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Rota-Baxter operators and more generally $\mathcal{O}$-operators play acrucial role in broad areas of mathematics and physics, such as integrablesystems, the Yang-Baxter equation and pre-Lie algebras. The main objects ofstudy in the paper are certain $\mathcal{O}$-operator morphisms on associativealgebras. The cohomology theory of an associative $\mathcal{O}$-operatormorphism is established. In development, we give the Cohomology ComparisonTheorem 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 CohomologyComparison Theorem of a Rota-Baxter operator morphism (of weight zero) and anassociative $r$-matrice weak morphism as a particular case of$\mathcal{O}$-operator morphisms.

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