On jets, extensions and characteristic classes II

TL;DR

This paper introduces generalized Atiyah classes for quasi-coherent sheaves over arbitrary scheme morphisms, extending classical concepts and providing new characteristic classes with potential geometric applications.

Contribution

It defines generalized jet bundles and Atiyah classes for sheaves relative to scheme morphisms, expanding the theoretical framework of characteristic classes.

Findings

Existence of examples with trivial and non-trivial Atiyah classes.

Generalized jet bundle J(E) as a bimodule over a sheaf of rings.

The Atiyah class c(E) characterizes the isomorphism between left and right module structures.

Abstract

In this paper we define and study generalized Atiyah classes for quasi coherent sheaves relative to arbitrary morphisms of schemes. We use derivations and quasi coherent sheaves of left and right O-modules to define a generalized first order jet bundle J(E) and a generalized Atiyah sequence for E. The generalized jet bundle J(E) is a left and right module over a sheaf J of associative rings on X. The sheaf J is an extension of O with a sheaf I of two sided ideals of square zero. The Atiyah sequence gives rise to a generalized Atiyah class c(E) with the property that c(E)=0 if and only if the left structure on J(E) is O-isomorphic to the right structure on J(E). We give examples where c(E)=0 and c(E)\neq 0 hence the class c(E) is a non trivial characteristic class.