A Pure Subalgebra of a Finitely Generated Algebra Is Finitely Generated

We prove the following. Let R be a Noetherian commutative ring, B a finitely generated R-algebra, and A a pure R-subalgebra of B. Then A is finitely generated over R. In this paper, all rings are commutative. Let A be a ring and B an A-algebra. We say that A → B is pure, or A is a pure subring of B, if for any A-module M , the map M = M ⊗A A → M ⊗A B is injective. Considering the case M = A/I, where I is an ideal of A, we immediately have that IB ∩A = I. There have been a number of cases where it has been shown that if B has a good property and A is a pure subring of B, then A has a good property. If B is a regular Noetherian ring containing a field, then A is Cohen-Macaulay [5], [4]. If k is a field of characteristic zero, A and B are essentially of finite type over k, and B has at most rational singularities, then A has at most rational singularities [1]. In this paper, we prove the following. Theorem 1. Let R be a Noetherian ring, B a finitely generated R-algebra, and A a pure R-subalgebra of B. Then A is finitely generated over R. The case that B is A-flat is proved in [3, Corollary 2.6]. This theorem is on the same line as the finite generation results in [3]. To prove the theorem, we need the following, which is a special case of a theorem of Raynaud-Gruson [7], [8]. Theorem 2. Let A → B be a homomorphism of Noetherian rings, and φ : X → Y the associated morphism of affine schemes. Let U ⊂ Y be an open subset, and assume that φ : φ−1(U) → U is flat. Then there exists some ideal I of A such that V (I) ∩ U = ∅, and such that the morphism Φ: ProjRB(BI) → ProjRA(I), determined by the associated morphism of the Rees algebras RA(I) := A[tI] → RB(BI) := B[tBI], is flat. The morphism Φ in the theorem is called a flattening of φ. Proof of Theorem 1. Note that for any A-algebra A′, the homomorphism A′ → B ⊗A A′ is pure. Received by the editors December 30, 2003. 2000 Mathematics Subject Classification. Primary 13E15.