339 research outputs found
Presheaves of triangulated categories and reconstruction of schemes
To any triangulated category with tensor product , we associate
a topological space , by means of thick subcategories of , a
la Hopkins-Neeman-Thomason. Moreover, to each open subset of
, we associate a triangulated category , producing what
could be thought of as a presheaf of triangulated categories. Applying this to
the derived category of perfect
complexes on a noetherian scheme , the topological space
turns out to be the underlying topological space of ; moreover, for each
open , the category is naturally equivalent to
.
As an application, we give a method to reconstruct any reduced noetherian
scheme from its derived category of perfect complexes ,
considering the latter as a tensor triangulated category with .Comment: 18 pages; minor change
Stacks of group representations
We start with a small paradigm shift about group representations, namely the
observation that restriction to a subgroup can be understood as an
extension-of-scalars. We deduce that, given a group , the derived and the
stable categories of representations of a subgroup can be constructed out
of the corresponding category for by a purely triangulated-categorical
construction, analogous to \'etale extension in algebraic geometry.
In the case of finite groups, we then use descent methods to investigate when
modular representations of the subgroup can be extended to . We show
that the presheaves of plain, derived and stable representations all form
stacks on the category of finite -sets (or the orbit category of ), with
respect to a suitable Grothendieck topology that we call the sipp topology.
When contains a Sylow subgroup of , we use sipp Cech cohomology to
describe the kernel and the image of the homomorphism , where
denotes the group of endotrivial representations.Comment: Slightly revised version of the 2012 June 21 versio
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
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