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ADDITIONAL NOTES in ALGEBRAIC GEOMETRY, II 1. Basic Properties

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ADDITIONAL NOTES in ALGEBRAIC GEOMETRY, II 1. Basic Properties ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II ANDREAS LEOPOLD KNUTSEN Abstract. These notes supplement [Hart, Chapter II] and form part of the syl- labus in the course MAT321-Introduction to sheaves and schemes, taught at the University of Bergen, Spring 2016. 1. Basic properties of coverings and morphisms of schemes This section supplements [Hart, II, x2]. The following summarizes basic properties of an affine scheme that in particular are used in the proof of [Hart, II, Prop. 3.2]: Lemma 1.1. Let X = Spec A be an affine scheme. Then the following holds: (a) The open sets of the form D(α), for α 2 A, form a basis for the topology and (D(α); OX jD(α)) ' Spec Aα: (b) For any affine subscheme Spec B ⊆ X, we have (1) D(α) \ Spec B = D(α); where α = '(α), with ' : A ! B corresponding to the inclusion morphism Spec B ! Spec A as in the proof of [Hart, II,Prop. 2.3] (see also below). In particular, if D(α) ⊆ Spec B, then Aα ' Bα. (c) We have (2) Spec A = [i2I D(αi) () (fαi j i 2 Ig) = A: (d) Given (2), we have Spec A = D(α1) [···[ D(αr); for finitely many αi. In particular, sp(X) is quasi-compact ([Hart, II, Exc. 2.13(b)]). (e) sp(X) is not necessarly noetherian, but is noetherian if A is noetherian ([Hart, II, Exc. 2.13(b,c)]). Proof. (a) The first part is proved in [Hart, p. 70-71] (bottom and start of pages) and the second is [Hart, II, Exc. 2.1] and is proved in the same way as [Hart, II, Prop. 2.5(b)] (and is partially proved in [Hart, II, Prop. 2.2(b,c)]). (b) We have D(α) \ Spec B = fp 2 Spec A j α 62 pg \ Spec B = fq 2 Spec B j α 62 '−1qg = fq 2 Spec B j '(α) 62 qg = D(α); Date: April 18th, 2016. 1 2 ANDREAS LEOPOLD KNUTSEN as desired. (This will be generalized in Remark 2.4 below). In particular, if D(α) ⊆ Spec B, then D(α) = D(α), so that, by (a), we have Spec Aα ' Spec Bα, so that Aα ' Bα by (the proof of) [Hart, II, Prop. 2.3]. (c) We have [i2I D(αi) = X − V ((fαi j i 2 Ig)) by [Hart, II, Lemma 2.1(b)] and V ((fαi j i 2 Ig)) = ; if and only if (fαi j i 2 Ig) = A. (d) (fαi j i 2 Ig) = A means that we can write X 1A = tiαi; for some ti 2 A: i2I But this must hold for finitely many i's, so that (α1; : : : ; αr) = A, which means that Spec A = D(α1) [···[ D(αr). By (a), this implies that sp(X) is quasi-compact. (e) One easily sees that Spec k[x1; x2;:::] is not noetherian. However, if A is noetherian and V (a1) % V (a2) % ··· is a descending chain of closed subsets, then the corresponding chain p p a1 ⊆ a2 ··· p must eventually stabilize, as A is noetherian. Now the rest follows as V (ai) = V ( ai). An almost immediate consequence of Lemma 1.1 is that any open subset U ⊂ X is itself a scheme, with structure sheaf OU = OX jU , cf. [Hart, II, Exc. 2.2]. (Indeed, X can be covered by open affine subschemes Spec A, each of which has a basis for the topology consisting of open sets that are spectra of rings, whence also U is a union of such spectra of rings.) We next study morphisms between affine schemes. Let X = Spec A and Y = Spec B be affine schemes and (f;f ]) Y / X p / '−1p be a morphism, corresponding to a ring homomorphism ' A / B; as given in the proof of [Hart, II,Prop. 2.3] (so that ' is given by f ](X) A ' Γ(X; OX ) / Γ(Y; OY ) ' B; (using [Hart, II, Prop. 2.2(c)]). In particular, we have that for any y 2 Y , the induced morphism on the stalks ] fy (3) OX;f(y) / OY;y ADDITIONAL NOTES IN ALGEBRAIC GEOMETRY, II 3 is just the natural localization of ': 'py −1 (4) A' py = Af(py) / Bpy ; where py 2 Spec B is the prime ideal corresponding to y. It was noted in the proof of [Hart, II, Prop. 2.3] that (5) f −1(V (a)) = V ('(a)) for any ideal a ⊆ A: Conversely, it is easy to check that (6) f(V (b)) = V ('−1(b)) for any ideal b ⊆ B: The following properties of morphisms is for instance used in [Hart, II, Example 3.2.3]: Lemma 1.2. ([Hart, II, Exc. 2.18(b,c)]) Let X = Spec A and Y = Spec B be affine schemes and f, f ] and ' as in the beginning of the section. Then ] (i) ' is injective if and only if f : OX ! f∗OY is injective. Moreover, f is dominant in this case. (ii) If ' is surjective, then f is a closed immersion 1. Proof. (i) If ' is injective, then also (4) is injective for every py 2 Spec B, whence also (3), and therefore also f ] is injective. Conversely, if f ] is injective, then also ' = f ](X) is. The fact that f is dominant in this case follows from (6): (∗) f(Y ) = f(Spec B) = f(V ((0))) = V ('−1((0))) = V ((0)) = Spec A where the injectivity of ' is used in (∗). (ii) If ' is surjective, then B ∼= A= ker ', and, as is well-known from commutative algebra, there is a one-to-one correspondence (given by '−1) between prime ideals in B and prime ideals in A containing ker '. This turns (6) into (7) f(V (b)) = V ('−1(b)) for any ideal b ⊆ B and yields that f(Y ) = f(Spec B) = f(V ((0))) = V (ker '); and f induces a bijection between Y = Spec B and the closed subset V (ker ') ⊆ A. It is continuous as f is, and it is bicontinuous since f maps closed sets into closed sets by (7). The surjectivity of f ] can be checked on stalks, where it follows since (4) and (3) are the same map. There are similar results concerning the schemes Proj S associated to graded rings, as we will now see. We use the same notation as in [Hart, p. 76-77]. Proposition 1.3. ([Hart, II, Exc. 2.14(b-c) and Exc. 3.12(a)]) Let ' : S ! T be a graded homomorphism of graded rings. Let U := fp 2 Proj T j '(S+) 6⊆ pg. 1By definition (see [Hart, Def. p. 85]) this means that f is a homeomorphism onto a closed subset of X (necessarily equal to V (ker '), as seen in the proof) and f ] is surjective. 4 ANDREAS LEOPOLD KNUTSEN (a) U is an open subset of Proj T and ' determines a natural morphism f : U ! Proj S (given on the topological spaces by f(p) = '−1p). (b) If ' is surjective, then U = Proj T and f : Proj T ! Proj S is a closed immersion. (c) If 'd : Sd ! Td is an isomorphism for all d ≥ d0, for some d0 2 Z, then U = Proj T and f : Proj T ! Proj S is an isomorphism. Proof. (a) The ideal '(S+)T in T is homogeneous, as it is generated by the homoge- neous elements '(α) for homogeneous α 2 S+. Hence, as (8) Proj T − U = V ('(S+)T ); the set U is an open subset of Proj T . −1 −1 We note that if p 2 U, then S+ 6⊆ ' p, whence ' p 2 Proj S and there is a map f : U ! Proj S on the topological spaces by f(p) = '−1p. If V (a) ⊆ Proj S is a closed set, for a homogeneous ideal a ⊂ S, then f −1V (a) = fp 2 U j '−1p ⊇ ag = V ('(a)T ); whence f is continuous. ] We have left to define the morphism of sheaves f : OProj S ! f∗OU. We proceed as in the proof of [Hart, II, Prop. 2.3]. For each p 2 U, we obtain a local homomorphism of local rings 'p : S('−1p) ! T(p). For any open set V ⊆ Proj S we thus obtain a homomorphism of rings f ](V ) −1 OProj S(V ) / OU(f V ) s / t[p2f −1V ]'p ◦ s ◦ f; ] ] ] such that ff(p) = 'p on the stalks. This defines f and proves that (f; f ) is a morphism. (b) Surjectivity of ' implies that '(S+) = T+, proving that U = Proj T . As T ' S= ker ', we have that f(Proj T ) = V (ker '), which is closed in Proj S. Finally, the surjectivity of f ] can be proved on the stalks, where we get the local homomorphisms ''−1p : S('−1p) ! T(p), which are surjective by hypothesis. Hence f is a closed immersion. (c) Assume that 'd : Sd ! Td is an isomorphism for all d ≥ d0. If p 2 Proj T − U, d0 then '(S+) ⊆ p, whence ⊕d≥d0 Td ⊂ p. Therefore (T+) ⊆ p. As p is prime, we must have T+ ⊂ p, a contradiction. Thus U = Proj T . We now prove that f is injective. If f(p) = f(q), then '−1p = '−1q, whence p \ Td = q \ Td for all d ≥ d0. Thus, if a 2 p is a homogeneous element, then n n a 2 p \ Td for some n and d ≥ d0. Hence a 2 q, which implies a 2 q, as q is prime. This proves that p ⊆ q. By symmetry, we have p = q, which proves that f is injective.
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