29 research outputs found
Geometric description of the connecting homomorphism for Witt groups
We give a geometric setup in which the connecting homomorphism in the
localization long exact sequence for Witt groups decomposes as the pull-back to
the exceptional fiber of a suitable blow-up followed by a push-forward.Comment: 19 pages, minor details added, reference to published paper adde
Witt groups of Grassmann varieties
We compute the total Witt groups of (split) Grassmann varieties, over any
regular base X. The answer is a free module over the total Witt ring of X. We
provide an explicit basis for this free module, which is indexed by a special
class of Young diagrams, that we call even Young diagrams.Comment: 31 pages, 16 figures. Final version. Complies with the new formalism
on total Witt groups recently developed by the authors (arXiv:1104.5051
Trivial Witt groups of flag varieties
Let G be a split semi-simple linear algebraic group over a field, let P be a
parabolic subgroup and let L be a line bundle on the projective homogeneous
variety G/P. We give a simple condition on the class of L in Pic(G/P)/2 in
terms of Dynkin diagrams implying that the Witt groups W^i(G/P,L) are zero for
all integers i. In particular, if B is a Borel subgroup, then W^i(G/B,L) is
zero unless L is trivial in Pic(G/B)/2.Comment: 3 pages, 1 figur
Invariants, torsion indices and oriented cohomology of complete flags
In the present notes we generalize the classical work of Demazure [Invariants
sym\'etriques entiers des groupes de Weyl et torsion] to arbitrary oriented
cohomology theories and formal group laws. Let G be a split semisemiple linear
algebraic group over a field and let T be its split maximal torus. We construct
a generalized characteristic map relating the so called formal group ring of
the character group of T with the cohomology of the variety of Borel subgroups
of G. The main result of the paper says that the kernel of this map is
generated by W-invariant elements, where W is the Weyl group of G. As one of
the applications we provide an algorithm (realized as a Macaulau2 package)
which can be used to compute the ring structure of an oriented cohomology
(algebraic cobordism, Morava -theories, connective K-theory, Chow groups,
K_0, etc.) of a complete flag variety.Comment: 36pp. xypi
A coproduct structure on the formal affine Demazure algebra
In the present paper we generalize the coproduct structure on nil Hecke rings
introduced and studied by Kostant-Kumar to the context of an arbitrary
algebraic oriented cohomology theory and its associated formal group law. We
then construct an algebraic model of the T-equivariant oriented cohomology of
the variety of complete flags.Comment: 28 pages; minor revision of the previous versio