[Paper Review] Splitting Algebras II: The Cohomology Algebra

Brad Shelton|arXiv (Cornell University)|Aug 10, 2012

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper computes the cohomology and coalgebra of the associated graded splitting algebra $A'_{ ext{\Gamma}}$ associated to a finite ranked poset $\Gamma$, showing it is isomorphic to the reduced cohomology of order complexes $\Delta(\Gamma_{b,q})$. The key result establishes that $A'_{\Gamma}$ is Koszul (resp. quadratic) if and only if $\Gamma$ is Cohen-Macaulay (resp. uniform), providing a new homological characterization of these poset properties via algebraic topology.

ABSTRACT

Gelfand, Retakh, Serconek and Wilson, in \cite{GRSW}, defined a graded algebra $A_Γ$ attached to any finite ranked poset $Γ$ - a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of $Γ$. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra is denoted here by $A'_Γ$. We calculate the cohomology algebra (and coalgebra) of $A'_Γ$ explicitly. As a corollary to this calculation we have a proof that $A'_Γ$ is Koszul (respectively quadratic) if and only if $Γ$ is Cohen-Macaulay (respectively uniform). We show by example that the cohomology algebra (resp. coalgebra) of $A_Γ$ may be strictly smaller that the cohomology algebra (resp. coalgebra) of $A'_Γ$.

Motivation & Objective

• To compute the cohomology and coalgebra of the associated graded splitting algebra $A'_{\Gamma}$ for a finite ranked poset $\Gamma$.
• To establish a homological characterization of the Cohen-Macaulay and uniform properties of $\Gamma$ via the Koszul and quadratic properties of $A'_{\Gamma}$.
• To demonstrate that the cohomology of $A_{\Gamma}$ may be strictly smaller than that of $A'_{\Gamma}$, showing the filtration does not always preserve cohomological structure.
• To unify and generalize known results on splitting algebras by linking them to topological invariants of posets.

Proposed method

• The cohomology of $A'_{\Gamma}$ is computed via a spectral sequence and a canonical basis construction from [3], which allows defining a homotopy on a chain complex.
• The cohomology groups $\mathrm{Ext}^{p,q}_{A'_{\Gamma}}(\mathbb{F},\mathbb{F})$ are shown to be isomorphic to the reduced cohomology $\tilde{H}^{p-2}(\Delta(\Gamma_{b,q}))$ over all $b \in \Gamma$ with $rk(b) \geq q$.
• The coalgebra structure $\mathrm{Tor}^{A'_{\Gamma}}_{p,q}(\mathbb{F},\mathbb{F})$ is similarly expressed as a direct sum of reduced homology groups of order complexes.
• The proof relies on the canonical basis of $A_{\Gamma}$ from [3], which is used not for counting but to define a chain homotopy, enabling the cohomology computation.
• The authors use the standard conventions for reduced cohomology of the empty complex: $\tilde{H}^{-1}(\Delta(\emptyset)) = \mathbb{F}$, and $\tilde{H}^n(\Delta(\emptyset)) = 0$ for $n \neq -1$.
• Counterexamples are constructed to show that $gr(\mathrm{Ext}_{A_{\Gamma}}(\mathbb{F},\mathbb{F})) \neq \mathrm{Ext}_{A'_{\Gamma}}(\mathbb{F},\mathbb{F})$, proving the cohomology of $A_{\Gamma}$ can be strictly smaller than that of $A'_{\Gamma}$.

Experimental results

Research questions

• RQ1 What is the cohomology algebra of the associated graded splitting algebra $A'_{\Gamma}$ for a finite ranked poset $\Gamma$?
• RQ2 How does the cohomology of $A_{\Gamma}$ compare to that of $A'_{\Gamma}$, and under what conditions do they differ?
• RQ3 Can the Koszul and quadratic properties of $A'_{\Gamma}$ be characterized in terms of topological properties of $\Gamma$, such as Cohen-Macaulayness or uniformity?
• RQ4 Is there a homological invariant of $A_{\Gamma}$ that captures the topology of $\Gamma$'s order complex?

Key findings

• The cohomology algebra $\mathrm{Ext}^{p,q}_{A'_{\Gamma}}(\mathbb{F},\mathbb{F})$ is isomorphic to the direct sum of reduced cohomology groups $\tilde{H}^{p-2}(\Delta(\Gamma_{b,q}))$ over all $b \in \Gamma$ with $rk(b) \geq q$.
• The coalgebra $\mathrm{Tor}^{A'_{\Gamma}}_{p,q}(\mathbb{F},\mathbb{F})$ is isomorphic to the direct sum of reduced homology groups $\tilde{H}_{p-2}(\Delta(\Gamma_{b,q}))$ over the same index set.
• The algebra $A'_{\Gamma}$ is quadratic if and only if $\Gamma$ is uniform, and Koszul if and only if $\Gamma$ is Cohen-Macaulay.
• There exist finite ranked posets $\Gamma$ such that $gr(\mathrm{Ext}_{A_{\Gamma}}(\mathbb{F},\mathbb{F})) \neq \mathrm{Ext}_{A'_{\Gamma}}(\mathbb{F},\mathbb{F})$, proving the cohomology of $A_{\Gamma}$ can be strictly smaller than that of $A'_{\Gamma}$.
• The Hilbert series of $A_{\Gamma}$ is recovered as a corollary of the cohomology computation, providing a new proof of the result in [8].
• The results provide a new proof of the equivalence between $\Gamma$ being Cohen-Macaulay and $R_{\Gamma}$ being Koszul, as established in [5], by linking it to the cohomology of $A'_{\Gamma}$.