Some results concerning the local analytic branches of an algebraic variety

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.

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Graded Complexes Over Power Series Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1986-008-8 ◽

1986 ◽

Vol 38 (1) ◽

pp. 158-178 ◽

Cited By ~ 1

Author(s):

Paul Roberts

Keyword(s):

Power Series ◽

Local Ring ◽

Simple Structure ◽

Free Resolution ◽

Koszul Complex ◽

Local Rings ◽

Power Series Ring ◽

Series Ring ◽

Noetherian Local Ring ◽

Power Series Rings

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.

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On strongly J-Noetherian rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501444 ◽

2021 ◽

pp. 2250144

Author(s):

Esmaeil Rostami

Keyword(s):

Power Series ◽

Commutative Rings ◽

The Other ◽

Noetherian Rings ◽

Power Series Rings

In this paper, we introduce a class of commutative rings which is a generalization of ZD-rings and rings with Noetherian spectrum. A ring [Formula: see text] is called strongly[Formula: see text]-Noetherian whenever the ring [Formula: see text] is [Formula: see text]-Noetherian for every non-nilpotent [Formula: see text]. We give some characterizations for strongly [Formula: see text]-Noetherian rings and, among the other results, we show that if [Formula: see text] is strongly [Formula: see text]-Noetherian, then [Formula: see text] has Noetherian spectrum, which is a generalization of Theorem 2 in Gilmer and Heinzer [The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980) 13–16].

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