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 the Rlinear map
n
1
q
q
q
q
q
q
[q]
yields cr = r (1)
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 characteristic = 3.
e
The matrix has as entries the trac
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 that is not in P .
[q]
If u N M , we have that cuq NM fo
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 [q] + mn ) = I q .
Hence, u I .
2. u I i cuq I [q] for
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 of the kind of theorem
one can prove about symbolic pow
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 R (e S S N ) (by
=
the associativity of ) M R (e S S N
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 p > 0, and let x1 , . . . , xn
be a system of paramete
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 for every q, R1/q is
e
contained in the normalization o
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 Noetherian ring R and let M be any Rmodule, not necessar
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 of its localizations are
weakly Fregular.
It is an ope