Elementary proof of Krull’s intersection theorem

When doing my 3rd-year project, I found I needed a classical theorem in commutative algebra called Krull’s Intersection Theorem. It turned out to be very useful; I used it in three proofs involving formal power series, and it also gave me some intuition for DVRs. However, despite being an intuitive result, the proofs I found in Zariski–Samuel and Atiyah–MacDonald required quite a bit more commutative algebra knowledge than I currently possess. Thankfully, a much simpler proof was found in 2004 by Herv√© Perdry. Here is a presentation of that proof, which I originally found in Milne’s Primer and to whom I give the credit. I post it here simply to make this useful theorem more widely accessible.

Theorem

Let \m be an ideal in the Jacobson radical (ie. the intersection of all maximal ideals) in a Noetherian ring R. Then

    \[ \bigcap_{n=1}^\infty \m^n = (0).\]

As most sources are quick to point out, this “intuitive” fact can fail if the ring is not Noetherian.

Proof

We first wish to prove that

    \[\m \cdot \bigcap_{n=1}^\infty \m^n = \bigcap_{n=1}^\infty \m^n.\]

The inclusion \subset is obvious. For the reverse inclusion, let a_1, \dots, a_r generate \m, and let S_m be the set of all homogeneous polynomials in r letters over R of degree m such that for f \in S_m, we have that f(a_1, \dots, a_r) \in \bigcap_{n=1}^\infty \m^n. Generate an ideal \mathfrak{b} from \bigcup_{m=1}^\infty S_m, and choose a finite generating subset f_1, \dots, f_s \in \bigcup_{m=1}^\infty S_m.

Let d be the maximum degree of all the f_i‘s, and let x \in \bigcap_{n=1}^\infty \subset \m^{d+1}. Then x can be represented as F(a_1, \dots, a_r) where F is a homogeneous polynomial over R of degree d+1. Therefore

    \[F = g_1 f_1 +  g_2 f_2 + \dots + g_s f_s, \quad g_i \in R[t_1, \dots, t_r].\]

Since F and each f_i are homogeneous, we can take g_i‘s to be homogeneous too, and of degree F - f_i, since any terms not of this degree will cancel anyway and so are redundant. It follows that no g_i is constant since F was specifically chosen to be higher degree than any f_i. Thus

    \[ x= F(a_1, \dots, a_r) = \sum_{i=1}^s (g_i \cdot f_i)(a_1, \dots, a_r).\]

We have that g_i(a_1, \dots, a_r) \in \m and f_i(a_1, \dots, a_r) \in \bigcap_{n=1}^\infty \m^n. Therefore x \in \m \cdot \bigcap_{n=1}^\infty \m^n.

Now use Nakayama’s lemma to deduce the result. \square

Original source: http://www.jmilne.org/math/xnotes/ca.html


Next semester I enter my final year. This week, I selected the courses I’ll be taking.

  • Masters project: Not quite sure what this will involve yet. The supervisor says he will like me to be introduced to sheaf cohomology but hasn’t decided where to take that.
  • Algebraic Geometry: Natch. The syllabus changed a lot this year because there is a new lecturer taking it, but it looks great.
  • Manifolds, Homology and Morse Theory: Interesting looking course with one of my favourite lecturers.
  • Number Theory
  • Higher Arithmetic:¬†When I mention this one people think I’m taking a class in multiplying very large numbers.
  • Measure Theory: I felt I could not leave my undergraduate without learning about measure and Lebesgue integration.
  • Complex Dynamical Systems: My university has a dynamical systems research group and the lecturer is very passionate about the subject, so this should be good.

A bit of a resolution is to take better advantage of office hours. I’m one of those students who maybe for reasons of vanity (fear of looking stupid, for example; totally irrational, I know) doesn’t make much use of office hours, but not taking advantage of access to professional mathematicians is a great way to hobble oneself.

Leave a comment about what you’ll be doing next year!