22 research outputs found
Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture
We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is
the first step of Voevodsky's proof of this conjecture for arbitrary prime l)
in a rather clear and elementary way. Assuming this conjecture, we construct a
6-term exact sequence of Galois cohomology with cyclotomic coefficients for any
finite extension of fields whose Galois group has an exact quadruple of
permutational representations over it. Examples include cyclic groups, dihedral
groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4.
Several exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn
are proven in this way. In addition, we introduce a more sophisticated version
of the classical argument known as "Bass-Tate lemma". Some results about
annihilator ideals in Milnor rings are deduced as corollaries.Comment: LaTeX 2e, 17 pages. V5: Updated to the published version + small
mistake corrected in Section 5. Submitted also to K-theory electronic
preprint archives at http://www.math.uiuc.edu/K-theory/0589
The Pure Virtual Braid Group Is Quadratic
If an augmented algebra K over Q is filtered by powers of its augmentation
ideal I, the associated graded algebra grK need not in general be quadratic:
although it is generated in degree 1, its relations may not be generated by
homogeneous relations of degree 2. In this paper we give a sufficient criterion
(called the PVH Criterion) for grK to be quadratic. When K is the group algebra
of a group G, quadraticity is known to be equivalent to the existence of a (not
necessarily homomorphic) universal finite type invariant for G. Thus the PVH
Criterion also implies the existence of such a universal finite type invariant
for the group G. We apply the PVH Criterion to the group algebra of the pure
virtual braid group (also known as the quasi-triangular group), and show that
the corresponding associated graded algebra is quadratic, and hence that these
groups have a (not necessarily homomorphic) universal finite type invariant.Comment: 53 pages, 15 figures. Some clarifications added and inaccuracies
corrected, reflecting suggestions made by the referee of the published
version of the pape
From Koszul duality to Poincar\'e duality
We discuss the notion of Poincar\'e duality for graded algebras and its
connections with the Koszul duality for quadratic Koszul algebras. The
relevance of the Poincar\'e duality is pointed out for the existence of twisted
potentials associated to Koszul algebras as well as for the extraction of a
good generalization of Lie algebras among the quadratic-linear algebras.Comment: Dedicated to Raymond Stora. 27 page
Syzygy algebras for the Segre embeddings
We describe the syzygy spaces for the Segre embedding
in terms of
representations of and construct the minimal
resolutions of the sheaves
in for and . Also we prove some
property of multiplication on syzygy spaces of the Segre embedding.Comment: 17 pages, 11 picture