Marve Grönblad Vesterinen: Dual Elements in Posets and their Connection to the Homological Algebra of Vector Space Representations

Abstract.

We prove that the cohomology of the local Koszul cochain complex of finite dimensional vector space representations of finite posets computes the injective Betti diagrams of such representations. Furthermore, we introduce the notion of dual elements in posets, and prove that, if the poset is finite, there is a close connection between the injective and projective Betti diagrams evaluated at such elements. Also, we provide a simple classification of dual elements for the case where the underlying poset is a distributive lattice.