Universal homogeneous derivations of graded ε-commutative algebras

Let K be a commutative ring, let Delta be an abelian group, and let epsilon : Delta x Delta -> K be a commutation factor over Delta. A Delta-graded K-algebra is said to be epsilon-commutative if its epsilon-bracket is identically zero. (K, epsilon)-derivations from a given a-commutative a-graded K-algebra A into bimodules are studied. It is proved that for each lambda epsilon Delta there exists a universal initial (K, epsilon)-derivation of degree lambda of A. For each lambda epsilon Delta a natural module of (K, epsilon, lambda)-differentials of A along with a differential map is constructed. It is proved that each derivation of A canonically equipps this module with a structure of differential module. Applications and examples are given. It is shown that the first order exterior differentials which are known from the theory of smooth graded manifolds are universal initial homogeneous derivations of the sort considered hereby.