Crystalline cohomology

Definition

Crystalline cohomology is a cohomology theory for schemes over a field of positive characteristic, primarily used to study the properties of algebraic varieties in the context of p-adic numbers. It provides insights into the structure of these varieties by connecting their geometric and arithmetic aspects through a framework that incorporates both algebraic and topological methods.

5 Must Know Facts For Your Next Test

1. Crystalline cohomology is particularly significant for its ability to connect algebraic geometry over fields of positive characteristic with its p-adic counterpart, enhancing the understanding of both realms.
2. One key feature is that crystalline cohomology can be computed using the formalism of divided power, which allows for the handling of singularities and other complexities in algebraic varieties.
3. This cohomology theory is closely related to étale cohomology, but it retains unique properties that make it suitable for analyzing the behavior of varieties under reduction modulo p.
4. Crystalline cohomology provides tools to study the semistable reduction theorem, which describes how algebraic varieties behave when reduced modulo a prime.
5. The comparison between crystalline and de Rham cohomology establishes deep connections between differential forms and algebraic geometry, allowing for insights into both smooth and singular cases.

Review Questions

• How does crystalline cohomology relate to p-adic numbers and why is this connection significant?
  • Crystalline cohomology is deeply intertwined with p-adic numbers as it serves as a bridge between algebraic geometry over positive characteristic fields and p-adic analysis. This connection is significant because it allows mathematicians to apply techniques from p-adic analysis to solve problems in algebraic geometry, providing a richer understanding of the structure and properties of varieties defined over these fields.
• Discuss the role of the Frobenius morphism in the context of crystalline cohomology and its implications for algebraic varieties.
  • The Frobenius morphism plays a crucial role in crystalline cohomology as it reflects how functions behave under raising to the p-th power. This morphism helps categorize algebraic varieties according to their behavior when viewed through crystalline cohomology. Understanding its action allows researchers to gain insights into the geometric structures and potential singularities present in these varieties, revealing important properties related to their deformation and reduction.
• Evaluate the impact of crystalline cohomology on understanding semistable reduction and its relevance in modern arithmetic geometry.
  • Crystalline cohomology has a profound impact on understanding semistable reduction by providing tools to analyze how algebraic varieties transition when reduced modulo a prime. This theory aids in classifying different types of reductions and examining their geometric properties, which is essential for modern arithmetic geometry. The insights gained from crystalline cohomology not only enhance our grasp of variety behavior under reductions but also facilitate advancements in areas like moduli problems and the study of rational points on varieties.

Related terms

p-adic numbers: A system of numbers that extends the ordinary arithmetic of the integers to include solutions to polynomial equations, allowing for a form of analysis that is particularly useful in number theory.

de Rham cohomology: A type of cohomology that applies to smooth manifolds, using differential forms to study topological properties, often compared with crystalline cohomology in understanding their relationships.

Frobenius morphism: A morphism defined in algebraic geometry that raises functions to their p-th power, playing a crucial role in the study of varieties over fields of positive characteristic.