Artin–Tate lemma

In algebra, the Artin–Tate lemma, named after John Tate and his former advisor Emil Artin, states:[1]

Let A be a commutative Noetherian ring and B ⊂ C {\displaystyle B\subset C} commutative algebras over A. If C is of finite type over A and if C is finite over B, then B is of finite type over A.

(Here, "of finite type" means "finitely generated algebra" and "finite" means "finitely generated module".) The lemma was introduced by E. Artin and J. Tate in 1951[2] to give a proof of Hilbert's Nullstellensatz.

The lemma is similar to the Eakin–Nagata theorem, which says: if C is finite over B and C is a Noetherian ring, then B is a Noetherian ring.

Proof

The following proof can be found in Atiyah–MacDonald.[3] Let x 1 , … , x m {\displaystyle x_{1},\ldots ,x_{m}} generate C {\displaystyle C} as an A {\displaystyle A} -algebra and let y 1 , … , y n {\displaystyle y_{1},\ldots ,y_{n}} generate C {\displaystyle C} as a B {\displaystyle B} -module. Then we can write

x i = ∑ j b i j y j and y i y j = ∑ k b i j k y k {\displaystyle x_{i}=\sum _{j}b_{ij}y_{j}\quad {\text{and}}\quad y_{i}y_{j}=\sum _{k}b_{ijk}y_{k}}

with b i j , b i j k ∈ B {\displaystyle b_{ij},b_{ijk}\in B} . Then C {\displaystyle C} is finite over the A {\displaystyle A} -algebra B 0 {\displaystyle B_{0}} generated by the b i j , b i j k {\displaystyle b_{ij},b_{ijk}} . Using that A {\displaystyle A} and hence B 0 {\displaystyle B_{0}} is Noetherian, also B {\displaystyle B} is finite over B 0 {\displaystyle B_{0}} . Since B 0 {\displaystyle B_{0}} is a finitely generated A {\displaystyle A} -algebra, also B {\displaystyle B} is a finitely generated A {\displaystyle A} -algebra.

Noetherian necessary

Without the assumption that A is Noetherian, the statement of the Artin–Tate lemma is no longer true. Indeed, for any non-Noetherian ring A we can define an A-algebra structure on C = A ⊕ A {\displaystyle C=A\oplus A} by declaring ( a , x ) ( b , y ) = ( a b , b x + a y ) {\displaystyle (a,x)(b,y)=(ab,bx+ay)} . Then for any ideal I ⊂ A {\displaystyle I\subset A} which is not finitely generated, B = A ⊕ I ⊂ C {\displaystyle B=A\oplus I\subset C} is not of finite type over A, but all conditions as in the lemma are satisfied.

References

  1. ^ Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8, Exercise 4.32
  2. ^ E Artin, J.T Tate, "A note on finite ring extensions," J. Math. Soc Japan, Volume 3, 1951, pp. 74–77
  3. ^ M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0-201-40751-5. Proposition 7.8
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