## Date of Submission

3-28-2001

## Date of Award

3-28-2002

## 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

Srinivas, V.

## Abstract (Summary of the Work)

The main theme of this thesis is to study the theory of algebraic cycles on singular varieties over a field. This has been studied before extensively by Collins, Barbieri-Viale, Levine, Srinivas among several others. Our interest in this thesis is to address some well known problems in the theory of zero-cycles over nominal varieties. The use of K- theoretic techniques in our proofs illustrate the interplay between the study of algebraic cycles and algebraic K-theory.For a quasi-projective surface X over a field k, we define FA,(X) to be the subgroup of the Grothendieck group Ko(X) of vector bundies generated by the classes of smooth codimension 2 points of X. By a result of Levine [L3] (see also BS), one knows that if k is algebraically closed, then F2Ko(N) is naturally isomorphic to CH2(X), the (cohomological) Chow group of zero cycles on X modulo rational equivalence. We recall here that zero cycles are elements of the free abelian group on smooth. codimension 2 points of V. Following Levine and Weibel [LW], the cycles rationally equivalent to 0 are defined to be sums of divisors of suitable rational functions on Cartier curves in X. It is known by the work of Bloch and Levine [L2 that CH2(.N) Nv) for any quasi-projective surface Y over an algebraically closed field.For any closed sub-scheme Z of N, we denote by KolX, Z), the relative K-group as defined, for example, in [CS), and let F2Ko(.X, Z) be the subgroup of Ko(X, Z) defined by the classes of smooth points of X \\Z as in [CS].These definitions are particular instances of the more general theory of [TT]. where for example the negative relative K-groups are defined as well.In Chapter 1, we set up various definitions, notations etc. and discuss some background material. We also prove some lemmas and other results needed for the rest of this thesis. In Chapter 2. we consider the problem of understanding the relation between the Chow group of 0-cycles on a normal quasi-projective surface X over a field, and that on its desingularization. Let *:Aâ†’X be a resolution of singularities of X, with reduced exceptional divisor E. We prove a formula, conjectured by Bloch and Srinivas. This formula describes the Chow group of 0-cycles CH2(X) of X, as an inverse limit of the relative Chow groups F2Ko(X,nE) of relative to multiples of E. Later in Chapter 4, we prove a result, which will imply that, if the ground field k is algebraically closed of characteristic p>t 0, then the relative Chow groups of X relative to all non-zero multiples of E are same. Hence in this case, CH2 (X) is isomorphic to the relative Chow group F2Ko(Ã‘,E) of relative to E. In the end of Chapter 2, we demonstrate the necessity of taking higher multiples of the exceptional divisor in characteristic 0 (see (S5).

## Control Number

ISILib-TH121

## 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

Krishna, Amalendu Dr., "Zero-Cycles and K-Theory on Normal Surfaces." (2002). *Doctoral Theses*. 87.

https://digitalcommons.isical.ac.in/doctoral-theses/87

## 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:28842863