Math 711: Lecture of September 10, 2007
In order to give our next characterization of tight closure, we need to discuss a theory
of multiplicities suggested by work of Kunz and developed much further by P. Monsky.
We use (M ) for the length of a nite leng
Math 711: Lecture of September 24, 2007
Flat base change and Hom
We want to discuss in some detail when a short exact sequence splits. The following
result is very useful.
Theorem (Hom commutes with at base change). If S is a at R-algebra and M , N
are R-
Math 711: Lecture of September 21, 2007
F-nite rings
Let R be a Noetherian ring of prime characteristic p > 0. R is called F-nite if the
Frobenius endomorphism F : R R makes R into a module-nite R-algebra. This is
equivalent to the assertion that R is mod
Math 711: Lecture of September 12, 2007
In our treatment of tight closure for modules it will be convenient to use the Frobenius
functors, which we view as special cases of base change. We rst review some basic facts
about base change.
Base change
from
If
Math 711: Lecture of September 19, 2007
Earlier (see the Lecture of September 7, p. 7) we discussed very briey the class of
excellent Noetherian rings. The condition that a ring be excellent or, at least, locally excellent, is the right hypothesis for man
Math 711: Lecture of October 3, 2007
Koszul homology
We dene the i th Koszul homology module Hi (x1 , . . . , xn ; M ) of M with respect to
x1 , . . . , xn as the i th homology module Hi K (x1 , . . . , xn ; M ) of the Koszul complex.
We note the followin
Math 711: Lecture of September 5, 2007
Throughout these lecture notes all given rings are assumed commutative, associative,
with identity and modules are assumed unital. Homomorphisms are assumed to preserve
the identity. With a few exceptions that will b
Math 711: Lecture of October 1, 2007
In the proof of the Radu-Andr Theorem we will need the result just below. A more
e
general theorem may be found in [H. Matsumura, Commutative Algebra, W. A. Benjamin,
New York, 1 970], Ch. 8 (20.C) Theorem 49, p. 146,
Math 711: Lecture of September 14, 2007
The following result is very useful in thinking about tight closure.
Proposition. Let R be a Noetherian ring of prime characteristic p > 0, let N M be
R-modules, and let u M . Then u NM if and only if the image u of
Math 711: Lecture of September 26, 2007
We want to use the theory of strongly F-regular F-nite rings to prove the existence of
test elements.
We rst prove two preliminary results:
Lemma. Let R be an F-nite reduced ring and c R be such that Rc is F-split (
Math 711: Lecture of September 28, 2007
We next want to note some elementary connections between properties of regular sequences and the vanishing of Tor.
Proposition. Let x1 , . . . , xn R and let M be an R-module. Suppose that x1 , . . . , xn is
a possi