93 research outputs found
The Schur multiplier of finite symplectic groups
We show that the Schur multiplier of is
, when is divisible by 4.Comment: Bull. Soc. Math. France, to appea
Hochster duality in derived categories and point-free reconstruction of schemes
For a commutative ring , we exploit localization techniques and point-free
topology to give an explicit realization of both the Zariski frame of (the
frame of radical ideals in ) and its Hochster dual frame, as lattices in the
poset of localizing subcategories of the unbounded derived category .
This yields new conceptual proofs of the classical theorems of Hopkins-Neeman
and Thomason. Next we revisit and simplify Balmer's theory of spectra and
supports for tensor triangulated categories from the viewpoint of frames and
Hochster duality. Finally we exploit our results to show how a coherent scheme
can be reconstructed from the tensor triangulated structure
of its derived category of perfect complexes.Comment: v5:Minoir typos corrected the proof of tensor nilpotence is made
totally point-free and self-contained; some simplifications and expository
improvements; section on preliminaries shortened; 50pp. To appear in Trans.
AM
Homology fibrations and "group-completion" revisited
We give a proof of the Jardine-Tillmann generalized group completion theorem.
It is much in the spirit of the original homology fibration approach by McDuff
and Segal, but follows a modern treatment of homotopy colimits, using as little
simplicial technology as possible. We compare simplicial and topological
definitions of homology fibrations.Comment: 13 page
Some remarks on the Maslov index
It is a classical fact that Wall's index of three Lagrangians in a symplectic
space over a field defines a -cocycle on the associated
symplectic group with values into the Witt group of . Moreover, module the
square of the fundamental ideal this is a trivial -cocycle. In this work we
revisit this fact from the viewpoint of the theory of Sturm sequences and
Sylvester matrices developed by J.~Barge and J.~Lannes in their book ``Suites
de Sturm, indice de Maslov et p\'eriodicit\'e de Bott. (French) [Sturm
sequences, Maslov index and Bott periodicity]'' Progress in Mathematics, 267.
This allows us in particular to give an explicit formula for the coboundary
associated to the mod reduction of the cocycle which is valid on
any field of characteristic different from Comment: 19 page
Finite quotients of symplectic groups vs mapping class groups
We give alternative computations of the Schur multiplier of , when is divisible by 4 and : using K-theory
arguments based on the work of Barge and Lannes and a second one based on the
Weil representations of symplectic groups arising in abelian Chern-Simons
theory. We can also retrieve this way Deligne's non-residual finiteness of the
universal central extension . We prove then that
the image of the second homology into finite quotients of symplectic groups
over a Dedekind domain of arithmetic type are torsion groups of uniformly
bounded size. In contrast, quantum representations produce for every prime ,
finite quotients of the mapping class group of genus whose second
homology image has -torsion. We further derive that all central extensions
of the mapping class group are residually finite and deduce that mapping class
groups have Serre's property for trivial modules, contrary to symplectic
groups. Eventually we compute the module of coinvariants
.Comment: 40p., 3 figures, former arxiv:1103.1855 is now split into two
separate papers, the actual arxiv:1103.1855 and the present on
Short incompressible graphs and -free groups
Consider a finite connected -complex endowed with a piecewise
Riemannian metric and whose fundamental group is freely indecomposable, of rank
at least , and in which every -generated subgroup is free. In this paper
we show that we can always find a connected graph such that
(in short, a
-incompressible graph) whose length satisfies the following curvature-free
inequality: . This generalizes a
previous inequality proved by Gromov for closed Riemannian surfaces with
negative Euler characteristic. As a consequence we obtain that the volume
entropy of such -complexes with unit area is always bounded away from zero.Comment: 11 pages, 2 figure
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