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 Hervé 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.
Let be an ideal in the Jacobson radical (ie. the intersection of all maximal ideals) in a Noetherian ring . Then
As most sources are quick to point out, this “intuitive” fact can fail if the ring is not Noetherian.
We first wish to prove that
The inclusion is obvious. For the reverse inclusion, let generate , and let be the set of all homogeneous polynomials in letters over of degree such that for , we have that . Generate an ideal from , and choose a finite generating subset .
Let be the maximum degree of all the ‘s, and let . Then can be represented as where is a homogeneous polynomial over of degree . Therefore
Since and each are homogeneous, we can take ‘s to be homogeneous too, and of degree , since any terms not of this degree will cancel anyway and so are redundant. It follows that no is constant since was specifically chosen to be higher degree than any . Thus
We have that and . Therefore .
Now use Nakayama’s lemma to deduce the result.
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!