A method for integral cohomology of posets

We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is based on homological algebra arguments in the category of functors and on a spectral sequence built upon the poset. We show its relation to discrete Morse theory. As application we give an alternative proof of Webb’s conjecture for saturated fusion systems and we compute the cohomology of Coxeter complexes for finite and infinite Coxeter groups.

💡 Research Summary

The paper introduces a systematic method for computing the integral cohomology of a poset (P). The approach hinges on two structural conditions on the lower and upper intervals of each element: (C1) the reduced cohomology of the lower interval (P_{<x}) vanishes except possibly in degree (\operatorname{rk}(x)-1), and (C2) the reduced cohomology of the upper interval (P_{>x}) vanishes except possibly in degree (\operatorname{rk}_{\max}-\operatorname{rk}(x)-1). These conditions are satisfied by a broad class of posets, notably simplicial complexes, “simplex‑like” posets, and the face posets of Coxeter complexes.

The authors work in the functor category (\mathbf{Fun}(P,\mathbf{Ab})). For each element (x) they define a standard projective functor (F_x) that is (\mathbb Z) on all objects (y\ge x) and zero otherwise. The constant functor (\underline{\mathbb Z}) admits a projective resolution built from the direct sum of these (F_x)’s. The resulting cochain complex (C^\bullet(P)) has the form
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