仿射对偶辫群：有限核、例外丛复形与Koszul分解
Affine Dual Braid Monoids: Finite Cores, Exceptional Cluster Complexes, and Koszul Resolutions
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中文总结 AI 辅助
本研究针对有限秩晶体学仿射Coxeter系统，构造了平凡模的极小线性分次自由分解，证明仿射对偶辫群代数是Koszul的，揭示了例外丛复形的拓扑性质与同调意义。
中文摘要 AI 辅助
针对每个有限秩晶体学仿射Coxeter系统$(W,S)$和Coxeter元$c$，我们构造了平凡模在$k[M([1,c]_T)]$上的极小线性分次自由分解，该分解以校正后的例外丛复形为支撑。由此可得仿射对偶辫群代数在任意域$k$上都是Koszul的。\n 引入例外复形是为了恢复直接的Reading--Stella标记所缺失的主纤维拓扑。半轨道校正将可递根标记替换为普通例外模，使得面$F$确定一个例外宽子范畴，且内蕴权满足\n \\[\n ω(F)=\operatorname{cox}(\operatorname{wide}\langle F\rangle).\n \\] 所得的主纤维是诱导子复形，并可典范分裂为附着于连通Dynkin块和仿射块的子复形的并；这些子复形都是可缩的。\n 仿射非格可除性产生了真正的非主情形。McCammond--Sulway完备化表明，当不存在最大区间右除子时，所有极大区间右除子都共享一个公共的非平凡完全有限Coxeter分支。在相关的例外宽分解中，这个公共Coxeter分支是某个Dynkin块的Coxeter元，且附着于该块的子复形作为公共可缩并因子出现。因此每个非单位纤维都是可缩的，且加权面复形是正合的、极小的且线性的。特别地，$\operatorname{Tor}^{A_c}_q(k,k)$由内蕴次数为$q$的$q$顶点例外丛面索引，且$\operatorname{pd}_{A_c}k=|S|$。
英文摘要
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, $\operatorname{Tor}^{A_c}_q(k,k)$ is indexed by $q$-vertex exceptional cluster faces in internal degree $q$, and $\operatorname{pd}_{A_c}k=|S|$.
