Universal homogeneous derivations of graded ϵ-commutative algebras

Abstract

Let K be a commutative ring, let ▵ be an abelian group, and let ϵ:▵x▵→K be a commutation factor over ▵.A ▵ graded K-algebra is said to be ϵ-commutative if its ϵ-bracket is identically zero, (K,ϵ) derivations from a given ϵ-commutative ▵-graded K-algebra A into bimodules are studied. It is proved that for each λϵ▵ there exists a universal initial (k,ϵ)-derivation of degree λ of A. For each λϵ▵ a natural module of (K, ϵ, λ)-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.

1991 Mathematics Subject Classifications:

• 13N05
• 16U80
• 16W50
• 58A10
• 58A50
• Modules of differentials
• Generations of commutativity
• Graded rings and modules
• Differentials forms
• Supermanifolds and graded manifolds

^*Partially supported by CONACyT grants 28491-E, E130.1880, and MB-RLM grant 1411-97/98. Martially supported by DGICyT grant PB94-1196, and Comissionat per a Universitats i Recerca de la Generalitat de Catalunya.

^†Partially supported by DGICYT grant PB94-1196, and Comissionat per a Universitats i Recerca de la Generalitat de Catalunya.

^*Partially supported by CONACyT grants 28491-E, E130.1880, and MB-RLM grant 1411-97/98. Martially supported by DGICyT grant PB94-1196, and Comissionat per a Universitats i Recerca de la Generalitat de Catalunya.

^†Partially supported by DGICYT grant PB94-1196, and Comissionat per a Universitats i Recerca de la Generalitat de Catalunya.

Notes

^*Partially supported by CONACyT grants 28491-E, E130.1880, and MB-RLM grant 1411-97/98. Martially supported by DGICyT grant PB94-1196, and Comissionat per a Universitats i Recerca de la Generalitat de Catalunya.

^†Partially supported by DGICYT grant PB94-1196, and Comissionat per a Universitats i Recerca de la Generalitat de Catalunya.