Characterization of modules of finite projective dimension over complete intersections

Let be a finitely generated module over a local complete intersection of characteristic . The property that has finite projective dimension can be characterized by the vanishing of for some and for some .

     

On modules of finite projective dimension over complete intersections

Recently Avramov and Miller proved that over a local complete intersection ring in characteristic 0$">, a finitely generated module has finite projective dimension if for some 0$"> and for some 0$">, Tor}^{R}_{i}(M,f^{n}_{R})=0-f^{n}$"> being the frobenius map repeated times. They used the notion of ``complexity' and several related theorems. Here we offer a very simple proof of the above theorem without using ``complexity' at all.

     

A Traffic Splitting Algorithm for Load Balancing in Tor

As the most popular anonymous communication system, Tor provides anonymous protection for users by sending their messages through a series of relays. Due to the use of the bandwidth-weighted path selection algorithm, many more users choose routers with high bandwidth as relays. This will cause the utilization of high bandwidth routers to be much higher than that of low bandwidth routers, which will bring congestion risk. The Quality of Service (QoS) is difficult to guarantee for users who need delay-sensitive services such as web browsing and instant messaging. To reduce the average load of routers and improve the network throughput, we propose a circuit construction method with multiple parallel middle relays and conduct a dynamic load allocation method. The experiment demonstrates that our proposed method can provide better load balancing. Compared with other multipath anonymous communication networks, our proposed method can provide better anonymity.     

The General Structure of Inverse Polynomial Modules

In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as -modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.     

Tor gives the inverse to the Hilbert function of a graded algebra

The Hilbert generating function of Tor^U(A, A) gives the inverse of the Hilbert generating function of U where U is a graded A algebra.     

Tor与Ext的对偶关系

Tor与Ext是模范畴中重要的导出函子.本文首先讨论它们的某些性质,然后讨论它们之间的对偶关系.     

Characterizations of regular local rings in positive characteristics

In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher Tor}$"> counterparts Tor}_i(k,{}^{f^n} R))/p^{nd}$">. We also apply the characterizations to improve a recent result by Bridgeland and Iyengar in the characteristic case. Our proof avoids using the existence of big Cohen-Macaulay modules, which is the major tool in the proof of Bridgeland and Iyengar.

     

Homology with Models and Tor

We consider the extension of the notion of a projective module to that of a projective functor relative to a model set (as in Dold, MacLane, Oberst, 1967). Then taking projective resolutions of functors, we consider the usual associated homology.We show that in some cases, including the classical simplicial homology of topological spaces, the model set can be replaced by a model set having only one element. We show that when the model set consists of a single element the homology modules can be interpreted as values of the Torsion functor. In the case of simplicial homology of topological spaces these Tors will be shown to be analogous to the Tors which occur in group homology.