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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On maximality of Gorenstein sequences


Author: Maria Grazia Marinari
Journal: Proc. Amer. Math. Soc. 59 (1976), 33-38
MSC: Primary 13H10
DOI: https://doi.org/10.1090/S0002-9939-1976-0441956-7
MathSciNet review: 0441956
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Abstract: It is well known that if $ A$ is a Gorenstein ring, then every ideal generated by a regular sequence $ {\text{x}} \subset A$ has irreducible (minimal) primary components. This feature led us to define a Gorenstein sequence of a ring $ A$ to be any ordered regular sequence $ {\text{x}} = \{ {x_1}, \ldots ,{x_r}\} \subset A$ such that for every $ i \in \{ 1, \ldots ,r\} $ the ideal $ ({x_1}, \ldots ,{x_i})$ has irreducible minimal primary components. We showed for Gorenstein sequences ( $ {\mathbf{G}}$-sequences for short) some parallels of well-known properties of regular sequences and moreover by means of $ {\mathbf{G}}$-sequences we gave the following natural characterization of local Gorenstein rings: ``A local ring $ (A,\;\mathfrak{m})$ is Gorenstein iff $ \mathfrak{m}$ contains a $ {\mathbf{G}}{\text{ - sequence of length = }}K{\text{ - }}\dim A$".

In this note we are going to give some information about ``maximality'' of $ {\mathbf{G}}$-sequences in a local ring $ A$, producing sufficient conditions on $ A$ in order that the maximal $ {\mathbf{G}}$-sequences of $ A$ all have the same length, i.e. in order to give a ``good'' definition of $ {\mathbf{G}}$-depth $ A$. Furthermore, we will state some results about the $ {\mathbf{G}}$-depth behavior with respect to local flat ring homomorphisms.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0441956-7
Keywords: Commutative noetherian Cohen-Macaulay Gorenstein ring, Gorenstein sequence, canonical module
Article copyright: © Copyright 1976 American Mathematical Society