Math 615: Lecture of February 2, 2007
Regular sequences in the monomial case
We want to analyze what it means for a sequence of monomials 1 , . . . , k in R =
K[x1 , . . . , xn ] to be a regular sequence on F/M when F is a nitely generated free Rmodule an
Math 615: Lecture of January 12, 2007
Example. Now consider
f = x1 x2 x3 , g1 = x1 x2 + x2 , g2 = x1 x3 + x2
3
2
in F = R = K[x1 , x2 , x3 ] with hlex as the monomial order. On the one hand,
f = x3 g1 + 0 g2 x3
3
is a standard expression with remainder x3
Math 615: Lecture of January 19, 2007
We give one example of how one starts to calculate a Grbner basis in a specic ino
stance. Let g1 = x2 x2 x4 + x4 and g2 = x1 x3 x2 + x4 be generators for an ideal I in
1
3
4
4
R = K[x1 , x2 , x3 , x4 ] and suppose tha
Math 615: Lecture of January 24, 2007
Hilbert functions
Let M be a nitely generated graded module over R = K[x1 , . . . , xn ], a polynomial
ring over a eld. The Hilbert function HilbM of M is dened by the formula
HilbM (d) = dimK ([M ]d )
for all d Z. It
Math 615: Lecture of January 10, 2007
The denition of lexicographic order is quite simple, but the totally ordered set that
one gets is not even if there are only two variables one has
1 < x2 < x2 < < xn <
2
2
< x1 < x1 x2 < x1 x2 < < x1 xn <
2
2
< x2 <
Math 615: Lecture of February 5, 2007
Associated primes and primary decompostion for modules
Throughout this section R is a Noetherian ring and M an R-module. Recall that P is
an associated prime of M if, equivalently
(1) There is an injection R/P M .
(2)
Math 615: Lecture of January 22, 2007
Proof. Because the maps are degree preserving, it suces to prove that the complex is
exact in each degree . In fact, the full complex
0 Fk1 F0 I 0
is the direct sum of the homogeneous subcomplexes
( )
0 [Fk1 ] [F0 ] [
Math 615: Lecture of January 8, 2007
Monomial Ideals and Submodules
Let R = K[x1 , . . . , xn ] be a polynomial ring over a eld K. When a free R-module
F is given, it will typically be assumed to be nitely generated with an ordered free
basis b1 , . . . ,
Math 615: Lecture of January 26, 2007
Review of the theory of Krull dimension
We recall that the (Krull) dimension of a ring R, which need not be Noetherian, is the
supremum of lengths k of strictly increasing chains P0 P1 Pk1 Pk of chains
of prime ideals
Math 615: Lecture of January 5, 2007
This course will deal with several topics in the theory of commutative Noetherian rings,
including the following:
(1) The theory of Grbner bases and applications: a lot more about this momentarily.
o
(2) The structure
Math 615: Lecture of January 31, 2007
Invariant Theory
We want to present some examples from classical invariant theory to which one can
apply the Theorem on the Cohen-Macaulay property for rings of invariants stated at the
end of the Lecture Notes of Jan
Math 615: Lecture of January 17, 2007
The notion of Grbner basis is non-trivial and of some interest even when there are no
o
indeterminates, i.e., when R = K is a eld, and F = K s .
Consider an r s matrix A = ai,j over a eld K. The leftmost nonzero entry
Math 615: Lecture of January 29, 2007
We next want to discuss the notion of a regular sequence in a ring or on a module. We
are aiming to discuss criteria, using revlex, for a sequence to be regular on F/M . However,
we also want to discuss some theorems