Cohomology of diagrams of algebras

We consider cohomology of diagrams of algebras by Beck’s approach, using comonads. We then apply this theory to computing the cohomology of $\Psi$-rings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra.

💡 Research Summary

The paper develops a cohomology theory for diagrams of algebras by extending Beck’s comonadic approach. Starting from a variety of algebras 𝔙 equipped with the usual free‑algebra/forgetful adjunction (F ⊣ U), the authors recall the associated comonad G on 𝔙 and the standard G‑cohomology obtained from the derived functors of the comonad. They then consider a small indexing category I and a diagram F : I → 𝔙, i.e., a functor that assigns to each object of I an algebra in 𝔙 and to each morphism a homomorphism. By lifting the comonad G to the functor category 𝔙^I, they construct a new comonad G^I. The G^I‑cohomology of the diagram with coefficients in a G‑module M is defined as the derived functors of the functor of G^I‑invariants, and it can be computed via the G^I‑resolution (the comonadic bar construction) applied objectwise.

The central technical achievement is a Grothendieck‑type spectral sequence that relates the cohomology of the whole diagram to the ordinary cohomology of its components. For each object i∈I, one computes the usual G‑cohomology groups H^q(F(i),M). The higher derived limits lim^p over the diagram then assemble these groups into the E₂‑page: \