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 $M$.
On the Local Homology Theory for Artinian Modules
Let A be a non-zero Artinian R-module. For an arbitrary ideal I of R, we show that the local homology module Hpx(A) is independent of the choice of x whenever 0 : AI = 0 : A(x1,...,xr). Thus, identifying these modules, we write HpI(A). In this paper we prove that there is a certain duality between HiI(A) and the local cohomology modules and provide some information about the vanishing local homology module HiI(A) which gives an improved form of the main results of [22].
Yet another application of the Gauss-Bonnet Theorem for the sphere
where K and dAg denote the Gaussian curvature and the area element of g, respectively. In particular, this implies the existence of at least an elliptic point of (S, g). In this short note we prove the Fundamental Theorem of Algebra as a consequence of this fact. As far as the authors know, this proof is new. Moreover, it does have the advantage of being directly based on the work of Gauss, who was the first mathematician giving an universally accepted proof of the Fundamental Theorem of Algebra (see [2], [3]). Hence our argument can also have some historical and pedagogical interest. As it is well known, the stereographic projection allows to identify the sphere S with the one-point compactification Ĉ = C ∪ {∞} of the complex plane C. Let us take p(z) = a0 + a1z+ · · ·+ anz n with a0 an 6= 0 and let us assume that p(z) 6= 0 for all z ∈ C. We set p∗(z) = an + an−1z + · · ·a0z , that is, the i-th coefficient of p∗(z) is the (n− i)-th coefficient of p(z). Clearly, p∗(z) = z n p(1/z) for all z ∈ C \ {0}. Put