[Paper Review] Cohomology of diagrams of algebras

Michael Robinson|ArXiv.org|Feb 25, 2008

Advanced Topics in Algebra11 references5 citations

TL;DR

This paper develops a unified cohomology theory for diagrams of algebras using Beck's comonad approach, establishing a spectral sequence that relates the cohomology of a diagram to the cohomology of its underlying algebras. The key contribution is a local-to-global spectral sequence connecting Baues-Wirsching cohomology of the indexing category with comonad cohomology of the diagram, applicable to $ψ$-rings, $Π$-algebras, and diagrams of groups.

ABSTRACT

We consider cohomology of diagrams of algebras by Beck's approach, using comonads. We then apply this theory to computing the cohomology of $Ψ$-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.

Motivation & Objective

• To unify existing cohomology theories for diagrams of algebras—such as groups, Lie algebras, associative algebras, and $ψ$-rings—under a single framework based on comonads.
• To establish a computational tool for diagram cohomology by constructing a local-to-global spectral sequence that relates the cohomology of the diagram to the cohomology of its constituent algebras.
• To apply the theory to $ψ$-rings, showing that their cohomology fits into the comonad framework and admits a spectral sequence decomposition.
• To demonstrate that the proposed cohomology theory generalizes and unifies prior constructions, such as those by Cegarra for diagrams of groups and Blanc-Johnson-Turner for $Π$-algebras.
• To show that the cohomology of diagrams of algebras via comonads is isomorphic to known theories when appropriate monads are chosen, thereby validating the framework.

Proposed method

• Use Beck's comonad approach: given a monad $T$ on $Σετσ$, define the category of $T$-algebras $\mathfrak{Alg}(T)$, and construct the comonad $\mathbb{G} = FU$ from the adjunction $F \dashv U$.
• Define the cohomology of a diagram $A: I \to \mathfrak{Alg}(T)$ with coefficients in an $A$-module $M$ as $H^*_{I}(A,M)$, using the comonad resolution and derived functors.
• Construct a natural system $\mathcal{H}^q_{\mathbb{G}}(A,M)$ on the indexing category $I$, where for each morphism $\alpha: i \to j$, the value is $H^q_{\mathbb{G}}(A(i), \alpha^*M(j))$.
• Apply Baues-Wirsching cohomology $H^*_{BW}(I, \mathcal{H}^q_{\mathbb{G}}(A,M))$ to the natural system to build a spectral sequence.
• Establish the spectral sequence: $E^{pq}_2 = H^p_{BW}(I, \mathcal{H}^q_{\mathbb{G}}(A,M)) \Rightarrow H^{p+q}_I(A,M)$, which is the central computational tool.
• Verify that the spectral sequence recovers known results in special cases, such as diagrams of groups (Cegarra) and $Π$-algebras (Blanc-Johnson-Turner), by choosing appropriate monads and categories.

Experimental results

Research questions

• RQ1 Can a single cohomology theory unify the cohomology of diagrams of various algebraic structures—such as groups, Lie algebras, associative algebras, and $ψ$-rings—using comonads?
• RQ2 Does a spectral sequence exist that relates the cohomology of a diagram of algebras to the cohomology of its individual components via a local-to-global principle?
• RQ3 How does the cohomology of $ψ$-rings, which are diagrams of commutative rings with $Π$-actions, fit into the comonad framework and admit spectral sequence decomposition?
• RQ4 To what extent does the proposed comonad-based cohomology of diagrams recover or generalize existing cohomology theories for diagrams of groups and $Π$-algebras?
• RQ5 Is the natural system $\mathcal{H}^q_{\mathbb{G}}(A,M)$ on the indexing category $I$ sufficient to reconstruct the global diagram cohomology via Baues-Wirsching cohomology?

Key findings

• A spectral sequence $E^{pq}_2 = H^p_{BW}(I, \mathcal{H}^q_{\mathbb{G}}(A,M)) \Rightarrow H^{p+q}_I(A,M)$ is constructed, providing a computational bridge between local cohomology of individual algebras and global cohomology of the diagram.
• For $ψ$-rings, the spectral sequence takes the form $E^{p,q}_2 = H^p_{BW}(\mathbb{N}^{\text{mult}}, H^{q}_{AQ}(A,M)) \Rightarrow H^{p+q}_{\Psi}(A,M)$, linking the cohomology of $ψ$-rings to Quillen-Arthur cohomology.
• The cohomology theory for diagrams of algebras via comonads is isomorphic to existing theories when appropriate monads are selected, such as those for groups, $Π$-algebras, and associative algebras.
• The spectral sequence for diagrams of groups recovers Cegarra's result as a special case when the indexing category $I$ has two objects and one non-identity morphism.
• The theory applies to $ψ$-rings by modeling them as diagrams of commutative rings indexed by the multiplicative monoid $\mathbb{N}^{\text{mult}}$, with $ψ^n$-maps as structure morphisms.
• The natural system $\mathcal{H}^q_{\mathbb{G}}(A,M)$ on $I$ is well-defined and enables the construction of the spectral sequence, with values $H^q_{\mathbb{G}}(A(i), \alpha^*M(j))$ for each morphism $\alpha: i \to j$.