Homological Algebra on Graded Posets

TL;DR

This paper explores homological algebra in graded posets, introducing pseudo-projectivity, and provides methods for computing integral cohomology, with applications to simplicial complexes, Quillen's complex, and Morse theory.

Contribution

It introduces pseudo-projectivity for functors on graded posets and develops new methods for computing their integral cohomology.

Findings

Pseudo-projectivity ensures vanishing of derived limits.

Generalized Whitehead theorem for pushouts.

Two cohomology computation methods: local and global.

Abstract

We describe the projectives in the category of functors from a graded poset to abelian groups. Based on this description we define a related condition, pseudo-projectivity, and we prove that this condition is enough for the vanishing of the derived direct limits. We apply this result to deduce a generalized version of a theorem of Whitehead for the pushout. The dual results for inverse limits are also considered. We present two methods to compute integral cohomology of posets, a local one a and a global one. The local method is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes, simplex-like posets and for Quillen's complex of a finite group. The global method is related to Morse Theory.