1-Dimensional Cohen-Macaulay Rings
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327 Eben Matlis Northwestern University, Evanston, IL/USA
1-Dimensional Cohen-Macaulay Rings
Springer-Verlag Berlin. Heidelberg • New York 1973
A M S Subject Classifications (1970): 13-02, 13Cxx, 13E05, 1 3 E l 0 , 13F05, 1 3 H x x
I S B N 3-540-06327-7 Springer-Verlag B e r l i n . H e i d e l b e r g . N e w Y o r k I S B N 0-387-06327-7 Springer-Verlag N e w Y o r k " H e i d e l b e r g " Berlin This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher, © by Springer-Verlag Berlin - Heidelberg 1973. Library of Congress Catalog Card Number 73-80869. Printed in Germany. Offsetdruck: Julius Bettz, Hemsbach/Bergstr.
TABLE Chapter
I
h-Divisible Chapter
and C o t o r s i o n
Chapter
Extensions
11
. . . . . . . . . . . . . . . . . . . . . .
28
IV
Localizations Chapter
. . . . . . . . . . . . . . . . . . . . . . . . . .
34
V
Artinian Chapter
Divisible
Modules . . . . . . . . . . . . . . . . . . . .
42
Vi
Strongly Chapter
Unramified
Ring Extensions
. . . . . . . . . . . . . . .
56
VII
Closed
Chapter
Components
of R . . . . . . . . . . . . . . . . . . . .
68
Modules . . . . . . . . . . . . . . . . . . . . .
79
VIII
Simple
Divisible
Chapter
IX
Semi-Simple Chapter
and U n i s e r i a l
Divisible
Modules
. . . . . . . . . . .
85
X
Integral
Chapter
Closure . . . . . . . . . . . . . . . . . . . . . . .
90
XI
Primary
Chapter
Decomposition
. . . . . . . . . . . . . . . . . . . .
96
Ring . . . . . . . . . . . . . . . . . . .
103
XII
The F i r s t Chapter
Neighbourhood
XIII
Gorenstein Chapter
Rings . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
116
XIV
Multiplicities
The
I
III
Chapter
The
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
Compatible
The
Modules
II
Completions
The
OF C O N T E N T S
. . . . . . . . . . . . . . . . . . . . . . . . . .
126
Ideal
141
XV
Canonical
of R . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
153
Index
155
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
To
Margarita Hill Matlis
INTRODUCTION The purpose Artinian ring,
of these notes
modules
is to present
over a 1-dimensional,
a structure
Noetherian,
and to show that m a n y of the properties
derived
from a knowledge
to recapture Macaulay
of this theory.
m a n y of the known
results
rings by these techniques,
Thus we have a new and unifying &mining
a category
of modules
theory for
Cohen-Macaulay
of such a ring may be
In fact, we shall be able
about
1-dimensional
and find
Cohen-
some new ones
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