Bloch's Conjecture

Definition

Bloch's Conjecture posits a connection between the Milnor K-theory of a field and its étale cohomology, specifically suggesting that for a field, the n-th Milnor K-group can be viewed as a quotient of the n-th étale cohomology group with rational coefficients. This conjecture highlights the relationship between different algebraic structures and has implications in understanding the behavior of algebraic varieties over fields.

5 Must Know Facts For Your Next Test

1. Bloch's Conjecture links two significant areas: Milnor K-theory and étale cohomology, providing a bridge between them.
2. The conjecture suggests that Milnor K-groups can be interpreted in terms of étale cohomology groups, making it easier to understand field extensions.
3. It plays a crucial role in number theory and algebraic geometry by helping to unravel properties of algebraic varieties over different fields.
4. While it remains a conjecture for general fields, it has been proven for specific cases, notably for fields of characteristic 0.
5. Understanding Bloch's Conjecture can lead to deeper insights into other conjectures in K-theory and arithmetic geometry.

Review Questions

• How does Bloch's Conjecture connect Milnor K-theory with étale cohomology?
  • Bloch's Conjecture establishes a relationship where the n-th Milnor K-group of a field can be expressed as a quotient of its n-th étale cohomology group with rational coefficients. This connection is essential because it allows mathematicians to utilize tools from both theories to gain insights into the structure and properties of fields. By showing that these two seemingly separate areas are linked, it opens up new pathways for research in algebraic geometry and number theory.
• Discuss the implications of Bloch's Conjecture on our understanding of algebraic varieties over fields.
  • The implications of Bloch's Conjecture are profound as it suggests that properties of algebraic varieties can be better understood by studying their associated Milnor K-groups. This connection means that researchers can apply techniques from étale cohomology to derive new results about these varieties, especially concerning their rational points and how they behave under field extensions. As such, Bloch's Conjecture not only aids in theoretical pursuits but also has practical applications in arithmetic geometry.
• Evaluate the significance of Bloch's Conjecture within the broader context of algebraic K-theory and its related conjectures.
  • Bloch's Conjecture is significant as it positions itself at a crossroads within algebraic K-theory and offers a vital link to étale cohomology, enriching our understanding of both fields. Its relevance extends to several other conjectures, such as the Gersten Conjecture and the Milnor Conjecture, since resolving Bloch's ideas may yield insights into these as well. The ongoing efforts to prove or disprove Bloch's Conjecture showcase its central role in current mathematical research and highlight its potential to unify various concepts across algebraic structures.

Related terms

Milnor K-theory: A branch of algebraic K-theory that focuses on the study of the Milnor K-groups, which are constructed using the field's elements and their relations to algebraic cycles.

Étale Cohomology: A cohomology theory used in algebraic geometry that allows one to study the properties of schemes using sheaves defined on étale sites.

Quillen's K-theory: A type of K-theory developed by Daniel Quillen, which connects algebraic topology, algebraic geometry, and abstract algebra through the study of projective modules and their classes.