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We shall show that M/I n M is flat over A/I n . For 0 ≤ i ≤ n − 1 we have a commutative diagram I i+1 /I n ⊗A M −−−−→ I i /I n ⊗A M −−−−→ I i /I i+1 ⊗A M −−−−→ 0       αi+1 γi α i 0 −−−−→ I i+1 M/I n M −−−−→ I i M/I n M −−−−→ I i M/I i+1 M −−−−→ 0. 8 Local criteria of flatness 3 with exact rows. It follows from the assumptions that γi is an isomorphism for i = 0, 1, . . By descending induction on i, starting with αn = 0, it follows that αi is an isomorphism for i = 0, . . , n. In particular we have that α1 : I/I n ⊗A M = I/I n ⊗A/I n M/I n M → IM/I n M is an isomorphism.

Given a map ϕ: F → G of finite complexes of A-modules of finite length. Then we have that A (ϕ) → = A (G) − A (F ). Proof. ). 6) Proposition. Given a commutative diagram 0 −−−−→ F −−−−→ F −−−−→ F −−−−→ 0       ϕ ϕ ϕ 0 −−−−→ G −−−−→ G −−−−→ G of complexes of A-modules. If two of the lengths defined, then the third is, and we have that A (ϕ) → = A (ϕ )+ A (ϕ −−−−→ 0 A (ϕ ), A (ϕ) and A (ϕ ) are ). Proof. That the third length is defined when the two others are is an immediate consequence of the properties of length of modules.

9 Generic flatness 2 Proof. Let K be the quotient field of A. Then B ⊗A K is a K-algebra of finite type and N ⊗A K is a B ⊗A K-module of finite type. Let s = dim supp(N ⊗A K) be the Krull dimension of the support of N ⊗A K in Spec(B ⊗A K). We shall prove the Lemma by induction on s. When s < 0 we have that N ⊗A K = 0. Since K is flat over A we have that N ⊗A K = 0 implies that each element in N has A torsion, and since N is a finitely generated B-module there is an element f ∈ A such that f N = 0.

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Intersection theory by Dan Laksov

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Categories: Algebraic Geometry