## Doctoral Theses

### Study on Algebras with Retractions and Planes over a DVR.

12-22-2009

12-22-2010

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Dutta, Amartya Kumar (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Aim:The main aim of this thesis is to study the following problems:1. For a Noetherian ring R, to find a set of minimal sufficient fibre conditions for an R-algebra with a retraction to R to be an A1-fibration over R.2. To investigate sufficient conditions for a factorial A1-form, with a retraction to the base ring, to be A1.3. To investigate whether planes of the form b(X, Y)Zn â€“ a(X, Y) are co- ordinate planes in the polynomial ring in three variables X, Y and Z over a discrete valuation ring.The 1st problem will be discussed in Chapter 3 entitled Codimension- one A-fibration with retraction, the 2nd problem will be studied in Chapter 4 under the heading A1-form with retraction and the 3rd problem will be investigated in Chapter 5 which has the title Planes of the form b(X, Y)Zn - a(X, Y) over a DVR.Brief introductions to the topics of the problems and precise statements of the main results obtained are given below:âˆ™Codimension-one A1-fibration with retractionLet R be a ring. A1 finitely generated flat R-algebra A is said to be an A1- fibration over R ifR k(P)[1]=k(P)[1]for all prime ideals P of R. A very nteresting and important phenomenon is that the generic and codimension- one fibres determine an A1-fibration. To get a feel for this striking feature of A1-fibration, here is a nice result by Bhatwadekar-Dutta( [BD95):Theorem 1.0.1.Let R be a Noetherian domain with field of fractions K and A an R-subalgebra of R[T1, T2,,T) such that A is flat over R, A OR K = K and A OR k(P) is an integral domain for every prime ideal P in R of height one. Then (i) If R is normal, then A SymR(I) for an invertible ideal I of R.(ii) If R contains Q, then A is an A1-fibration over R.(iii) If R is seminormal and contains Q, then Aâ‰Œ Symp(I) for an invertible ideal I of R.An analogous result has also been obtained by Dutta ( [Dut95]) for finitely generated faithfully flat R-subalgebras:Theorem 1.0.2.Let R be a Noetherian domain with field of fractions K and A a faithfully flat finitely generated R-algebra such that A âŠ—R K = K[1] and AÂ®RK(P) is geometrically integral for every prime ideal P in R of height one. Then(i) If R is normal, then Aâ‰ŒSymR(I) for an invertible ideal I of R.(ii) If R contains Q, then A is an A-fibration over R.(iii) If R is seminormal and contains Q, then Aâ‰ŒSymR(I) for an invertible ideal I of R.We will call an R-algebra A a Codimension-one A1-fibration if AâŠ— Rk(P) = k(P)â‰¤[1] for each prime ideal P of R with ht(P) < 1. In view of the above theorems it is easy to see that1. For a Noetherian normal domain R or a Noetherian domain R containing Q, a flat R-subalgebra A of a polynomial algebra over R is an A1-fibration over R if and only if A is a codimension-one A1-fibration over R.

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:28842824

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