Math 8212, Spring 2020

Homework 7

Due Monday, May 4

Reading: Class notes. Eisenbud: Sections 19.1-19.3, 18.1 (skipping 18.1.1), 18.2, 20.1, 20.2.

Exercises: Eisenbud:   17.3, 18.2, 18.6, 19.7, 19.13, 20.16 (Figure 20.2 contains typos: it should be R³, rather than R² in both cases. The diagonal arrows denote multiplication of row vectors by the given column vectors on the right. (x,y,z) in the wording of the problem is meant to be (x₁, x₂, x₃).)

A few hints

17.3(b):
Here are some useful results we discussed in class in February, when classes were still held in Vincent Hall. If you took notes, you may find these statements there. Let n = dim R, m the max. ideal.

1. (See Corollary 10.7) TFAE:

• x_1, ..., x_n is a system of parameters;
• the maximal ideal m is a minimal prime of the ideal (x_1, ..., x_n);
• rad (x_1, ..., x_n) = m;
• (x_1, ..., x_n) is m-primary.

2. One cannot have fewer than n elements generating an ideal whose radical is m, for then dim(R) would be < n. Elements x1, ..., xk ∈ m can be extended to a system of parameters for R if and only if dim R/(x1, . . . , xk) ≤ n - k, in which case dim R/(x1, . . . , xk) = n - k.

3. In particular, x = x1 is part of a system of parameters iff x is not in any minimal prime P of R such that dim(R/P) = n.

4. Elements y1, ... ,y_{n-k} extend x1, ..., xk to a system of parameters for R iff their images in R/(x1, . . . , xk) are a system of parameters for R/(x1, ..., xk).

18.2:
Try to use Krull's PIT for m and the inequality between depth and dim to estimate the number of elements in this regular sequence.

19.7:
Show one of two things (your choice): P consists of nilpotent elements or P consists of zero divisors.

19.13:
Try to use the Auslander-Buchsbaum formula and the behavior of depth under localization...

Last modified: (2020-05-03 15:51:25 CDT)