On approximations of the de Rham complex and their cohomology

ABSTRACT

For a commutative algebra R, its de Rham cohomology is an important invariant of R. In the paper, an infinite chain of de Rham-like complexes is introduced where the first member of the chain is the de Rham complex. The complexes are called approximations of the de Rham complex. Their cohomologies are found for polynomial rings and algebras of power series over a field of characteristic zero.

KEYWORDS:

• Algebra of power series
• approximations
• differentials
• the de Rham cohomology
• the de Rham complex
• polynomial algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D03
• 13N05
• 13N10
• 13N15

Acknowledgments

The authors were supported by ‘BAP’ of the Hacettepe University (the project number FBD-2017-14108) during a visit of the first author to the Hacettepe University. The second author was supported by TUBITAK during her visit to the University of Sheffield. The hospitality of the Hacettepe University and the University of Sheffield is greatly acknowledged.