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

💡 Research Summary

The paper revisits the Gerstenhaber‑Schack Invariance Theorem, which asserts that for a diagram of algebras A over a small category C, the subdivision functor d*:A‑mod → A′‑mod induces an isomorphism on Yoneda cohomology. The authors aim to lift this result from the level of cohomology groups to the level of derived categories, thereby obtaining a more robust and functorial form of the General Cohomology Comparison Theorem (CCT).

The authors begin by fixing a commutative base ring k and a small category C. A diagram of k‑algebras A:Cᵒᵖ→k‑alg is regarded as a presheaf, and the category A‑mod of left A‑modules is an abelian, complete, and cocomplete category. They introduce the notion of an “allowable” morphism η:M→N: for each object i∈C the component ηᵢ admits a k‑module splitting (not required to be natural). This concept is crucial because Yoneda cohomology is a relative theory over a possibly non‑field base ring. An A‑module P is called relatively projective if every allowable epimorphism M→N induces a surjection Hom_A(P,M)→Hom_A(P,N). Gerstenhaber‑Schack’s generalized simplicial bar (GSB) resolution provides, for any A‑module, a relative projective allowable resolution.

Next, the authors construct a “relative derived category” Dₖ(A‑mod). They start with the homotopy category K(A‑mod) of bounded‑below complexes of A‑modules. A morphism f:M•→N• is declared a relative quasi‑isomorphism if, for each i∈C, the cone of the component map f_i is contractible when regarded as a complex of k‑modules. Proposition 3.2 shows that this condition is equivalent to the existence of a homotopy inverse in K(k‑mod). Proposition 3.3 proves that the class Σ of relative quasi‑isomorphisms is localizing (satisfies the two‑out‑of‑three property, admits extensions, and left‑right equivalence). Hence one can form the Verdier quotient Dₖ(A‑mod)=K(A‑mod)