As an algebraic structure underlying the differential calculus and differential equations, a differential algebra is an associative algebra equipped with a linear map satisfying the Leibniz rule. The subject has been studied for about a century and has become an important area of mathematics. In recent years the area has been expanded to the noncommutative associative and Lie algebra contexts and to the case when the defining operator identity has a weight in order to include difference operators. This paper provides a cohomology theory for differential algebras of arbitrary weight, via a uniform approach to cover both the zero weight case which is similar to the earlier study of differential Lie algebras, and the non-zero weight case which poses challenges. The cohomology of a differential algebra is related to the Hochschild cohomology by a type of long exact sequence for relative homology. As an application, abelian extensions of a differential algebra are classified by the second cohomology group. Furthermore, formal deformations of a differential algebra are characterized by the second cohomology group and the rigidity of a differential algebra is characterized by the vanishing of the second cohomology group.
Keywords: cohomology, extension, deformation, differential algebra, difference algebra, derivation
2020 MSC: 16E40,16S80,12H05,12H10,16W25,16S70
Theory and Applications of Categories, Vol. 38, 2022, No. 37, pp 1409-1433.