G\'eom\'etrie non-commutative, formule des traces et conducteur de Bloch

TL;DR

This paper explores Bloch's conductor formula, a conjecture linking topology changes in algebraic varieties to non-commutative and derived geometric techniques, aiming to resolve Bloch's conjecture.

Contribution

It introduces a novel approach combining non-commutative and derived geometry to address Bloch's conductor conjecture.

Findings

Proposes a general framework for Bloch's conjecture

Connects non-commutative geometry with algebraic topology

Lays groundwork for future proofs of the conjecture

Abstract

This text is based on a talk by the first named author at the first congress of the SMF (Tours, 2016). We present Bloch's conductor formula, which is a conjectural formula describing the change of topology in a family of algebraic varieties when the parameter specialises to a critical value. The main objective of this paper is to describe a general approach to the resolution of Bloch's conjecture based on techniques from both non-commutative geometry and derived geometry.