Modules with regular singularities over filtered rings (Q580441)

The theory of holonomic systems with regular singularities is a generalization to several variables of the classical theory of ordinary differential equations of Fuchsian type [cf. e.g. \textit{M. Kashiwara} and \textit{T. Kawai}, Publ. Res. Inst. Math. Sci. 17, 813-979 (1981; Zbl 0505.58033)]. An important property of such systems is the following: They can be globally endowed with a good filtration. Many of the results in this theory are proved using micro-local analysis. In the paper under review there is developed a purely algebraic theory of modules with regular singularities over a large class of filtered rings, including the rings of differential operators in the above mentioned paper by Kashiwara and Kawai. The main result (theorem 7.3) gives several equivalent descriptions of the notion of a holonomic A-module M with regular singularities. One of them is the existence of a very good filtration on M, which makes the link with the results of Kashiwara's paper. The paper under review builds heavily on the author's paper on algebraic micro- localization [Commun. Algebra 14, 971-1000 (1986; Zbl 0607.16001)]. The author works mainly with filtered rings A such that gr(A) is a commutative \({\mathbb{Q}}\)-algebra and the filtration is noetherian. For such rings, the notion of holonomic A-module is introduced, and many of their properties are derived. A special class of filtered rings, called E-rings by the author, is studied in {\S} 4; and in {\S} 6 a formalism is developed which makes it possible to obtain results for arbitrary filtered rings from results on E-rings, thus proving theorem 7.3. Reviewer's remark: Most of the contents of section 9.4 can be found, e.g. in the book by \textit{D. G. Northcott} [``Lessons on rings, modules and multiplicities'' (1968; Zbl 0159.330)].