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Section 2.7.1 - Blowing Up

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Section 2.7.1 - Blowing Up Section 2.7.1 - Blowing Up Daniel Murfet October 5, 2006 Now we come to the generalised notion of blowing up. In (I, §4) we defined the blowing up of a variety with respect to a point. Now we will define the blowing up of a noetherian scheme with respect to any closed subscheme. Since a closed subscheme corresponds to a coherent sheaf of ideals, we may as well speak of blowing up a coherent sheaf of ideals. The necessary background for this section includes (MRS,Section 1.9), (MOS,Section 2) and (SSA,Section 6). Lemma 1. Let X be a noetherian scheme and J a coherent sheaf of ideals on X. Then B(J ) = L n n≥0 J is a commutative quasi-coherent sheaf of graded OX -algebras locally finitely generated 1 0 0 by B (J ) as a B (J )-algebra with B (J ) = OX . 0 Proof. The sheaf of graded OX -algebras B(J ) is clearly quasi-coherent, and has B (J ) = OX by construction. The other properties follow from (SSA,Corollary 55). Definition 1. Let X be a noetherian scheme and J a coherent sheaf of ideals on X. We define Xe to be the scheme ProjB(J ). This is a noetherian scheme with proper structural morphism π : Xe −→ X and twisting sheaf O(1). We call Xe the blowing-up of X with respect to the coherent sheaf of ideals J . If Y −→ X is a closed immersion with sheaf of ideals J , then we also call Xe the blowing-up of X along Y or with center Y . If J = 0 then Xe is the empty scheme. Proposition 2. If X is an integral noetherian scheme and J a nonzero coherent sheaf of ideals then the blowing-up of J is an integral scheme. Proof. By (TRPC,Proposition 5) it suffices to show that B(J ) is relevant, in the sense of (TRPC,Definition 2). It follows from (SSA,Proposition 54) that B(J ) is locally an integral domain. Let Λ be the set of all nonempty affine open subsets U ⊆ X with J (U) 6= 0. Since J 6= 0 the set Λ is clearly nonempty. Given U, V ∈ Λ let f : U −→ SpecOX (U) be the canonical isomorphism. Since X is integral, U ∩ V is nonempty and we can find f ∈ OX (U) with W = D(f) ∼ nonempty and W ⊆ U ∩ V . Then Γ(D(f), J ) = Γ(U, J )f . Since OX (U) is an integral domain and f 6= 0 this latter ideal cannot be zero. Therefore W ∈ Λ and the proof is complete. Proposition 3. Let X be a noetherian scheme, J a coherent sheaf of ideals, and let π : Xe −→ X be the blowing-up of J . Then (a) The inverse image ideal sheaf = ·O is an invertible sheaf on X. Jf J Xe e (b) If Y is the closed subset corresponding to J then π−1U −→ U is an isomorphism, where U = X \ Y . Proof. (a) It suffices to show that Jfis invertible on a neighborhood of every point of Xe. Given x ∈ Xe find an affine open subset U ⊆ X with π(x) ∈ U. Using (SSA,Proposition 54) there is an isomorphism of schemes over U −1 π U =∼ P rojB(J )(U) =∼ P rojB(J (U)) By (MRS,Lemma 48) we have | −1 = | ·O | −1 and therefore by (MRS,Lemma 51) it Jfπ U J U Xe π U suffices to show that J |U ·OP rojB(J (U)) is an invertible sheaf on P rojB(J (U)). In fact we will show that there is a canonical isomorphism of sheaves of modules on P rojB(J (U)) ∼ J |U ·OP rojB(J (U)) = O(1) (1) 1 By (MRS,Lemma 51), (MOS,Lemma 9)(b) and (MOS,Lemma 10), J |U ·OP rojB(J (U)) is the sheaf of ideals corresponding to the homogenous ideal J (U)B(J (U)). The graded modules J (U)B(J (U)) and B(J )(U)(1) are quasi-isomorphic, so by (MPS,Proposition 18) we obtain ˙ the desired isomorphism (1). Note that for a ∈ J (U), (1) maps the global section ai/1 of ˙ OP rojB(J (U)) (here ai has grade zero) to the global section ai/1 of O(1) (here ai has grade one). (b) By the closed subset corresponding to J we mean Y = Supp(OX /J ), which is the closed image of the closed subscheme of X with ideal sheaf J . To show π−1U −→ U is an isomorphism it suffices to show that for every open affine V ⊆ U, π−1V −→ V is an isomorphism. Using the argument of part (a) we see that π−1V is isomorphic as a scheme over V to P rojB(J (V )). So it suffices to show that the structural morphism P rojB(J (V )) −→ SpecOX (V ) is an iso- morphism. But since Y = Supp(OX /J ) we have J |U = OX |U (MRS,Lemma 8) and therefore ∼ J (V ) = OX (V ). But B(OX (V )) = OX (V )[x] as graded OX (V )-algebras (SSA,Proposition 56) so P rojB(J (V )) −→ SpecOX (V ) is an isomorphism, as required. Remark 1. If X is a noetherian scheme and J a coherent sheaf of ideals, then we associate with an affine open subset U ⊆ X the open immersion P rojB(J (U)) −→ Xe given in the previous result. This is the composite P rojB(J (U)) =∼ P rojB(J )(U) −→ Xe, so there is a pullback diagram P rojB(J (U)) / Xe U / X Lemma 4. Let f : X −→ Y be a morphism of ringed spaces and J a sheaf of ideals on Y # # generated by global sections a1, . , an. Then the global sections fY (a1), . , fY (an) generate the ideal sheaf J ·OX . Proof. We know from (PM,Lemma 1) that the global sections [Y, ai] ⊗˙ 1 generate the sheaf of modules f ∗J on X. By construction there is an epimorphism of sheaves of modules f ∗J −→ ˙ # J ·OX mappng [Y, ai] ⊗ 1 to fY (ai), so the latter sections clearly generate J ·OX . Lemma 5. Let X be scheme and F a quasi-coherent sheaf of modules on X. Let U ⊆ X be an open affine subset and {ai}i∈I a nonempty subset of F (U). Then the elements ai generate F (U) as an OX (U)-module if and only if the global sections ai generate F |U . Proof. We may assume U = X. Let f : X −→ SpecOX (X) be the canonical isomorphism. Then ∼ ˙ since f∗F = F (X)e it suffices to show the global sections ai/1 generate F (X)e if and only if the ˙ ai generate F (X) as a OX (X)-module. The reverse implication is clear, so suppose that the ai/1 generate F (X)e and let G be the OX (X)-submodule of F (X) generated by the ai. By hypothesis for every prime ideal p we have Gp = F (X)p, and therefore G = F (X) as required. Lemma 6. Let X be a ringed space and J an invertible sheaf of ideals on X. Then for any d > 1 there is a canonical isomorphism of sheaves of modules J ⊗d −→ J d a1 ⊗˙ · · · ⊗˙ ad 7→ a1 ··· ad In particular the ideal sheaf J d is invertible. Proof. We prove the case d = 2, with other cases following by a simple induction. Since J is 2 ∼ invertible it is flat, so J is the image of the monomorphism J ⊗ J −→ OX ⊗ J = J . Therefore J ⊗ J =∼ J 2, and so J 2 is invertible as required. We need two more small results before embarking on the proof of the next Theorem. Intuitively the Lemmas prove various naturality properties of the morphisms produced by (PM,Corollary 4). The first proves naturality with respect to adding superfluous generators while the second proves naturality with respect to restriction. 2 Lemma 7. Let A be a ring, set X = SpecA and let f : Z −→ X a scheme over A. Let L be an invertible sheaf of ideals on Z generated by global sections a0, . , an ∈ A. Suppose b1, . , bm ∈ A are linear combinations of the ai bj = λj0a0 + ··· + λjnan λjk ∈ A and let φ, φ0, φ00 be the morphisms of A-schemes determined by the following tuples φ :(L , a0, . , an) 0 φ :(L , a0, . , an, b1, . , bm) 00 φ :(L , b1, . , bm, a0, . , an) Then the following diagrams commute φ n φ n Z / PA Z / PA CC CC CC CC C Ψ C Ψ0 0 C 00 C φ CC φ CC ! n+ m ! n+ m PA PA where the vertical morphisms are induced by the following morphisms of graded A-algebras n X ψ : A[x0, . , xn, y1, . , ym] −→ A[x0, . , xn] xi 7→ xi, yj 7→ λjixi i=0 n 0 X ψ : A[y1, . , ym, x0, . , xn] −→ A[x0, . , xn] yj 7→ λjixi, xi 7→ xi i=0 Proof. The correspondence between tuples and morphisms we refer to is the one defined in (PM,Corollary 4). We show that the first diagram commutes, since the proof for the second diagram is the same. It suffices to show that there is an isomorphism of sheaves (Ψφ)∗O(1) =∼ L ∗ ∗ that identifies (Ψφ) (xi) with ai and (Ψφ) (yj) with bj. The isomorphism is given by (Ψφ)∗O(1) =∼ φ∗Ψ∗O(1) =∼ φ∗O(1) =∼ L Using (MRS,Remark 8), (MPS,Proposition 13) and the unique isomorphism α : φ∗O(1) =∼ L ∗ identifying φ (xi) with ai, it is not hard to see this isomorphism has the desired property.
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