Math 8212, Spring 2020

Homework 3

Due Friday, February 28

The problem set is due by the beginning of Friday class.

Reading: Class notes. Eisenbud: Section 9.1 and Chapter 10 through Section 10.2.

Exercises: Eisenbud: 9.2, 9.5, 10.2 (You do not need to prove that there are infinitely many primes in R[x] of type b.)

Exercise A: Show that Ext_Rⁿ(M,N) can be computed from a projective resolution of M or an injective resolution of N. Hint: Use spectral sequences, as we did for Tor.

Exercise B: Show that Krull dimension satisfies the first half of Axiom D1 and Axiom D2.

Exercise C: If R ⊂ S is a finite extension of Noetherian rings, then the induced map Spec S → Spec R has finite fibers.

Last modified: (2020-02-17 00:37:45 CST)