Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings

AbstractAn algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. Suppose A is a homogeneous algebra over an infinite field k. Let La denote left multiplication by any nonzero element a ∈ A. Several results are proved concerning the structure of A in terms of La. In particular, it is shown that A decomposes as the direct sum A = ker La Im La. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.

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A lattice of extension rings for a commutative ring

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055067 ◽

1978 ◽

Vol 84 (2) ◽

pp. 225-234 ◽

Cited By ~ 2

Author(s):

D. Kirby ◽

M. R. Adranghi

Keyword(s):

Point Of View ◽

Local Rings ◽

Algebraic Point ◽

Multiple Points ◽

One Dimensional ◽

Maximal Ideals ◽

Blowing Up ◽

Abstract Case ◽

Algebroid Curve

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).

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Dilatation properties of regular local rings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100033648 ◽

1959 ◽

Vol 55 (1) ◽

pp. 1-9

Author(s):

D. G. Northcott

Keyword(s):

Dimensional Space ◽

Geometric Theory ◽

Commutative Rings ◽

Ideal Theory ◽

Local Rings ◽

Abstract Form ◽

One Dimensional ◽

Regular Local Rings ◽

Dimensional Projective Space ◽

The Ideal

The results and methods of algebraic geometry, when analysed in terms of modern algebra, have revealed on several occasions algebraic principles of surprising generality. Recently it has become apparent that the geometric theory of infinitely near points has, as it were, an abstract form which forms part of the ideal theory of commutative rings, but there are many details which have yet to be worked out. Roughly speaking, one may say that what corresponds to the theory of the sequence of points on a curve branch is now known in some detail, and forms a substantial addition to our knowledge of the properties of one-dimensional local rings†; but the construction of an abstract theory similarly related to the theory of neighbourhoods in n-dimensional projective space can hardly be said to have been started. A number of necessary preliminary steps were taken by the author in (3)—in the process of providing algebraic foundations for certain applications of dilatation theory—and later some applications were made to 2-dimensional problems. However, the present paper should be regarded as an attempt to initiate a dilatation theory of regular local rings to run parallel to the general theory of infinitely near points in n-dimensional space.

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