1,005 research outputs found
Quadratic forms and Clifford algebras on derived stacks
In this paper we present an approach to quadratic structures in derived
algebraic geometry. We define derived n-shifted quadratic complexes, over
derived affine stacks and over general derived stacks, and give several
examples of those. We define the associated notion of derived Clifford algebra,
in all these contexts, and compare it with its classical version, when they
both apply. Finally, we prove three main existence results for derived shifted
quadratic forms over derived stacks, define a derived version of the
Grothendieck-Witt group of a derived stack, and compare it to the classical
one.Comment: 42 pages; revised version to appear in Advances in Mat
A remark on K-theory and S-categories
It is now well known that the K-theory of a Waldhausen category depends on
more than just its (triangulated) homotopy category (see [Schlichting]). The
purpose of this note is to show that the K-theory spectrum of a (good)
Waldhausen category is completely determined by its Dwyer-Kan simplicial
localization, without any additional structure. As the simplicial localization
is a refined version of the homotopy category which also determines the
triangulated structure, our result is a possible answer to the general
question: ``To which extent -theory is not an invariant of triangulated
derived categories ?''Comment: 23 pages; final version, accepted for publication in 'Topology
Higher algebraic K-theory for actions of diagonalizable groups
We study the K-theory of actions of diagonalizable group schemes on
noetherian regular separated algebraic spaces: our main result shows how to
reconstruct the K-theory ring of such an action from the K-theory rings of the
loci where the stabilizers have constant dimension. We apply this to the
calculation of the equivariant K-theory of toric varieties, and give conditions
under which the Merkurjev spectral sequence degenerates, so that the
equivariant K-theory ring determines the ordinary K-theory ring. We also prove
a very refined localization theorem for actions of this type.Comment: Addendum contains mainly a corrected definition of specialization
maps, the previous one being wrong as noticed by A. Neeman. All the other
results (in particular the main results) still hold. Several other typos also
correcte
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