Artin-Tate lemma

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This article is about the statement of a simple but indispensable lemma in commutative algebra
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This article defines a result where the base ring (or one or more of the rings involved) is Noetherian
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Statement

Suppose A, B, C are commutative unital rings, such that:

Then: B is finitely generated as an A-algebra

Explanation

The Artin-Tate lemma is related to the general philosophy that if an object is finitely generated, then any subobject of it is also finitely generated provided the subobject is almost the whole object. In our case, C is finitely generated as an A-algebra, but it is not necessary that all subalgebras of C are finitely generated as A-algebras. However, if the subalgebra is large enough inside C, i.e. it is not too deep, then it behaves like C, and we can try to show it is finitely generated.

An analogous result in group theory is the statement that any subgroup of finite index in a finitely generated group, is also finitely generated.

Definitions used

Facts used

Proof

Find a finitely generated subalgebra of B

Pick a finite generating set c_1, c_2, \ldots, c_r of C as an algebra over A.

Since C is finitely generated as a B-module, every element in C is integral over B. In particular, every c_i is integral over B. Choose a monic polynomial satisfied by c_i over B for each i, and consider the A-algebra B_0 generated by the coefficients of all such monic polynomials.

Clearly B_0 satisfies the following:

  • B_0 is an A-subalgebra of B
  • B_0 is finitely generated as an A-algebra
  • C is finitely generated as a B_0-algebra (since in fact C is finitely generated as an A-algebra)
  • C is integral over B_0

Going down

Since C is both finitely generated as a B_0-algebra and integral over B_0, we see that C is finitely generated as a B_0-module.

We now use the fact that A is Noetherian, and B_0 is finitely generated as an A-algebra, to conclude that B_0 is Noetherian. Thus C is a finitely generated module over a Noetherian ring B_0. Hence, the submodule B is also finitely generated as a B_0-module. In particular, B is finitely generated as a B_0-algebra.

Transitivity

We know that B_0 is finitely generated as an A-algebra, and we just proved that B is finitely generated as a B_0-algebra. Thus, B is finitely generated as an A-algebra.