# Artin-Tate lemma

This article is about the statement of a simple but indispensable lemma in commutative algebra

View other indispensable lemmata

This article defines a result where the base ring (or one or more of the rings involved) is Noetherian

View more results involving Noetherianness or Read a survey article on applying Noetherianness

## Contents

## Statement

Suppose are commutative unital rings, such that:

- is Noetherian
- and are -algebras, with a subalgebra of
- is finitely generated as an -algebra (i.e. is an algebra of finite type over )
- is finitely generated as a -module (i.e. is finite as an algebra over )

Then: is finitely generated as an -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, is finitely generated as an -algebra, but it is not necessary that *all* subalgebras of are finitely generated as -algebras. However, if the subalgebra is *large* enough inside , i.e. it is *not too deep*, then it behaves like , 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

- Any submodule of a finitely generated module over a Noetherian ring, is finitely generated
- Any finitely generated algebra over a finitely generated algebra is finitely generated
- Finitely generated and integral implies finite: A finitely generated algebra over a ring, such that every element in it is integral over the ring, is finite as an algebra i.e. is finitely generated as a module over the base ring.
- Any finitely generated algebra over a Noetherian ring is Noetherian. This follows from two facts: any polynomial ring over a Noetherian ring is Noetherian, and Any quotient ring of a Noetherian ring is Noetherian.

## Proof

### Find a finitely generated subalgebra of

Pick a finite generating set of as an algebra over .

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

Clearly satisfies the following:

- is an -subalgebra of
- is finitely generated as an -algebra
- is finitely generated as a -algebra (since in fact is finitely generated as an -algebra)
- is integral over

### Going down

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

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

### Transitivity

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