A pure subalgebra of a finitely generated algebra is finitely generated

We prove the following. 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 $R$.

In this paper, all rings are commutative. Let A be a ring and B an Aalgebra. 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.
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]. 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]. 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 Since B is finitely generated over R, it is Noetherian. Since A is a pure subring of B, A is also Noetherian. So if A red is finitely generated, then so is A. Replacing A by A red and B by B ⊗ A A red , we may assume that A is reduced.
Since A → P ∈Min(A) A/P is finite and injective, it suffices to prove that each A/P is finitely generated for P ∈ Min(A), where Min(A) denotes the set of minimal primes of A. By the base change, we may assume that A is a domain.
There exists some minimal prime P of B such that P ∩ A = 0. Assume the contrary. Then take a P ∈ P ∩ A \ {0} for each P ∈ Min(B). Then P a P must be nilpotent, which contradicts to our assumption that A is a domain.
So by [6, (2.11) and (2.20)], A is a finitely generated R-algebra if and only if A p is a finitely generated R p -algebra for each p ∈ Spec R. So we may assume that R is a local ring.
By the descent argument [2, (2.7.1)],R ⊗ R A is a finitely generatedRalgebra if and only if A is a finitely generated R-algebra, whereR is the completion of R. So we may assume that R is a complete local ring. We may lose the assumption that A is a domain (even if A is a domain,R ⊗ R A may not be a domain). However, doing the same reduction argument as above if necessary, we may still assume that A is a domain.
Let ϕ : X → Y be a morphism of affine schemes associated with the map A → B. Note that ϕ is a morphism of finite type between Noetherian schemes. We denote the flat locus of ϕ by Flat(ϕ). Then ϕ(X \ Flat(ϕ)) is a constructible set of Y not containing the generic point. So U = Y \ ϕ(X \ Flat(ϕ)) is a dense open subset of Y , and ϕ : ϕ −1 (U) → U is flat. By Theorem 2, there exists some nonzero ideal I of A such that Φ : Proj R B (BI) → Proj R A (I) is flat.
If J is a homogeneous ideal of R A (I), then we have an expression J = n≥0 J n t n (J n ⊂ I n ). Since A is a pure subalgebra of B, we have J n B ∩ I n = J n for each n. Since JR B (BI) = n≥0 (J n B)t n , we have that JR B (BI) ∩ R A (I) = J. Namely, any homogeneous ideal of R A (I) is contracted from R B (BI).
Let P be a homogeneous prime ideal of R A (I). Then there exists some minimal prime Q of P R B (BI) such that Q ∩ R A (AI) = P . Assume the contrary. Then for each minimal prime Q of P R B (BI), there exists some a Q ∈ (Q ∩ R A (AI)) \ P . Then a Q ∈ P R B (BI) ∩ R A (AI) \ P . However, we have and this is a contradiction. Hence Φ : Proj R B (BI) → Proj R A (I) is faithfully flat.
Since Proj R B (BI) is of finite type over R and Φ is faithfully flat, we have that Proj R A (I) is of finite type by [3,Corollary 2.6]. Note that the blow-up Proj R A (I) → Y is proper surjective. Since R is excellent, Y is of finite type over R by [3,Theorem 4.2]. Namely, A is a finitely generated R-algebra.