X F x X, (1) F x := lim U x F(U) F x stalk., x. (2) F x x germ... x X F germ, U x s F(U) (U, s), U, s. (U, s) (V, t) W x W U V s W = t W.. Spec(A). Spec(A) {D(f) f A}. 1.2.8 D(f) Spec(A f ). 1.5.1. Spec(A) Ã Ã(D(f)) = A f.. A Spec(A) Ã (Spec(A), Ã) A (affine scheme). Ã (structure sheaf). 1.5.1. D(f) = D(g) A f A g. D(f) = D(g) = D(fg), A f A fg a/f m ag m /(f m g m )., Hilbert 1.4.2 (3) D(f) = D(g) (f) = (g). D(f) D(g) = D(fg), r D(f),D(fg) : A f A fg a/f m ag m /(f m g m ). *8 stalk, germ,. *9 (colimit), colim.

2018/10/04 IV/ IV 9/12 U Spec(A) U = i I D(f i ). s = (s i ) i I, s i A fi, s i D(f i f j ) = s j D(f i f j ) i, j I. U = j J D(g j ) t = (t j ) j J. s t s i D(f i g j ) = t j D(f i g j ) i I, j J, Ã(U) := { s}/.., U = D(f) D(f) = i I D(f i ). 2. a A f i I a D(f i ) = 0 a = 0 A f. {a i A fi i I} i I, j J a i D(f i f j ) = a j D(f i f j ), a A f a i = a D(f i ). [H77, p.71, Proposition 2.2, Chap.II 2] * 10. A Spec(A) Ã, A Spec(A) Ã. Ã 1.5.2. X, O X. X O (sheaf of O-modules) M, X 2. U X M(U) O(U). O(U)., ru,v O O, rm U,V M, a O(U) m M(U) ru,v O (a).rm U,V (m) = rm U,V (a.m). Ã 1.. A M f A, S := {1, f, f 2,...} M S 1 M M f. 1.2.3 M f A f. 1.5.3. M A. Ã M M(D(f)) = M f.. 1.5.1, U = i I D(f i ) M(U) := {(m i ) i I m i M fi, m i D(f i f j ) = m j D(f i f j )}/. Ã M f A f. Ã M stalk. p Spec(A) A p ( 1.3.2),. p Spec(A), S := A \ p A M S 1 M M p. *10 [H77].

2018/10/04 IV/ IV 10/12 1.5.4. p Spec(A) Ã Ã M stalk (Ã) p = A p, ( M) p = M p. 1.3.3 A p ( )... X O (X, O) (ringed space). (X, O), x X stalk O x, (locally ringed space). 1.8 ( ).,.. 1.6, ( ) ( )... 1.6.1. X. (1) F G X. (morphism of presheaves) θ : F G, F, G., U X θ(u) : F(U) G(U), U V r G U,V θ(u) = θ(v ) rf U,V. (2) F G X. (morphism of sheaves) θ : F G, F, G, θ. 1.6.2. 1.6.1, x X, stalk θ x : F x G x, θ θ x (functorial). id : F = F id x, E σ F θ F (θ σ) x = θ x σ x. 1.6.3. 1.6.2, (natural) θ x. 1.6.4. f : X Y, F X. Y f F F f (direct image). (f F)(U) := F(f 1 (U)). 1.6.5. 1.6.4, x X, y := f(x), stalk (f F) y F x ( 1.6.3 ).. stalk (f F) y = lim(f F)(U) = lim F(f 1 (U)). U y U y f 1 (U) x, F(f 1 (U)) F x., (f F) y F x.,.

2018/10/04 IV/ IV 11/12. (X, O X ) (Y, O Y ). (morphism of ringed spaces) (X, O X ) (Y, O Y ), f : X Y Y f : O Y f O X (f, f ). (f,, f ), f f. 1.6.6. (1) (X, O X ) (Y, O Y ). (morphism of locally ringed spaces) (X, O X ) (Y, O Y ) (f, f ), x X f x : O Y,y (f ) y (f O X ) y O X,x. (f ) y 1.6.2, 2 1.6.5. (2) (isomorphism of locally ringed spaces)... A m, B n. φ : A B φ(m) n (local).. (morphism of affine schemes). (isomorphism of affine schemes). φ : A B a φ : Spec(B) Spec(A), 1.6.7.. φ : A B, f A. φ A f B φ(f) φ f. 1.6.7. φ : A B., f A ( a φ, φ) : (Spec(B), B) (Spec(A), Ã) φ(d(f)) : Ã(D(f)) = A f ( ( a φ) B) (D(f)) = B(D(φ(f))) = Bφ(f) φ f. φ ( a φ, φ).. φ, Ã., U Spec(A) U = i D(f i ), D(f i ) φ fi, φ(u) : Ã(U) ( a φ) B(U)., f, g A Ã(D(f)) A f φ f B φ(f) ( ( a φ) B) (D(f)) r D(f),D(fg) r D(f),D(fg) Ã(D(fg)) A fg φ fg B φ(fg) ( ( a φ) B) (D(fg)). ( a φ, φ).

2018/10/04 IV/ IV 12/12 p Spec(B), q := ( a φ)(p) = φ 1 (p), stalk ψ := φ p : Ãq B p. 1.5.4 Ãq = A q, ψ A q = lim A f lim B φ(f) B p f / q, φ., a q φ(a) p f / q, ψ(qa q ) pb p. ψ.. 1.6.8. (f, θ) : (Spec(A), Ã) (Spec(B), B), φ : B A ( a φ, φ).. [H77, p.73, Proposition 2.3 (c)]. 1.6.7 1.6.8, 1.6.9. Spec. [H77] R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977;,, 1,2,3,, 2008..

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