Local Homology and Local Cohomology
Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak m}}(M)$ is co-Cohen-Macaulay of N.dimension $d$ and $H^{\underline{x}}_d(H^d_{{\frak m}}(M))\cong\hat{M}$ where $\underline{x}=(x_1,...,x_d)$ is a system of parameters for $M$.
Some applications of a criterion for artinianness of a module
Abstract A criterion previously shown by the author for a module to be artinian is first applied to quickly deduce that certain local cohomology modules are artinian and thereafter to determine the first nonartinian local cohomology module using the concept of filter-depth. Next homomorphisms having finite index are considered. The logarithmic law is proved for such homomorphisms, and the index of an endomorphism on a noetherian (artinian) module is shown to be nonnegative (resp. nonpositive). Finally finiteness results for Koszul (co) homology modules are given, which are extensions of Matijevic's classical theorem on maximal ideal transforms.
Maximal ideal transforms of Noetherian rings
Let J? be a commutative Noetherian ring with unit. Let T be the set of all elements of the total quotient ring of R whose conductor to R contains a power of a finite product of maximal ideals of R. If A is any ring such that R C A C T, then A/xA is a finite R module for any non-zerodivisor x in R. It follows that if, in addition, R has no nonzero nilpotent elements, then any ring A such that R C A C T is Noetherian. Let fi be a commutative Noetherian domain with unit of Krull dimension one. The Krull-Akizuki theorem [6, Theorem 33.2] states that if Pis an integral domain containing R and contained in a finite algebraic extension of the quotient field of R then P is Noetherian. By adjoining a finite number of elements to R and letting this new ring be called R, one proves the theorem by proving the following: Any ring A between a Noetherian domain R of Krull dimension one and its quotient field is Noetherian. This is equivalent to showing that if x is any nonzero element of R then A/xA is a finite R module. We shall restate this reduction of the Krull-Akizuki theorem in such a way that the final statement is true for any Noetherian ring of any Krull dimension. In order to do this we first characterize the relationship between a one dimensional Noetherian domain R and its quotient field P. If y belongs to P, then its conductor to R, or its denominator ideal contains some powered product of a finite number of maximal ideals of R. In the special case that R has precisely one maximal ideal M, T is the set of all elements whose conductor to R contains a power of M. If B is any commutative Noetherian ring with unit and / is any ideal of B that contains a non-zero-divisor, the Ptransform of B is defined to be the set of all elements of the total quotient ring whose conductor to B contains a power of I. The Ptransform is a ring between B and its total quotient ring. If B is a Noetherian domain of Krull dimension one and B has one maximal ideal M, then the quotient field is the A/-transform of B. Now if R is any commutative Noetherian ring with unit, we call P the global transform of R if P is the set of all elements of the total quotient ring whose conductor to R contains a power of a finite product of maximal ideals of R. If M is any maximal ideal of R, TM = P ®R RM, where RM is the localization of R at M, is the A/fi -transform of RM. Also P contains the M-transform of R for M any maximal ideal of R. If R is a domain of Krull dimension one, we Received by the editors May 16, 1974 and, in revised form, November 4, 1974. AMS (MOS) subject classifications (1970). Primary 13E05, 13E10; Secondary 13B99.