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Relative Leray numbers via spectral sequences

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 $i \geq d$ and $S \subset V$, the induced complex $X[S]$ satisfies $\tilde{H}_i(X[S];\mathbb{F})=0$. 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 $\tilde{H}_i(X[V \setminus σ];\mathbb{F})=0$ for all $i \geq d$ and $σ\in Y$. 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.