On the Cohomology of an associative $\mathcal{O}$-operator morphism

Rota-Baxter operators and more generally $\mathcal{O}$-operators play a crucial role in broad areas of mathematics and physics, such as integrable systems, the Yang-Baxter equation and pre-Lie algebras. The main objects of study in the paper are certain $\mathcal{O}$-operator morphisms on associative algebras. The cohomology theory of an associative $\mathcal{O}$-operator morphism is established. In development, we give the Cohomology Comparison Theorem of an $\mathcal{O}$-operator morphism, that is, the cohomology of an $\mathcal{O}$-operator morphism is isomorphic to the cohomology of an auxiliary $\mathcal{O}$-operator. As applications, we also study the Cohomology Comparison Theorem of a Rota-Baxter operator morphism (of weight zero) and an associative $r$-matrice weak morphism as a particular case of $\mathcal{O}$-operator morphisms.