Math 711, Fall 2007
Problem Set #1 Solutions
1. Let (1) = c = 0. If r IS R, we have r = f1 s1 + fn sn with r R, the fi I,
q
q
and s1 , . . . , sn S. Then for all q, rq = f1 sq + fn sq , and applying t
Math 711, Fall 2007
Problem Set #5 Solutions
1. (a) The extension is modulenite since the equation is monic in Z, and generically
tale since adjoining a cube root gives a separable eld extension in c
Math 711, Fall 2007
Problem Set #3 Solutions
1. Since the test ideal (R) has height two, it cannot be contained in the union of P and
the minimal primes of R. Hence, we can choose a test element c R t
Math 711, Fall 2007
Problem Set #2 Solutions
1. is clear. To prove , if c is a test element and u n (I + mn ) then for all q and
n, cuq (I + mn )[q] = I [q] + (mn )[q] I q + mn . Fix q. Then cuq n (I
Math 711: Lecture of September 7, 2007
Symbolic powers
We want to make a number of comments about the behavior of symbolic powers of
prime ideals in Noetherian rings, and to give at least one example
Math 711, Fall 2007
Problem Set #4 Solutions
e
1. Let e R indicate R and viewed as an Ralgebra via FR and similarly for e S. We have
e
e
that FR (M ) R FS (N ) = (e R R M ) e R (e S S N ) (M R e R) e
Math 711: Lecture of October 24, 2007
The action of Frobenius on the injective hull of the
residue class eld of a Gorenstein local ring
Let (R, m, K) be a Gorenstein local ring of prime characteristic
Math 711: Lecture of November 16, 2007
We next observe:
Lemma. Let A be a regular Noetherian domain of prime characteristic p > 0, and let
A R be modulenite, torsionfree, and generically tale. Then
Math 711: Lecture of November 19, 2007

Summary of Local Cohomology Theory
The following material and more was discussed in seminar but not in class. We give a
summary here.
Let I be an ideal of a No
Math 711: Lecture of September 17, 2007
Denition. Let R be a Noetherian ring of prime characteristic p > 0. R is called weakly
Fregular if every ideal is tightly closed. R is called Fregular if all