AI 中文总结
本文针对有限域F_q上的子空间格，确定了非零d循环的最小支撑大小，证明了d≥4时的间隙稳定性结果，构造支撑可控锥并给出上同调扩张下界。
AI 中文摘要
经典几何结果表明，从[n]的d元子集到(d-1)元子集的模2关联映射的每个非零循环，其支撑至少为d+1，且该下界由d+1个顶点构成的单纯形的边界达到。我们针对F_q^n的子空间格证明了一个类似结果，确定了当特征p整除q+1时，域K上非零d循环的最小支撑大小。令人惊讶的是，(d+1)维空间的边界并不总是最优的：当d=1时存在更短的循环，当n≥4且d=2时也存在更短的循环；其他情况下，(d+1)维空间的边界是最短的。对于d≥4，我们证明了间隙稳定性结果：每个支撑小于最小值的(2-10/q)倍的循环都是(d+1)维空间边界的倍数。我们还构造了支撑可控锥，对子空间关联复形同调群消失的维数进行了直接几何分析，并给出了其上同调扩张的显式下界；1阶扩张估计是该稳定性定理的一个组成部分。
英文摘要
A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of $\mathbb{F}_q^n$, determining the minimum support size of a nonzero $d$-cycle over a field $K$ of characteristic $p \mid q+1$. Surprisingly, the boundary of a $(d+1)$-space is not always optimal. Shorter cycles occur for $d=1$, and for $d=2$ when $n\ge4$, and otherwise, the boundary of a $(d+1)$-space is shortest. For $d\ge4$, we prove a gap-stability result: every cycle with support less than $(2-10/q)$ times the minimum is a multiple of the boundary of a $d+1$ space. We also construct support-controlled cones, yielding a direct geometric analysis of the dimensions in which the homology groups of the subspace incidence complex vanish and explicit lower bounds on its coboundary expansion. The degree-$1$ expansion estimate is an ingredient in the stability theorem.
Comments16 pages. A previous version with major overlaps appeared as 2306.14317
