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The Invariance and the General CCT Theorems

The \begin{it} Invariance Theorem \end{it} of M. Gerstenhaber and S. D. Schack states that if $\mathbb{A}$ is a diagram of algebras then the subdivision functor induces a natural isomorphism between the Yoneda cohomologies of the category $\mathbb{A}$-$\mathbf{mod}$ and its subdivided category $\mathbb{A}'$-$\mathbf{mod}$. In this paper we generalize this result and show that the subdivision functor is a full and faithful functor between two suitable derived categories of $\mathbb{A}$-$\mathbf{mod}$ and $\mathbb{A}'$-$\mathbf{mod}$. This result combined with our work in [5] and [6], on the $Special$ $Cohomology$ $Comparison$ $Theorem$, constitutes a generalization of M. Gerstenhaber and S. D. Schack's $General$ $Cohomology$ $Comparison$ $Theorem$ ($\mathbf{CCT}$).