• Corpus ID: 118046155

Lecture Notes On Motivic Cohomology

  title={Lecture Notes On Motivic Cohomology},
  author={Carlo Mazza and Vladimir Voevodsky and Charles A. Weibel},
* Etale motivic theory: * Etale sheaves with transfers * The relative Picard group and Suslin's rigidity theorem * Derived tensor products $\mathbb{A}^1$-weak equivalence * Etale motivic cohomology and algebraic singular homology * Nisnevich sheaves with transfers: * Standard triples * Nisnevich sheaves* * Nisnevich sheaves with transfers * The triangulated category of motives: * The category of motives * The complex $\mathbb{Z}(n)$ and $\mathbb{P}^n$ Equidimensional cycles * Higher Chow groups… 
On Chow-Weight Homology of Motivic Complexes and Its Relation to Motivic Homology
We study in detail the so-called Chow-weight homology of Voevodsky motivic complexes and relate it to motivic homology. We generalize earlier results and prove that the vanishing of higher motivic
Thom isomorphisms in triangulated motivic categories
We show that a triangulated motivic category admits categorical Thom isomorphisms for vector bundles with an additional structure if and only if the generalized motivic cohomology theory represented
Framed and MW-transfers for homotopy modules
In the paper we use the theory of framed correspondences to construct Milnor–Witt transfers on homotopy modules. As a consequence we identify the zeroth stable $${\mathbb {A}}^1$$A1-homotopy sheaf of
Motivic spectral sequence for relative homotopy K-theory
We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the
Motivic cohomology of semistable varieties
We construct log-motivic cohomology groups for semistable varieties and study the p-adic deformation theory of log-motivic cohomology classes. Our main result is the deformational part of a p-adic
A commutative P^1-spectrum representing motivic cohomology over Dedekind domains
We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined
Algebraic Cobordism and Étale Cohomology
Thomason's \'{e}tale descent theorem for Bott periodic algebraic $K$-theory \cite{aktec} is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary
Mixed motives and geometric representation theory in equal characteristic
Let $$\mathbb {k}$$k be a field of characteristic p. We introduce a formalism of mixed sheaves with coefficients in $$\mathbb {k}$$k and apply it in representation theory. We construct a system of
E-motives and motivic stable homotopy
We introduce in this work the notion of the category of pure $\mathbf{E}$-Motives, where $\mathbf{E}$ is a motivic strict ring spectrum and construct twisted $\mathbf{E}$-cohomology by using six
On the Conservativity of Some Functors in Motivic Homotopy Theory
Given a 0-connective motivic spectrum $E \in SH(k)$ over a perfect field k, we determine $h_0$ of the associated motive $M E \in DM(k)$ in terms of $\pi_0 (E)$. Using this we show that if k has


Stable homotopy and generalised homology
Preface Pt. I: S.P. Novikov's Work on Operations on Complex Cobordism 2: Cobordism groups 3: Homology 4: The Conner-Floyd Chern classes 5: The Novikov operations 6: The algebra of all operations 7:
Cycles, Transfers And Motivic Homology Theories
Part I of Cohomological Theory of Presheaves with Transfers and Triangulated Categories of Motives Over a Field, and Higher Chow Groups and Etale Cohomology are presented.
Hodge Cycles, Motives, and Shimura Varieties
General Introduction.- Notations and Conventions.- Hodge Cycles on Abelian Varieties.- Tannakian Categories.- Langlands's Construction of the Taniyama Group.- Motifs et Groupes de Taniyama.-
$K$-theory of a space with coefficients in a (discrete) ring
In [2], [3], S. Gersten has introduced higher i£-groups of a ring which satisfy properties analogous to those of a generalized homology theory in a suitably defined homotopy category of rings [ l ] .
For rings with a large number of units the authors prove a strengthened theorem on homological stabilization: the homomorphism Hk(GLn(A)) → Hk(GL(A)) is surjective for n ≥ k + sr A – 1 and bijective
The K-theory of fields in characteristic p
Abstract.We show that for a field k of characteristic p, Hi(k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural
We show that, assuming a rather innocuous looking ”moving lemma” called Theorem A, this complex is an exact couple, and the resulting spectral sequence has the form (0.1.1). §2 §6 of the paper are
"Classical" algebraic K-theory, and connections with arithmetic
Some problems in "classical" algebraic K-theory.- Comparison of algebraic and topological K-theory.- Applications algebriques du tore dans la sphere et de Sp x Sq dans Sp+q.- On the Ko of certain