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 sequ
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 s
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 r
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
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
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 phan
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 , . .
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 suppo
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
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.
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-Lyube