**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Motivic cohomology

Summary

Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.
Let X be a scheme of finite type over a field k. A key goal of algebraic geometry is to compute the Chow groups of X, because they give strong information about all subvarieties of X. The Chow groups of X have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for a closed subscheme Z of X, there is an exact sequence of Chow groups, the localization sequence
whereas in topology this would be part of a long exact sequence.
This problem was resolved by generalizing Chow groups to a bigraded family of groups, (Borel–Moore) motivic homology groups (which were first called higher Chow groups by Bloch). Namely, for every scheme X of finite type over a field k and integers i and j, we have an abelian group Hi(X,Z(j)), with the usual Chow group being the special case
For a closed subscheme Z of a scheme X, there is a long exact localization sequence for motivic homology groups, ending with the localization sequence for Chow groups:
In fact, this is one of a family of four theories constructed by Voevodsky: motivic cohomology, motivic cohomology with compact support, Borel-Moore motivic homology (as above), and motivic homology with compact support. These theories have many of the formal properties of the corresponding theories in topology. For example, the motivic cohomology groups Hi(X,Z(j)) form a bigraded ring for every scheme X of finite type over a field. When X is smooth of dimension n over k, there is a Poincare duality isomorphism
In particular, the Chow group CHi(X) of codimension-i cycles is isomorphic to H2i(X,Z(i)) when X is smooth over k.
The motivic cohomology Hi(X, Z(j)) of a smooth scheme X over k is the cohomology of X in the Zariski topology with coefficients in a certain complex of sheaves Z(j) on X.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (15)

Related courses (5)

Motivic cohomology

Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology. Let X be a scheme of finite type over a field k. A key goal of algebraic geometry is to compute the Chow groups of X, because they give strong information about all subvarieties of X.

Motive (algebraic geometry)

In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

Chow group

In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product.

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

MATH-124: Geometry for architects I

Ce cours entend exposer les fondements de la géométrie à un triple titre :
1/ de technique mathématique essentielle au processus de conception du projet,
2/ d'objet privilégié des logiciels de concept

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous

Related lectures (21)

Graded Ring Structure on CohomologyMATH-506: Topology IV.b - cohomology rings

Explores the associative and commutative properties of the cup product in cohomology, with a focus on graded structures.

Cohomology: Cross ProductMATH-506: Topology IV.b - cohomology rings

Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.

Cohomology Groups: Hopf FormulaMATH-506: Topology IV.b - cohomology rings

Explores the Hopf formula in cohomology groups, emphasizing the 4-term exact sequence and its implications.