#### Date of Submission

7-28-2014

#### Date of Award

7-28-2015

#### 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

Das, Mrinal Kanti (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

ObjectiveThe main objectives of this thesis are the following:(i) To investigate the behaviour of the Euler class groups under integral and subintegral extensions. More precisely, given a subintegral (or integral) extension R+ S of Noetherian rings, we are interested in finding out the relationship between the Euler class group of R and the Euler class group of S.(ii) To develop a theory (namely, an extension of the theory of Euler class group to the Euler class group of R[T) relative to a projective R[T|-module L of rank 1) in order to detect the precise obstruction for a projective R[T]-module P of rank n with determinant L to split off a free summand of rank one, where n is the Krull dimension of the (Noetherian) ring R. The results on (i) will be discussed in Chapters 3, 4, 5 and the results on (ii) will be discussed in Chapters 6, 7 and 8. These results have been obtained in joint works with Mrinal Kanti Das.The results on (i) are based on the paper [D-Z 1] and the results on (ii) are based on the paper [D-Z 2).We now give brief introductions to the problems tackled in this thesis and the statements of the main results that we obtained.As both (i) and (ii) involve the theory of the Euler class groups, we start with a few words on its history and development.Obstruction theory and the Euler class groupLet A be a Noetherian ring of (Krull) dimension n. A classical result of Serre (Se] asserts that if rank(P) 2n+1, then P QO A for some A module Q (in other words, P splits off a free summand of rank one). There are well-known examples of rings A and indecomposable projective A-modules of rank < dim(A) to show that Serre's result is best possible. Most of the research in projective modules in last thirty years is centred around the following question.Question 1. Let A be a Noetherian ring of dimension d and P be a projective A-module of rank n ; d. What is the precise obstruction for P to split off a free summand of rank one?To tackle the above question one would like to find a suitable "obstruction group" G"(A) so that given a projective A-module P of rank n, an element r,(P)(A) can be associated such that zn(P) is trivial in Gn (A) if and only if P- QO A. This has been achievod in the case d= n through the following path-breaking works.

#### Control Number

ISILib-TH419

#### 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

Zinna, Md. Ali Dr., "Euler Class Groups of Polynomial and Sub Integral Extensions of a Noetherian Ring." (2015). *Doctoral Theses*. 366.

https://digitalcommons.isical.ac.in/doctoral-theses/366

## 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:28843715