Math Conjectures

Terao’s Freeness Conjecture

OPENMajorConjectureProposed 1983 · Standard version

Canonical statement

Let A\mathcal A be a finite central hyperplane arrangement in a finite-dimensional complex vector space VV. For each H∈AH\in\mathcal A, choose αH∈V∗\alpha_H\in V^* with H=ker⁡αHH=\ker\alpha_H, and define the module of logarithmic derivations
D(A)={θ∈Der⁡CC[V]:θ(αH)∈αHC[V] for every H∈A}. D(\mathcal A)=\{\theta\in\operatorname{Der}_{\mathbb C}\mathbb C[V]: \theta(\alpha_H)\in\alpha_H\mathbb C[V] \text{ for every }H\in\mathcal A\}.
Call A\mathcal A free when D(A)D(\mathcal A) is a free C[V]\mathbb C[V]-module, and let
L(A)={⋂H∈SH:S⊆A} L(\mathcal A)=\left\{\bigcap_{H\in S}H:S\subseteq\mathcal A\right\}
be its intersection lattice, ordered by reverse inclusion. If two such arrangements A\mathcal A and A′\mathcal A' have isomorphic intersection lattices, then A\mathcal A is free if and only if A′\mathcal A' is free.
View source LaTeX
Let \(\mathcal A\) be a finite central hyperplane arrangement in a finite-dimensional complex vector space \(V\). For each \(H\in\mathcal A\), choose \(\alpha_H\in V^*\) with \(H=\ker\alpha_H\), and define the module of logarithmic derivations
\[
 D(\mathcal A)=\{\theta\in\operatorname{Der}_{\mathbb C}\mathbb C[V]:
       \theta(\alpha_H)\in\alpha_H\mathbb C[V]
       \text{ for every }H\in\mathcal A\}.
\]
Call \(\mathcal A\) free when \(D(\mathcal A)\) is a free \(\mathbb C[V]\)-module, and let
\[
 L(\mathcal A)=\left\{\bigcap_{H\in S}H:S\subseteq\mathcal A\right\}
\]
be its intersection lattice, ordered by reverse inclusion. If two such arrangements \(\mathcal A\) and \(\mathcal A'\) have isomorphic intersection lattices, then \(\mathcal A\) is free if and only if \(\mathcal A'\) is free.
ALG-ARR-020:[Terao1983Exponents][OrlikTerao1992Arrangements][Yoshinaga2014Freeness]
Algebra and representation theoryCombinatorics

Notes

Freeness is combinatorial for many classes of arrangements, and addition–deletion and characteristic-polynomial criteria settle extensive families. In characteristic zero, no proof or counterexample is known for arbitrary arrangements with the same intersection lattice.
Counterexamples exist over fields of positive characteristic. This record fixes the classical characteristic-zero form over C\mathbb C.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.