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In Commutative Algebra certain /-adic filtrations of Noetherian rings, i.e. the so-called Zariski rings, are at the basis of singularity theory. Apart from that it is mainly in the context of Homological Algebra that filtered rings and the associated graded rings are being studied not in the least because of the importance of double complexes and their spectral sequences. Where non-commutative algebra is concerned, applications of the theory of filtrations were mainly restricted to the study of enveloping algebras of Lie algebras and, more extensively even, to the study of rings of differential operators. It is clear that the operation of completion at a filtration has an algebraic genotype but a topological fenotype and it is exactly the symbiosis of Algebra and Topology that works so well in the commutative case, e.g. ideles and adeles in number theory or the theory of local fields, Puisseux series etc, .... . In Non commutative algebra the bridge between Algebra and Analysis is much more narrow and it seems that many analytic techniques of the non-commutative kind are still to be developed. Nevertheless there is the magnificent example of the analytic theory of rings of differential operators and 1J-modules a la Kashiwara-Shapira.
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Bibliographic Information
Book Title: Zariskian Filtrations
Authors: Li Huishi, Freddy Oystaeyen
Series Title: K-Monographs in Mathematics
DOI: https://doi.org/10.1007/978-94-015-8759-4
Publisher: Springer Dordrecht
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eBook Packages: Springer Book Archive
Copyright Information: Springer Science+Business Media Dordrecht 1996
Hardcover ISBN: 978-0-7923-4184-0Published: 31 October 1996
Softcover ISBN: 978-90-481-4738-0Published: 15 December 2010
eBook ISBN: 978-94-015-8759-4Published: 17 April 2013
Series ISSN: 1386-2804
Edition Number: 1
Number of Pages: IX, 253
Topics: Associative Rings and Algebras, Category Theory, Homological Algebra, Algebraic Geometry, Partial Differential Equations, Elementary Particles, Quantum Field Theory