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Unmixed Local Rings of Type Two are Cohen‐Macaulay

Unmixed Local Rings of Type Two are Cohen‐Macaulay UNMIXED LOCAL RINGS OF TYPE TWO ARE COHEN-MACAULAY THOMAS MARLEY Let R be a local ring of dimension d with maximal ideal m and residue field k. For i ^ 0, the rth Bass number of R, denoted fJ. (R), is defined to be dim Ext (k, R). See t fc R [1] for the basic properties of Bass numbers. The type of R is defined to be fi (R). Answering a conjecture of Vasconcelos [5], Roberts [4] proved that local rings of type one are Cohen-Macaulay (CM) and hence Gorenstein. By modifying Roberts' argument, Costa, Huneke and Miller [2] showed that if R is a local ring of type two and its completion is a domain, then R is CM. After giving examples to show that local rings of type two are not in general CM, they posed a question as to whether there exists a complete, equidimensional, reduced local ring of type two which is not CM. We answer this question in the negative; in particular, by stretching Roberts' argument even further, we are able to prove the following result. THEOREM. Let R be an unmixed local ring of type two. Then R is CM. A local http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Unmixed Local Rings of Type Two are Cohen‐Macaulay

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/23.1.43
Publisher site
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Abstract

UNMIXED LOCAL RINGS OF TYPE TWO ARE COHEN-MACAULAY THOMAS MARLEY Let R be a local ring of dimension d with maximal ideal m and residue field k. For i ^ 0, the rth Bass number of R, denoted fJ. (R), is defined to be dim Ext (k, R). See t fc R [1] for the basic properties of Bass numbers. The type of R is defined to be fi (R). Answering a conjecture of Vasconcelos [5], Roberts [4] proved that local rings of type one are Cohen-Macaulay (CM) and hence Gorenstein. By modifying Roberts' argument, Costa, Huneke and Miller [2] showed that if R is a local ring of type two and its completion is a domain, then R is CM. After giving examples to show that local rings of type two are not in general CM, they posed a question as to whether there exists a complete, equidimensional, reduced local ring of type two which is not CM. We answer this question in the negative; in particular, by stretching Roberts' argument even further, we are able to prove the following result. THEOREM. Let R be an unmixed local ring of type two. Then R is CM. A local

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1991

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