# On jets, extensions and characteristic classes II

Helge {\O}ystein Maakestad

Arxiv ID: 1006.0593•Last updated: 11/13/2020

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.

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