AI 中文总结
该论文将 Kozlov 关于单纯复形 f-向量凸包的定理推广到外代数上分次自由模的商模 Hilbert 函数，证明其凸包为 r 个 Kozlov 单纯形的 Minkowski 和，并给出自包含证明及顶点结构刻画。
AI 中文摘要
设 E 是域上由 n 个生成元生成的外代数，F 是秩为 r 的分次自由 E-模，其生成元次数为 d_1 ≤ … ≤ d_r。我们确定了商模 F/M 的 Hilbert 函数集合的凸包，其中 M 遍历 Amata 和 Crupi 的单项式子模。该凸包是 r 个同一单纯形（即 Kozlov 单纯形）平移副本的 Minkowski 和；Kozlov 于 1997 年证明了该单纯形是 n 个顶点上单纯复形的 f-向量集合的凸包。当 r=1 时，该结论即为 Kozlov 定理，因此这是该定理的秩-r 推广。我们给出了自包含的证明，包括利用局部 Lubell-Yamamoto-Meshalkin 不等式对 Kozlov 定理的简短证明，以及 Kozlov 单纯形的显式面描述（该描述在文献中似乎未见记载）。在此过程中，我们分离出论证适用的模类——理想直和模，并通过例子表明，该类之外的分次子模的 Hilbert 函数不必是 f-向量的平移和。组合内容以不含代数的语言单独陈述，作为关于单纯复形的 r-向量及其 ff-向量的结果。最后，我们描述了 Minkowski 和的顶点结构——即和项顶点的哪些和仍为顶点——证明了在全 1 腿向量（任意秩）和 Kozlov 腿向量（秩二）情形下的答案。
英文摘要
Let $E$ be the exterior algebra on $n$ generators over a field, and let $F$ be a graded free $E$-module with $r$ generators, of degrees $d_1 \le \dots \le d_r$. We determine the convex hull of the set of Hilbert functions of the quotients $F/M$, where $M$ runs over the monomial submodules of Amata and Crupi. The hull is the Minkowski sum of $r$ shifted copies of one and the same simplex: the Kozlov simplex, which Kozlov proved in 1997 to be the convex hull of the $f$-vectors of simplicial complexes on $n$ vertices. For $r=1$ the statement is Kozlov's theorem, so this is a rank-$r$ generalization of it. We give a self-contained proof, including a short proof of Kozlov's theorem from the local Lubell-Yamamoto-Meshalkin inequality, and an explicit facet description of the Kozlov simplex that appears not to be recorded in the literature. Along the way we isolate the class of submodules for which the argument works, the ideal-direct-sum modules, and show by example that the Hilbert function of a graded submodule outside that class need not be a shifted sum of $f$-vectors at all. The combinatorial content is stated separately, in language that uses no algebra, as a result on $r$-vectors of simplicial complexes and their $ff$-vectors. Finally we describe the vertex structure of the Minkowski sum -- which sums of vertices of the summands survive as vertices -- proving the answer for the all-ones leg vector at every rank and for the Kozlov leg vector at rank two.
Comments53 pages. The combinatorial core is formalized in Lean 4 + Mathlib (no sorry, no extra axioms); the formalization and all verification code are in the anc directory and at https://gitlab.liu.se/jansn19/exterior-convex
