Research

      
Overview

My primary interest is in rings, ideals and modules .  I work with projective modules over commutative, Noetherian rings. A projective module is a direct summand of a free module. Projective modules can also be thought of as algebraic analogue of vector bundles. One example of a projective module which is not free is the tangent bundle of the two-dimensional real sphere (proof of which follows from the so called "Hairy ball theorem"). One central question is to find necessary and sufficient conditions for a projective module to split off a free summand of rank one (or, a vector bundle to split off a trivial bundle). In a situation when rank of the projective module is equal to the Krull dimension of the ring (on which the module is defined), there are some invariants which take care of the above question. For instance, if the ring is the coordinate ring of an affine variety (smooth) over an algebraically closed field, the Chern class of the projective module is the desired invariant, which vanishes in the Chow group if and only if the projective module splits off a free, rank one summand. But example of tangent bundle mentioned above shows that the Chern class may vanish without the projective module splitting a free summand of rank one, thus implying the requirement of a better invariant than Chern class in cases when the projective module is defined on an affine domain over a field which is not algebraically closed (for example, affine varieties over real numbers). Motivation and model of this invariant comes from topology. It is the Euler class. At present I work in this Euler class theory.


Job Portfolio
Curriculum Vitae

Research Statement

Teaching Statement


Publications

17) Orbit spaces of unimodular rows over smooth real affine algebras (with Soumi Tikader and  Md. Ali Zinna)
Inventiones Mathematicae  (accepted).

16)
"Strong" Euler class of a stably free module of odd rank  (with Md. Ali Zinna)
Journal of Algebra 432 (2015), 185-204.

15)
Efficient generation of ideals in overrings of polynomial rings (with Md Ali Zinna)
Journal of Pure and  Applied Algebra 219 (2015), 4016-4034.

14) Projective generation of curves (III) (with S. M. Bhatwadekar)
IMRN Vol. 2015, No. 4, 960-980.

13) The Euler class group of a polynomial algebra with coefficients in a line bundle
(with Md. Ali Zinna), Math. Zeitschrift. 276 (2014), 757-783.

12) On invariance of the Euler class groups under a subintegral base change
(with Md. Ali Zinna), Journal of Algebra 398 (2014), 131-155.

11)
On triviality of the Euler class group of a deleted neighbourhood
of a smooth local scheme

Transactions of the American Mathematical Society, 365 (2013), 3397-3411.

10)
A question of Nori, Segre classes of ideals and other applications
(with Manoj K. Keshari),
Journal of Pure and  Applied Algebra, 216 (2012), 2193-2203.

9) Revisiting Nori's question and homotopy invariance of Euler class groups
Journal of K-Theory, 8 (2011), 451-480.

8) Euler class groups and a theorem of Roitman (with Raja  Sridharan)
Journal of Pure and  Applied Algebra, 215 (2011), 1340-1347.

7) Good invariants for bad ideals (with Raja  Sridharan)
Journal of Algebra, 323 (2010), 3216-3229.

6)  Projective modules over smooth real affine varieties
(with S. M. Bhatwadekar and S. Mandal
Inventiones Mathematicae, 166 (2006), 151-184.

5)  The Euler class group of a polynomial algebra II
Journal of Algebra, 299 (2006), 94-114.

4)  A Riemann-Roch theorem (with S. Mandal
Journal of Algebra, 301 (2006), 148-164.

3)  Euler class constructions (with S. Mandal
Journal of Pure and Applied Algebra, 198 (2005) 93-104.

2)  The Euler class groups of polynomial rings and unimodular elements
in projective modules
(with Raja  Sridharan)
Journal of Pure and Applied Algebra, 185 (2003), 73-86.

1)  The Euler class group of a polynomial algebra
 Journal of Algebra, 264 (2003), 582-612.
 


Preprints/In Preparation
 
1) "$P^1$-gluing" for local complete intersections (with Soumi Tikader and Md Ali Zinna) (submitted).

2) On two conjectures of Murthy (submitted).