Artin–Tate lemma
In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states:[1]
- Let A be a commutative Noetherian ring and [math]\displaystyle{ B \sub C }[/math] 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 [math]\displaystyle{ x_1,\ldots, x_m }[/math] generate [math]\displaystyle{ C }[/math] as an [math]\displaystyle{ A }[/math]-algebra and let [math]\displaystyle{ y_1, \ldots, y_n }[/math] generate [math]\displaystyle{ C }[/math] as a [math]\displaystyle{ B }[/math]-module. Then we can write
- [math]\displaystyle{ x_i = \sum_j b_{ij}y_j \quad \text{and} \quad y_iy_j = \sum_{k}b_{ijk}y_k }[/math]
with [math]\displaystyle{ b_{ij},b_{ijk} \in B }[/math]. Then [math]\displaystyle{ C }[/math] is finite over the [math]\displaystyle{ A }[/math]-algebra [math]\displaystyle{ B_0 }[/math] generated by the [math]\displaystyle{ b_{ij},b_{ijk} }[/math]. Using that [math]\displaystyle{ A }[/math] and hence [math]\displaystyle{ B_0 }[/math] is Noetherian, also [math]\displaystyle{ B }[/math] is finite over [math]\displaystyle{ B_0 }[/math]. Since [math]\displaystyle{ B_0 }[/math] is a finitely generated [math]\displaystyle{ A }[/math]-algebra, also [math]\displaystyle{ B }[/math] is a finitely generated [math]\displaystyle{ A }[/math]-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 [math]\displaystyle{ C = A\oplus A }[/math] by declaring [math]\displaystyle{ (a,x)(b,y) = (ab,bx+ay) }[/math]. Then for any ideal [math]\displaystyle{ I \subset A }[/math] which is not finitely generated, [math]\displaystyle{ B = A \oplus I \subset C }[/math] is not of finite type over A, but all conditions as in the lemma are satisfied.
References
- ↑ 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
- ↑ E Artin, J.T Tate, "A note on finite ring extensions," J. Math. Soc Japan, Volume 3, 1951, pp. 74–77
- ↑ M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN:0-201-40751-5. Proposition 7.8
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