Affine Dual Braid Monoids: Finite Cores, Exceptional Cluster Complexes, and Koszul Resolutions

Abstract

For every finite-rank crystallographic affine Coxeter system $(W,S)$ and Coxeter element $c$, we construct a minimal linear graded free resolution of the trivial module over $k[M([1,c{]}_T)]$ supported on a rectified exceptional cluster complex. Hence the affine dual braid monoid algebra is Koszul over every field $k$. The exceptional complex is introduced to recover the principal-fibre topology missing from the direct Reading--Stella labelling. Half-orbit rectification replaces the transjective root labels by ordinary exceptional modules, so that a face $F$ determines an exceptional wide subcategory and the intrinsic weight \[ ω(F)=\operatorname{cox}(\operatorname{wide}\langle F\rangle). \] The resulting principal fibres are induced subcomplexes and split canonically as joins of subcomplexes attached to connected Dynkin and affine blocks; these subcomplexes are contractible. Affine non-lattice divisibility creates the genuinely nonprincipal case. The McCammond--Sulway completion shows that whenever no greatest interval right divisor exists, all maximal interval right divisors share a common nontrivial complete finite Coxeter component. In the associated exceptional-wide decompositions, this common Coxeter component is the Coxeter element of a Dynkin block, and the subcomplex attached to that block occurs as a common contractible join factor. Thus every nonidentity fibre is contractible, and the weighted-face complex is exact, minimal and linear. In particular, ${\mathrm{Tor}}_q^{A_c}(k,k)$ is indexed by $q$-vertex exceptional cluster faces in internal degree $q$, and ${\mathrm{pd}}_{A_c}k=|S|$.

35 pages