Math 615: Lecture of February 12, 2007
We next want to consider one example where the generic initial ideal depends on the
characteristic. The example also illustrates that, even when the given ideal is monomial,
the generic initial ideal can be rather di
HISTORY OF
RAWALPINDI.
Lets start with that.
Rawalpindi ! Rawalpindi (Punjabi, Urdu:, Rwalpi
), commonly known as Pindi (
Punjabi:),isacityinthe
i
iiii
i
ii
ii
ii
Punjab,
Pakistan. It is the 4th most populous city of Pakistan and 3rd most
populated metrop
Math 615: Lecture of March 5, 2007
We next want to prove that the algebraic torus GL(1, K)s , which we shall refer to
simply as a torus, is linearly reductive, as asserted earlier, over every algebraically closed
eld K, regardless of characteristic. The n
Math 615: Lecture of February 14, 2007
We postpone further consideration of Grbner bases to study some results in invariant
o
theory.
To keep prerequisites from algebraic geometry to a minimum, in our study we will take
the ground eld K to be an algebraic
Math 615: Lecture of February 9, 2007
Elementary matrices and unipotent matrices
U
U
We shall write Un Bn for the subgroup consisting of upper triangular matrices such
that all diagonal entries are equal to 1. This is the group of upper triangular unipote
Math 615: Lecture of March 7, 2007
We next want to consider when a K-subalgebra of S = K[x1 , 1/x1 , . . . , xn , 1/xn ] generated by monomials is normal. This is entirely a property of the semigroup of monomials
involved, and does not depend on the base
Math 615: Lecture of March 9, 2007
We now have established the results that we need about convex geometry over the
rational numbers, and we are ready to prove the Lemma from the top of p. 4 of the
Lecture Notes of March 7, which will also complete the pro
Math 615: Lecture of February 7, 2007
Remark. Let R S be a homomorphism of Noetherian rings, I an ideal of R, and M a
nitely generated S-module such that IM = M . Let x1 , . . . , xk I be a regular sequence
on M . Let J = (x1 , . . . , xk )R. Then depthI
Math 615: Lecture of March 14, 2007
Open questions: tight closure, plus closure, and localization
We want to consider some open questions in tight closure theory, and some related
problems about when rings split from their module-nite extension algebras.
Math 615: Lecture of March 12, 2007
Tight closure for modules
We want to extend tight closure theory to modules. Suppose we are given N M ,
nitely generated modules over a Noetherian ring R of prime characteristic p > 0. We can
e
e
pe
pe
dene v p for v Rh
Math 615: Lecture of February 21, 2007
We need the following:
Lemma. Let R S be at, and let I R, J R be ideals such that J = (f1 , . . . , fk )R
is nitely generated. Then (I :R J)S = IS :S JS.
Proof. Consider the map R (R/I)k that sends r (f1 r, . . . , f
Math 615: Lecture of February 16, 2007
The additive group G = (K, +) of the eld K may be identied with the group of upper
triangular 2 2 unipotent matrices
cfw_
1 a
0 1
: a K,
since
1 a
0 1
1
0
b
1
=
1 a+b
0
1
for all a, b K. This group is not linearly re
Math 615: Lecture of February 19, 2007
If R is a ring of prime characteristic p we write FR : R R for the Frobenius endoe
morphism: FR (r) = rp . If e N, we write FR for the composition of FR with itself e
e
e
times, the iterated Frobenius endomorphism. T