Koszulity of the splitting algebras A(Γ) associated to directed graphs: the Cohen–Macaulay converse

To every layered directed graph Γ (equivalently, the Hasse graph of a finite ranked poset with a unique minimal vertex) Gelfand, Retakh, Serconek and Wilson (2005) attached a graded algebra $A(Γ)$ generated by the edges of Γ, with relations equating the path polynomials $P_π(t)=(1-te_1)⋯(1-te_k)$ of any two directed paths with the same origin and terminus. The construction generalizes the universal algebra $Q_n$ of pseudo-roots of noncommutative polynomials, and the source paper announced the program of relating the structure of $A(Γ)$ to the geometry of the graph. Later literature resolved large parts of this program — the Hilbert series of $A(Γ)$ is computed by a graded Möbius function, and the construction was extended to all finite acyclic directed graphs — but its central homological question survived a retraction: the 2006 claim that $A(Γ)$ is Koszul for every uniform Γ is false (counterexamples by Cassidy–Shelton and Nacin). What is known is one direction only: if Γ is Cohen–Macaulay (and uniform), then the rank-filtration associated graded $A'_Γ$ is Koszul, and hence so is $A(Γ)$. The open problem is the converse: does Koszulity of $A(Γ)$ over the ground field imply that $A'_Γ$ is Koszul, equivalently that Γ is Cohen–Macaulay? An answer in either direction, combined with the known one-directional theorem, completes the combinatorial characterization of Koszulity of splitting algebras — the surviving open core of the 2005 program.

Background

Factorizing a polynomial over a noncommutative ring $R$ (with central variable $t$) is far from unique, and the roots of one factorization do not determine the others; each factorization produces pseudo-roots, elements $ξ∈R$ for which $P(t)=Q_1(t)(t-ξ)Q_2(t)$. A program of I. Gelfand's school encodes the universal rational relations among pseudo-roots in an associative algebra: for polynomials of degree $n$ this yields the algebra $Q_n$ (Gelfand–Retakh–Wilson 2001), which is quadratic, has exponential growth, yet is 'tame' — it has a known linear basis, a computed Hilbert series (the generating function of the dimensions of its graded pieces), and is a Koszul algebra (Serconek–Wilson 2004).

Recall the relevant homological definitions. A graded algebra $A=⊕A_i$ generated by $A_0=F$ (the ground field) and $A_1$ is quadratic if its ideal of relations is generated in degree 2, and is Koszul if the trivial module $F$ has a linear minimal free resolution, i.e. every differential of the minimal resolution raises degree by exactly 1; equivalently, the Yoneda algebra $\mathrm{Ext}_A(F,F)$ is generated in internal degrees 0 and 1. Koszul duality pairs a Koszul algebra with its quadratic dual $A^!$. Koszulity implies the numerical condition $H_A(-t)H_{A^!}(t)=1$ on Hilbert series ('numerical Koszulity'), which depends only on the dimensions of graded pieces, but this numerical condition does not imply Koszulity in general.

Gelfand, Retakh, Serconek and Wilson (2005) generalized $Q_n$ as follows. Let $Γ=(V,E)$ be a layered directed graph: vertices are partitioned into levels $V_0,…,V_n$ with $V_0=\{∗\}$ a unique minimal vertex, and every edge runs from level $i$ to level $i-1$ (equivalently, Γ is the Hasse graph of a finite ranked poset with unique minimal element, of height $n$). Let $T(E)$ be the tensor algebra on the vector space spanned by the edges. To each directed path $π=(e_1,…,e_k)$ one attaches the polynomial $P_π(t)=(1-te_1)(1-te_2)⋯(1-te_k)$ in $T(E)[t]$ truncated modulo $t^{n+1}$, and $A(Γ)$ is the quotient of $T(E)$ by the relations $P_π(t)=P_{π'}(t)$ for all pairs of directed paths with the same origin and terminus. For the graph of all subsets of an $n$-element set (the $n$-dimensional hypercube, the Hasse graph of the Boolean lattice) $A(Γ)$ is isomorphic to $Q_n$; further motivating examples are Hasse graphs of subspace lattices, partition lattices, abstract polytopes, and complexes. The source paper constructs a linear basis of $A(Γ)$ for layered graphs with a unique minimal vertex and closes its introduction with the program sentence: 'We believe that the same is true for algebras associated to other directed graphs. In particular, we are planning to relate the structure of the algebra $A(Γ)$ with the geometry of the graph.' — that is, to extend the structure theory of $Q_n$ (bases, Hilbert series, quadraticity, Koszulness) to $A(Γ)$ and to read the combinatorial geometry of Γ off the algebra.

The subsequent literature works in poset language with two companion algebras. Filtering $A(Γ)$ by the rank function on Γ yields an associated graded algebra $A'_Γ$, of which $A(Γ)$ is a filtered deformation, and when Γ satisfies a combinatorial condition called uniform, $A'_Γ$ is quadratic and its quadratic dual is the finite-dimensional algebra $R_Γ$ introduced by Cassidy–Phan–Shelton. Uniform means: for every element $x$ of Γ, any two elements covered by $x$ can be joined by a chain of elements sharing a common lower cover. A finite ranked poset Γ is Cohen–Macaulay over $F$ (a field-relative condition) if the order complex — the simplicial complex of chains — of every open subinterval $(a,b)$ has reduced homology concentrated in the single top degree $\dim(a,b)$. The key landmarks: (i) Γ uniform is equivalent to $A'_Γ$ being quadratic (Shelton 2012) and implies $A(Γ)$ is quadratic (Retakh–Serconek–Wilson 2006). (ii) Kloefkorn–Shelton (2015) proved that Γ is uniform with $R_Γ$ Koszul if and only if Γ is Cohen–Macaulay, and that Koszulity of $A'_Γ$ or $R_Γ$ implies Koszulity of $A(Γ)$ — the last implication is the standard filtered-deformation fact that Koszulity of an associated graded descends to the filtered algebra (Polishchuk–Positselski). Combining (i) and (ii): $A'_Γ$ is Koszul if and only if Γ is Cohen–Macaulay, and Cohen–Macaulayness of Γ is sufficient for Koszulity of $A(Γ)$. (iii) The Hilbert series of $A(Γ)$ is computed from a two-variable Möbius polynomial of Γ (Retakh–Serconek–Wilson 2007, 2012), so numerical Koszulity of $A(Γ)$ is decidable from combinatorial data; but Kloefkorn–Shelton (2015) constructed uniform posets with $R_Γ$ numerically Koszul and not Koszul, so numerics cannot decide the homological question. (iv) The history includes a retraction: the 2006 assertion that $A(Γ)$ is Koszul for every uniform Γ is false (Cassidy–Shelton's counterexample with 18 edges and 11 vertices; Nacin's minimal example with 13 edges and 9 vertices; explicit retraction in Retakh–Serconek–Wilson 2010). Thus, within the class of uniform posets, the only known implication runs from Cohen–Macaulayness to Koszulity of $A(Γ)$, and it is unknown whether the two conditions coincide. Both Koszulity and Cohen–Macaulayness are relative to the ground field (for example, face posets of the real projective plane behave differently in characteristic 2, as shown by Cassidy–Phan–Shelton).

Problem Statement

Let $F$ be a field and let Γ be a finite ranked poset with a unique minimal element (equivalently, a layered directed graph with a unique minimal vertex), with splitting algebra $A(Γ)$ as defined by Gelfand–Retakh–Serconek–Wilson: the quotient of the tensor algebra on the $F$-vector space spanned by the edges by the relations $P_π(t)=P_{π'}(t)$, where for a directed path $π=(e_1,…,e_k)$ one sets $P_π(t)=(1-te_1)⋯(1-te_k)$, computed in the tensor algebra over $F$ with a central variable $t$ adjoined and truncated modulo $t^{n+1}$ ($n$ the height of Γ), imposed for all pairs of directed paths $π,π'$ with the same origin and terminus. Let $A'_Γ$ denote the associated graded of $A(Γ)$ for the rank filtration, and $R_Γ$ the quadratic dual of $A'_Γ$ (defined when Γ is uniform, i.e. when for every element $x$ of Γ any two elements covered by $x$ can be joined by a chain of elements sharing a common lower cover). Recall that Γ is Cohen–Macaulay over $F$ if the order complex — the simplicial complex of chains — of every open subinterval of Γ has reduced homology concentrated in the single top degree; this is a field-relative condition. Known: if Γ is Cohen–Macaulay over $F$, then $A'_Γ$ and $R_Γ$ are Koszul and consequently $A(Γ)$ is Koszul; and $A'_Γ$ is Koszul only if Γ is Cohen–Macaulay. The open core of the problem, as stated by Kloefkorn–Shelton: it is not known whether the Koszul property for $A(Γ)$ implies the Koszul property for $A'_Γ$.

Decide the converse: if $A(Γ)$ is Koszul over $F$, must $A'_Γ$ be Koszul — equivalently, must Γ be Cohen–Macaulay over $F$? Both Koszulity and Cohen–Macaulayness are field-relative. An answer in either direction, combined with the known one-directional theorem, completes the combinatorial characterization of when the splitting algebra of a directed graph is Koszul — the surviving open part of the program announced in the 2005 source paper.

References

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