THE BIG CHERN CLASSES AND THE CHERN CHARACTER
Abstract
Let X be a smooth scheme over a field of characteristic 0. The Atiyah class of the tangent bundle TX of X equips TX[-1] with the structure of a Lie algebra object in the derived category D+(X) of bounded below complexes of 
modules with coherent cohomology [6]. We lift this structure to that of a Lie algebra object 
in the category of bounded below complexes of 
modules in Theorem 2. The "almost free" Lie algebra 
is equipped with Hochschild coboundary. There is a symmetrization map 
where 
is the complex of polydifferential operators with Hochschild coboundary. We prove a theorem (Theorem 1) that measures how I fails to commute with multiplication. Further, we show that 
is the universal enveloping algebra of 
in D+(X). This is used to interpret the Chern character of a vector bundle E on X as the "character of a representation" (Theorem 4). Theorems 4 and 1 are then exploited to give a formula for the big Chern classes in terms of the components of the Chern character.
