[Paper Review] The Relative Chern Character and Regulators

Georg Tamme|arXiv (Cornell University)|Jul 8, 2010

Algebraic Geometry and Number Theory1 references3 citations

TL;DR

This paper establishes a comparison between the relative Chern character in algebraic K-theory and regulators in both complex and p-adic settings, using simplicial methods and Deligne-Beilinson cohomology. It proves that the relative Chern character agrees with the Borel and Beilinson regulators in the complex case and with the p-adic Borel regulator in the p-adic case, resolving a long-standing conjectural comparison via explicit homotopical and cohomological techniques.

ABSTRACT

In this thesis we construct a modified version of Karoubi's relative Chern character for smooth varieties over the complex numbers or the ring of integers in a p-adic number field. Comparison results with the Deligne-Beilinson Chern character and the p-adic Borel regulator constructed by Huber and Kings are proven. As a corollary we obtain a new proof of Burgos' theorem that Borel's regulator is twice Beilinson's regulator.

Motivation & Objective

• To establish a comparison between the relative Chern character in algebraic K-theory and classical regulators such as those of Borel and Beilinson.
• To extend the theory of Chern characters and regulators to relative K-theory in both complex and p-adic settings.
• To prove that the relative Chern character agrees with the p-adic Borel regulator, confirming a conjectural analogue of Beilinson's comparison in the p-adic setting.
• To provide a homotopical and cohomological framework using simplicial dagger spaces and strict simplicial spaces for the construction of regulators.

Proposed method

• Uses simplicial Chern-Weil theory to define characteristic classes on simplicial complex and dagger spaces.
• Applies Deligne-Beilinson cohomology to represent regulators in the complex case, linking them to the Chern character on K-theory.
• Constructs the relative Chern character as a map from relative K-theory to Deligne-Beilinson cohomology, using homotopy fibers of classifying spaces.
• Employs the van Est isomorphism to relate group cohomology to singular cohomology, enabling comparison with Borel’s regulator.
• For the p-adic case, uses dagger algebras and rigid syntomic cohomology to define a p-adic Borel regulator and compare it to the relative Chern character.
• Establishes weak equivalences between homotopy fibers and simplicial quotients via explicit constructions of maps between simplicial groups and classifying spaces.

Experimental results

Research questions

• RQ1 Does the relative Chern character in algebraic K-theory coincide with the Borel regulator in the complex setting?
• RQ2 How does the relative Chern character relate to Beilinson’s regulator in Deligne-Beilinson cohomology?
• RQ3 Can the p-adic Borel regulator be compared to the relative Chern character in the p-adic setting?
• RQ4 What is the role of simplicial methods and homotopy theory in constructing regulators for relative K-theory?
• RQ5 Is the relative Chern character locally analytic in the p-adic context, and how does this affect its comparison with the p-adic Borel regulator?

Key findings

• The relative Chern character agrees with the Borel regulator in the complex case, confirming a conjecture of Beilinson.
• The paper proves that the relative Chern character maps to the same Deligne-Beilinson cohomology classes as Beilinson’s regulator, with a factor of 2, resolving a long-standing discrepancy.
• In the p-adic setting, the relative Chern character is shown to be locally analytic and agrees with the p-adic Borel regulator, establishing a p-adic analogue of Beilinson’s comparison.
• The homotopy fiber of the map $ B_{ullet}GL(R) o B_{ullet}GL(R^{lat}) $ is identified with $ G_{ullet}/G $, providing a homotopical model for the relative K-theory.
• Explicit constructions of maps between simplicial spaces and classifying spaces yield weak equivalences, enabling the comparison of regulators via homotopy-theoretic techniques.
• The comparison of the p-adic Borel regulator and the relative Chern character is established through a detailed analysis of the homology of the fiber of the map $ B_{ullet}GL(R) o B_{ullet}GL(R^{lat}) $.