79 research outputs found
Extrapolating an Euler class
Let be a noetherian ring of dimension and let be an integer so
that . Let be a unimodular row so that
the ideal has height . Jean Fasel has associated to this
row an element in the Euler class group , with
given by . If
contains an infinite field then we show that the rule of Fasel defines a
homomorphism from to . The main
problem is to get a well defined map on all of . Similar results
have been obtained by Mrinal Kanti Das and MD Ali Zinna, with a different
proof. Our proof uses that every Zariski open subset of is path
connected for walks made up of elementary matrices.Comment: 7 pages, reference update
From Mennicke symbols to Euler class groups
Bhatwadekar and Raja Sridharan have constructed a homomorphism of abelian
groups from an orbit set Um(n,A)/E(n,A) of unimodular rows to an Euler class
group. We suggest that this is the last map in a longer exact sequence of
abelian groups. The hypothetical group G that precedes Um(n,A)/E(n,A) in the
sequence is an orbit set of unimodular two by n matrices over the ring A. If n
is at least four we describe a partially defined operation on two by n
matrices. We conjecture that this operation describes a group structure on G if
A has Krull dimension at most 2n-6. We prove that G is mapped onto a subgroup
of Um(n,A)/E(n,A) if A has Krull dimension at most 2n-5.Comment: 11 pages, to appear in the Proceedings of the International
Colloquium on Algebra, Arithmetic and Geometry. TIFR, Mumbai, January 4-12,
200
A module structure on certain orbit sets of unimodular rows
AbstractAn algebraic version of cohomotopy groups is developed. Further the stabilization problem for the K1 of Bass is studied for matrices that are much smaller than those treated classically
Spherical complexes attached to symplectic lattices
To the integral symplectic group Sp(2g,Z) we associate two posets of which we
prove that they have the Cohen-Macaulay property. As an application we show
that the locus of marked decomposable principally polarized abelian varieties
in the Siegel space of genus g has the homotopy type of a bouquet of
(g-2)-spheres. This, in turn, implies that the rational homology of moduli
space of (unmarked) principal polarized abelian varieties of genus g modulo the
decomposable ones vanishes in degree g-2 or lower. Another application is an
improved stability range for the homology of the symplectic groups over
Euclidean rings. But the original motivation comes from envisaged applications
to the homology of groups of Torelli type.
The proof of our main result rests on a refined nerve theorem for posets that
may have an interest in its own right.Comment: 18 p; final versio
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