Traces of Differential Forms

Abstract

For a finite locally free algebra S/R the canonical trace σ_S/R: S → R and the canonical norm n_S/R: S → R are defined (F.3). It would be very usefull if for any pair (S/R, Ω) consisting of an algebra S/R as above and a differential algebra Ω of R a trace mapping

$$ \sigma _{S/R}^\Omega :{\Omega _S} \to \Omega $$

where Ω_S is the universal S-extension of Ω, could be constructed such that the following conditions (“trace axioms”) are satisfied:

• Tr1) (Linearity) If we consider Ω_S as a left Ω-module via the canonical map Ω → Ω_S, then σ ^Ω_S/R is Ω-linear and homogeneous of degree O (i.e. Ω ^p_S is mapped into Ω^P for each p ∈ IN).

• Tr2) (Relation to the canonical trace) The restriction $\sigma _{S/R}^\Omega \left| {_{\Omega _S^0}^{}} \right.:S \to R$ to the elements of degree O is the canonical trace σ_S/R.