14 research outputs found
Naive -connectedness of retract rational varieties
A smooth, proper, retract rational variety over a field is known to be
-connected. We improve on this result, in the case when is
infinite, showing that such varieties are naively -connected.Comment: 14 pages, comments are welcom
Geometric motivic integration on Artin n-stacks
We construct a measure on the Boolean algebra of sets of formal arcs on an Artin stack which are definable in the language of Denef-Pas. The measure takes its values in a ring that is obtained from the Grothendieck ring of Artin stacks over the residue field by a localization followed by a completion. This construction is analogous to the construction of motivic measure on varieties by Denef and Loeser. We also obtain a "change of base" formula which allows us to relate the motivic measure on different stacks
Remarks on iterations of the -chain connected components construction
We show that the sheaf of -connected components of a Nisnevich
sheaf of sets and its universal -invariant quotient (obtained by
iterating the -chain connected components construction and taking
the direct limit) agree on field-valued points. This establishes an explicit
formula for the field-valued points of the sheaf of -connected
components of any space. Given any natural number , we construct an -connected space on which the iterations of the naive -connected components construction do not stabilize before the th stage.Comment: 8 pages, comments are welcome; v2: minor modification
Strong -invariance of -connected components of reductive algebraic groups
We show that the sheaf of -connected components of a reductive
algebraic group over a perfect field is strongly -invariant. As a
consequence, torsors under such groups give rise to -fiber
sequences. We also show that sections of -connected components of
anisotropic, semisimple, simply connected algebraic groups over an arbitrary
field agree with their -equivalence classes, thereby removing the
perfectness assumption in the previously known results about the
characterization of isotropy in terms of affine homotopy invariance of
Nisnevich locally trivial torsors.Comment: 14 pages, comments are welcome, v3: minor changes, results are
unchange