The Invariance and the General CCT Theorems

TL;DR

This paper generalizes the Invariance Theorem and the Cohomology Comparison Theorem for diagrams of algebras, showing that subdivision induces full and faithful functors between derived categories, extending prior cohomology results.

Contribution

It extends the Invariance Theorem and the Cohomology Comparison Theorem by demonstrating subdivision functors as full and faithful between derived categories of algebra diagrams.

Findings

Subdivision functor induces a full and faithful functor between derived categories.

Generalization of the Cohomology Comparison Theorem.

Extension of the Invariance Theorem to broader contexts.

Abstract

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 $General$ $Cohomology$ $Comparison$ $Theorem$ ($\mathbf{CCT}$).