Let $\mathbb{F}$ be a fixed field and let X be a simplicial complex on the vertex set V. The Leray number $L(X;\mathbb{F})$ is the minimal d such that for all $i⩾d$ and $S\subset V$, the induced complex $X[S]$ satisfies ${\stackrel{\sim }{H}}_i(X[S];\mathbb{F})=0$. Leray numbers play a role in formulating and proving topological Helly-type theorems. For two complexes $X,Y$ on the same vertex set V, define the relative Leray number $L_Y(X;\mathbb{F})$ as the minimal d such that ${\stackrel{\sim }{H}}_i(X[V\setminus \tau ];\mathbb{F})=0$ for all $i⩾d$ and $\tau \in Y$. In this paper we extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice.

中文翻译：

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通过光谱序列的相对勒雷数

让 $\mathbb{F}$是一个固定域，让X是顶点集V上的单纯复形。勒雷数$升(X;\mathbb{F})$是最小的d使得对于所有$\mathrm{一世}⩾d$ 和 $秒\subset 伏$, 诱导复合体 $X[秒]$ 满足 ${\stackrel{～}{H}}_{\mathrm{一世}}(X[秒];\mathbb{F})=0$. Leray 数在制定和证明拓扑 Helly 型定理中发挥作用。对于两个复合体$X,是$在同一个顶点集V 上，定义相对 Leray 数${升}_{是}(X;\mathbb{F})$作为最小的d使得${\stackrel{～}{H}}_{\mathrm{一世}}(X[伏\setminus \tau ];\mathbb{F})=0$ 对所有人 $\mathrm{一世}⩾d$ 和 $\tau \in 是$. 在本文中，我们将拓扑彩色 Helly 定理扩展到相对设置。我们的主要工具是由几何格子索引的复合物交集的光谱序列。