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 , . . . , Yn ]/(g1 , . . . , gi ) is an integral
domain, na
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. Since R0 is a homomorphic image of R, obtained by killing t
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 example of a Noetherian discrete
valuation domain of c
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 zerodivisor on M/(x1 , . . . , xi )M . If, in addition
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 considered as an R-module. We shall see below that the com
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 the factors and the
maximal ideals of S. Therefore, if S
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 (In K[x, y]) is
(yf )K[x, y], which must contain the inte
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 the conductor, and
n
1
i
i
an1
generate it: if say xn do
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 xed. This
is similar to the behavior of A R B: in fact,
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. Many problems, even some that
seem global, can be attac
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 number of generators of the maximal ideal m, is
equal to
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 of R would be integral
over the specied ideal. Call the
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 separable over K.
Then R1/q R S R1/q [S] is faithfully at o
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. If t = 0, we
may the convention I0 (A) = R. If t > min c