Math 711: Lecture of December 7, 2007
We assume that we have the situation of the displayed paragraph near the bottom of
p. 6 of the Lecture Notes from December 5. Specically, let
R T0 T1 Tr
be a sequence of algebras obtained from a complete local domain
Math 711: Lecture of November 12, 2007
Before proceeding further with our treatment of test elements, we note the following
consequence of the theory of approximately Gorenstein rings. We shall need similar splitting results in the proof of the generaliza
Math 711: Lecture of December 3, 2007
Step 6. Proof that J+ /J has nite length when d is minimum. We rst prove that the
test ideal commutes with localization for a reduced excellent Gorenstein local ring of prime
characteristic p > 0. In order to do so, w
Math 711: Lecture of November 26, 2007
Etale and pointed tale homomorphisms and a
e
generalization of Artin approximation
Let R be a Noetherian ring. We shall say that R S is tale if S is essentially of nite
e
type over R, at, and the bers are tale eld ex
Math 711: Lecture of November 14, 2007
We continue to develop the preliminary results needed to prove the Theorem stated on
p. 2 of the Lecture Notes from November 12. Only one more is needed.
Theorem. Let (R, m, K) be a complete local domain of prime cha
Math 711: Lecture of December 10, 2007
We have dened an element of the homology or cohomology of a complex of nitely
generated modules over a Noetherian ring R of prime characteristic p > 0 to be phantom
if it is represented by a cycle (or cocycle) that i
Math 711: Lecture of November 30, 2007
Theorem (K. E. Smith). Let (R, m, K) be an excellent, reduced, equidimensional local
d
ring of Krull dimension d, and let H = Hm (R). Then 0 = 0fg . If x1 , . . . , xd is a system
H
H
of parameters for R, It = (xt ,
Math 711: Lecture of November 28, 2007
Step 3. Reduction to the complete local case. Now suppose that the result holds for ideals
of height k of the form (x1 , . . . , xk )R whenever k < d. Also suppose that (R, m, K)
is a normal excellent local ring of p
Math 711: Lecture of November 9, 2007
We note the following fact from eld theory:
Proposition. Let K be a eld of prime characteristic p > 0, let L be a separable algebraic
extension of K, and let F be a purely inseparable algebraic extension of K. Then th
Math 711: Lecture of December 5, 2007
From the local cohomology criterion for solidity we obtain:
Corollary. A big Cohen-Macaulay algebra (or module) B over a complete local domain
R is solid.
Proof. Let d = dim (R) and let x1 , . . . , xd be a system of
Math 711: Lecture of November 21, 2007
We are aiming to prove the Theorem stated at the bottom of the last page of the
Lecture Notes from November 19, which will complete the proof of the Huneke-Lyubeznik
Theorem. We rst want to make an observation about