262 research outputs found
Orientation theory in arithmetic geometry
This work is devoted to study orientation theory in arithmetic geometric
within the motivic homotopy theory of Morel and Voevodsky. The main tool is a
formulation of the absolute purity property for an \emph{arithmetic cohomology
theory}, either represented by a cartesian section of the stable homotopy
category or satisfying suitable axioms. We give many examples, formulate
conjectures and prove a useful property of analytical invariance. Within this
axiomatic, we thoroughly develop the theory of characteristic and fundamental
classes, Gysin and residue morphisms. This is used to prove Riemann-Roch
formulas, in Grothendieck style for arbitrary natural transformations of
cohomologies, and a new one for residue morphisms. They are applied to rational
motivic cohomology and \'etale rational -adic cohomology, as expected by
Grothendieck in \cite[XIV, 6.1]{SGA6}.Comment: 81 pages. Final version, to appear in the Actes of a 2016 conference
in the Tata Institute. Thanks a lot goes to the referee for his enormous work
(more than 100 comments) which was of great help. Among these corrections, he
indicated to me a sign mistake in formula (3.2.14.a) which was very hard to
detec
Triangulated categories of mixed motives
This book discusses the construction of triangulated categories of mixed
motives over a noetherian scheme of finite dimension, extending Voevodsky's
definition of motives over a field. In particular, it is shown that motives
with rational coefficients satisfy the formalism of the six operations of
Grothendieck. This is achieved by studying descent properties of motives, as
well as by comparing different presentations of these categories, following and
extending insights and constructions of Deligne, Beilinson, Bloch, Thomason,
Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r,
and others. In particular, the relation of motives with -theory is addressed
in full, and we prove the absolute purity theorem with rational coefficients,
using Quillen's localization theorem in algebraic -theory together with a
variation on the Grothendieck-Riemann-Roch theorem. Using resolution of
singularities via alterations of de Jong-Gabber, this leads to a version of
Grothendieck-Verdier duality for constructible motivic sheaves with rational
coefficients over rather general base schemes. We also study versions with
integral coefficients, constructed via sheaves with transfers, for which we
obtain partial results. Finally, we associate to any mixed Weil cohomology a
system of categories of coefficients and well behaved realization functors,
establishing a correspondence between mixed Weil cohomologies and suitable
systems of coefficients. The results of this book have already served as ground
reference in many subsequent works on motivic sheaves and their realizations,
and pointers to the most recent developments of the theory are given in the
introduction.Comment: This is the final version. To appear in the series Springer
Monographs in Mathematic
Mixed Weil cohomologies
We define, for a regular scheme and a given field of characteristic zero
\KK, the notion of \KK-linear mixed Weil cohomology on smooth -schemes
by a simple set of properties, mainly: Nisnevich descent, homotopy invariance,
stability (which means that the cohomology of \GG_{m} behaves correctly), and
K\"unneth formula. We prove that any mixed Weil cohomology defined on smooth
-schemes induces a symmetric monoidal realization of some suitable
triangulated category of motives over to the derived category of the field
\KK. This implies a finiteness theorem and a Poincar\'e duality theorem for
such a cohomology with respect to smooth and projective -schemes (which can
be extended to smooth -schemes when is the spectrum of a perfect field).
This formalism also provides a convenient tool to understand the comparison of
such cohomology theories. Our main examples are algebraic de Rham cohomology
and rigid cohomology, and the Berthelot-Ogus isomorphism relating them.Comment: update references; hopefully improve the expositio
Borel-Moore motivic homology and weight structure on mixed motives
By defining and studying functorial properties of the Borel-Moore motivic
homology, we identify the heart of Bondarko-H\'ebert's weight structure on
Beilinson motives with Corti-Hanamura's category of Chow motives over a base,
therefore answering a question of Bondarko
The rigid syntomic ring spectrum
The aim of this paper is to show that Besser syntomic cohomology is
representable by a rational ring spectrum in the motivic homotopical sense. In
fact, extending previous constructions, we exhibit a simple representability
criterion and we apply it to several cohomologies in order to get our central
result. This theorem gives new results for syntomic cohomology such as
h-descent and the compatibility of cycle classes with Gysin morphisms. Along
the way, we prove that motivic ring spectra induces a complete Bloch-Ogus
cohomological formalism and even more. Finally, following a general motivic
homotopical philosophy, we exhibit a natural notion of syntomic coefficients.Comment: Final version to appear in the Journal de l'institut des
Math\'ematiques de Jussieu. Many typos have been corrected and the exposition
has been improved according to the suggestions of the referees: we thank them
a lot
Modules homotopiques (Homotopy modules)
The proof of the coincidence of the Gysin morphism in motivic cohomology and
the usual pushout on Chow groups has been improved (see Lemma 3.3 and
Proposition 3.11
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