Study on Algorithms for D-Modules

D-模代表微分算子环上的模，它对应于线性偏微分方程组。D-模的理论主要是由M.Sato，M.Kashiwara和T. Kawai在20世纪70年代发展起来的。然而，对于D-模的实际计算还没有系统的研究。通过与N.Takayama的合作，我发现了D-模的算法，将Groebner基技术应用于D-模。特别是，我们已经找到了算法计算的上同调群与逆图像（限制）和直接图像（整合）的D-模块。在这些算法的基本成分之一是自由分辨率的D-模块，这是适应过滤（或权重向量）的计算。然而我们得到的自由分辨率往往太大而无法完成计算，为了克服这个瓶颈，我们引入了适应于过滤的最小自由分辨率的概念，并证明了这样的最小自由分辨率是可计算的。这种最小分辨率算法使我们能够计算，例如，代数de Rham上同调群和代数局部上同调群。N.Takayama在他的计算机代数系统Kan中实现了该算法，并通过互联网向公众开放。

D-modules stand for modules over the ring of differential operators, which corresponds to systems of linear partial differential equations. The theory of D-modules has been developed since 1970's mainly by M.Sato, M.Kashiwara and T.Kawai. However, there was no systematic study on actual computation for D-modules. By the collaboration With N.Takayama, I have found algorithms for D-modules applying Groebner basis techniques to D-modules. In particular, we have found algorithms for computing the cohomology groups associated with the inverse image (restriction) and the direct image (integration) of a D- module. One of the essential ingredients in these algorithsms is the computation of free resolutions for D-modules which are adapted to a filtration (or a weight vector). However free resolutions we obtained were often too big to complete the computation.In order to overcome this bottleneck, we have introduced the notion of minimal free resolution that is adapted to a filtration and have shown that such a minial free resolution is computable. This minimal resolution algorithm enables us to compute, e.g., algebraic de Rham cohomology groups and algebraic local cohomology groups more effciently. N.Takayama has implemented this algorithm in his computer algebra system Kan and has opened it to the public via internet.