Math 615, Fall 2014
Problem Set #4: Solutions
1. By a class theorem, S R[Y2 , . . . , Yn ]/I where I = (g1 , . . . , gn1 ) if we take gi =
=
x1 Yi+1 xi+1 for i 1. Since each of the quotients R[Y2 , .
Hilbert Functions
We recall that an N-graded ring R is Noetherian i R0 is Noetherian and R is nitely
generated over R0 . (The suciency of the condition is clear. Now suppose that R is
Noetherian. Sinc
Integral Closures in a Finite Separable Algebraic Extension
We want to prove that certain integral closures are module-nite: see the Theorem just
below. We also give, at the end of this supplement, an
Cohen-Macaulay rings
A sequence of elements x1 , . . . , xn of a ring R is called a possibly improper regular
sequence on the R-module M if x1 is not a zerodivisor on M and, for 1 i n 1, xi+1 is
not a
Faithful Flatness
We shall say that an R-module F is faithfully at if it is at if it is at and for every
nonzero R-module M , F R M = 0. An R-algebra S is faithfully at if it is faithfully at
when con
Module-nite Extensions of Complete Local Rings
Theorem. A module-nite extension S of a complete local ring (R, m, K) is a nite product of complete local rings: moreover, there is a bijection between t
Math 615, Fall 2014
Problem Set #2: Solutions
1. In S = K[x, y], we have that In = (y f, xn+1 ) (y f ) + mn , where m =
(x, y)K[x, y]. Since ideals of a complete local ring are m-adically closed, n (I
Math 615, Fall 2014
Problem Set #3: Solutions
1. R is the K-span of all monomials of even degree and those of odd degree that are
multiplies of x3 for some i. It follows that the x3 and x2 x2 are in t
The Functor Tor
Basic Properties of Tor
Let R be a commutative ring. The functors TorR (A, B) are functors of two variable
i
R-modules A and B that are covariant in each module when the other is held
The Structure Theory of Complete Local Rings
Introduction
In the study of commutative Noetherian rings, localization at a prime followed by completion at the resulting maximal ideal is a way of life.
Regular Rings, Finite Projective Resolutions, and Grothendieck Groups
Recall that a local ring (R, m, K) is regular if its embedding dimension, dimK (m/m2 ),
which may also be described as the least n
Math 615, Fall 2014
Problem Set #1: Solutions
1. By a class theorem, the integral closure of the ideal In is homogeneous: if it were not
integrally closed, a nonzero element of degree d n 1 element f
Test elements using the Lipman-Sathaye theorem
Theorem. Let R S be Noetherian domains of positive characteristic p with fraction
elds K L such that R is regular, S is module-nite over R, and L is sepa
Ideals of Minors and Fitting Invariants
Let A denote an s h matrix over a ring R. Let t be a nonnegative integer. We dene
It (A) to be the ideal generated by the t t minors of A if 1 t min cfw_s, h. I