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.