Morphisms cohomology and deformations of modified λ-differential algebras

As an algebraic study of differential equations, differential algebras have been studied for a century and become an important area of mathematics. This paper provides a cohomology theory for modified [Formula: see text]-differential algebra morphisms. As applications, formal deformations of modified [Formula: see text]-differential algebra morphisms are obtained and the rigidity of a modified [Formula: see text]-differential algebra morphisms is characterized by the vanishing of the second cohomology group. Furthermore, we give the cohomology comparison theorem of modified [Formula: see text]-differential algebra morphisms, that is, the cohomology of a morphism of modified [Formula: see text]-differential algebra is isomorphic to the cohomology of an auxiliary modified [Formula: see text]-differential algebra.