相对Hopf模和广义cleft扩张

本文研究了π-cleft扩张的余代数分解问题及对偶的η-cocleft扩张的代数分解问题.利用相对Hopf模结构定理,给出了一个条件,使得具有余代数结构的π-cleft扩张有较好的余代数分解形式.作为特例,可以重新得到Radford给出的具有投射的双代数的双积分解定理.     

OPENCORE NMR: open-source core modules for implementing an integrated FPGA-based NMR spectrometer

A tool kit for implementing an integrated FPGA-based NMR spectrometer [K. Takeda, A highly integrated FPGA-based nuclear magnetic resonance spectrometer, Rev. Sci. Instrum. 78 (2007) 033103], referred to as the OPENCORE NMR spectrometer, is open to public. The system is composed of an FPGA chip and several peripheral boards for USB communication, direct-digital synthesis (DDS), RF transmission, signal acquisition, etc. Inside the FPGA chip have been implemented a number of digital modules including three pulse programmers, the digital part of DDS, a digital quadrature demodulator, dual digital low-pass filters, and a PC interface. These FPGA core modules are written in VHDL, and their source codes are available on our website. This work aims at providing sufficient information with which one can, given some facility in circuit board manufacturing, reproduce the OPENCORE NMR spectrometer presented here. Also, the users are encouraged to modify the design of spectrometer according to their own specific needs. A home-built NMR spectrometer can serve complementary roles to a sophisticated commercial spectrometer, should one comes across such new ideas that require heavy modification to hardware inside the spectrometer. This work can lower the barrier of building a handmade NMR spectrometer in the laboratory, and promote novel and exciting NMR experiments.     

Remarks on derived complete modules and complexes

Let R be a commutative ring and $I\subset R$ a finitely generated ideal. We discuss two definitions of derived I-adically complete (also derived I-torsion) complexes of R-modules, which appear in the literature: the idealistic and the sequential ones. The two definitions are known to be equivalent for a weakly proregular ideal I; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider I-adically flat R-modules.     

An Alexandrov topology for maximal Cohen-Macaulay modules

Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies.