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Title of Thesis

Stanley Decompositions of Multigraded Modules and Reductions Modulo Regular Elements

Author(s)

Asia Rauf

Institute/University/Department Details
Abdus Salam School Of Mathematical Sciences / GC University, Lahore
Session
2010
Subject
Mathematics
Number of Pages
68
Keywords (Extracted from title, table of contents and abstract of thesis)
Stanley, Decompositions, Multigraded, Modules, Reductions, Modulo, Regular, Elements,  Algebra, tensor

Abstract
In Chapter 1, some necessary definitions and results from commutative algebra are given along with a description on the progress towards the Stanley decompositions of multigraded S-modules, where S = K[x1, . . . , xn] is a polynomial ring in n variables over a field K.
In Chapter 2, we study the behavior of Stanley decompositions and of pretty clean filtrations under reduction modulo a regular element. We prove that the Stanley depth of cyclic module drop by one under reduction modulo a regular element. We see that the cyclic module is pretty clean if and only if the reduction modulo a regular element is pretty clean. We also discuss the behavior of depth, Stanley depth and dimension on algebra tensor products.
In Chapter 3, it is shown that to what extend Stanley depth behaves like ordinary depth under reduction modulo an element and is clarified the different behavior of the two concepts by some examples. We discuss the behavior of depth, Stanley depth and dimension along short exact sequences of finitely generated multigraded S-modules.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

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1

PRELIMINARIES

1.1 Graded rings and modules

1.2 Monomial ideals

1.3 Primary decomposition of modules

1.4 Dimension theory

1.5 Regular sequences and depth

1.6 Koszul complex and Koszul homology

1.7 Prime filtrations

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3 2 STANLEY DECOMPOSITIONS, PRETTY CLEAN FILTRATIONS AND REDUCTIONS MODULO REGULAR ELEMENTS

2.1 Stanley decompositions of multigraded modules

2.2 Prime filtrations and Stanley decompositions

2.3 Stanley’s conjecture

2.4 Stanley decompositions and regular elements

2.5 Pretty clean filtrations and regular elements

2.6 Behavior of depth, Stanley depth and dimension on algebra tensor products

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4 3 DEPTH, STANLEY DEPTH AND DIMENSION OF MULTIGRADED MODULES

3.1 Behavior of depth, Stanley depth and dimension under reduction modulo an element

3.2 Behavior of depth, Stanley depth and dimension on short exact sequences

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5 4 BIBLIOGRAPHY

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