## Date of Submission

2-28-1982

## Date of Award

2-28-1983

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Mathematics

## Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

## Supervisor

Roy, Amit (TSMU-Kolkata; ISI)

## Abstract (Summary of the Work)

Unless otherwise stated all rings are assumed to be commutative and Noetherian with finite Krull dimension and modules are finitely generated. There are three related questions in commutative algebra. They are about the existence of unimodular elements in projective modules, cancellation of projective modules and minimal number of generators for finitely generated modules. In particular, determining the minimal number of generators of ideals is also of great interest in this area.In this thesis we shall be concerned with the above questions in the case of Laurent polynomial rings.The thesis consists of three Chapters. The main results are presented in Chapters II and III.We shall brifely recall some classical results in the abovementioned area. Serre (Sr) proved that : if P is a projective module over a ring R with rank P > dim R+1 then P has a unimodular element. Cancellation theorem of Bass (Ba-2] says that : if P is a projective module over R with rank P > dim R+1 then P has cancellation property. Forster [F] and Swan [Sw] proved that : for an R-module M, if n = max{u(p,M) + dim(R/p) |p is a prime ideal with M, + 0} then M is generated by n elements.A unified treatment to all these problems was given by Eisenbud and Evans [EE-2). They introduced the idea of basic elements for modules, extending the concept of unimodular elements for projective modules, and deduced all the results mentioned above.It was also known that the above mentioned results are the best possible in the general situation. However one expected to improve these results for special kinds of rings.Eisenbud and Evans [EE-1] suggested the following three conjectures for polynomial rings:EEC I. If M is a finitely generated R[T]- module such that u(p, M) > dim R +1 then M has a basic element.EEC II. If P is a finitely generated projective module over R[T] of rank > dim R +1 and if P' and Q are finitely generated projective modules such that POQ - P' OQ then P= P'EEC III. Let M be a finitely generated R[T]- module and let e(M) = max{ u(p,M) + dim(R[T]/p) |pe Spec (R[ T] )with dim(R[T]/p) < dim RJ. Then M is qenerated by e(M) elements.

## Control Number

ISILib-TH422

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

## Recommended Citation

Mandal, S. Dr., "Some Results About Basic Elements, Cancellation and Efficient Generation of Modules over Laurent Polynomial Rings." (1983). *Doctoral Theses*. 296.

https://digitalcommons.isical.ac.in/doctoral-theses/296

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843339