On the finiteness and stability of certain sets of associated primes ideals of local cohomology modules

Let $(R,\frak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and $N$ a finitely generated $R$-module. Let $k{\ge}-1$ be an integer and r=\depth_k(I,N) the length of a maximal $N$-sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan ({Comm. Algebra, 36 (2008), 1527-1536). For a subset S\subseteq \Spec R we set S_{{\ge}k}={\p\in S\mid\dim(R/\p){\ge}k}. We first prove in this paper that \Ass_R(H^j_I(N))_{\ge k} is a finite set for all $j{\le}r$}. Let \fN=\oplus_{n\ge 0}N_n be a finitely generated graded \fR-module, where \fR is a finitely generated standard graded algebra over $R_0=R$. Let $r$ be the eventual value of \depth_k(I,N_n). Then our second result says that for all $l{\le}r$ the sets \bigcup_{j{\le}l}\Ass_R(H^j_I(N_n))_{{\ge}k} are stable for large $n$.Comment: To appear in Communication in Algebr

High efficient infrared-light emission from silicon LEDs

Since 1965 the number of transistors on a single integrated circuit (IC) forminimum component costs has been almost doubling each 18 months, a phenomenon which is known as Moore’s law. The most advanced ICs contain more than a billion transistors and the line-width used in the next generation CMOS-processes is 32 nm. It is expected that in the coming decade the exponential increase in complexity will start saturating because we are approaching fundamental limits, and new approaches are being explored to push the development further after the\ud saturation of conventional microelectronic technology. Instead of electrical interconnects, optical integration is suggested to be used in microelectronics. Furthermore,\ud silicon nowadays remains the first material for IC technology. Meanwhile, a compact high-speed efficient silicon light source being suitable for on-chip integration\ud process is still missing. The aim of this research is to investigate this missing component.\ud This dissertation presents an exploration on infrared-light emitting probability in silicon light emitting devices (Si-LEDs) with new approaches to improve the\ud routinely low emission efficiency of silicon. It starts with a theoretical calculation of the possible internal quantum efficiency in bulk-Si LEDs and SOI-LEDs, after that a model for calculating the relationship between internal and external efficiencies is introduced. Following that the experimentally achieved results from\ud our realized LED structures implementing the theoretical predictions are exhibited subsequently

On the cofiniteness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules \displaystyle H^j_I(M,N)=\dlim\Ext^j_R(M/I^nM,N) of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal then $H^j_I(M,N)$ is $I$-cofinite for all $M, N$ and all $j$. Secondly, let $t$ be a non-negative integer such that \dim\Supp(H^j_I(M,N))\le 1 \text{for all} j Then $H^j_I(M,N)$ is $I$-cofinite for all $j<t$ and \Hom(R/I,H^t_I(M,N)) is finitely generated. Finally, we show that if $\dim(M)\le 2$ or $\dim(N)\le 2$ then $H^j_I(M,N)$ is $I$-cofinite for all $j$.Comment: 16 page