Some remarks on the Schweitzer complex

TL;DR

This paper proves the ellipticity of the Schweitzer complex, explores its cohomological properties, and derives results like finite dimensionality, Serre duality, deformation behavior, and an index formula related to Hirzebruch-Riemann-Roch.

Contribution

It establishes the ellipticity of the Schweitzer complex and connects its cohomology to classical index theorems, providing new insights into its structure.

Findings

Schweitzer complex is elliptic

Cohomologies are finite dimensional

Index formula aligns with Hirzebruch-Riemann-Roch

Abstract

We prove that the Schweitzer complex is elliptic and its cohomologies define cohomological functors. As applications, we obtain finite dimensionality, a version of Serre duality, restrictions of the behaviour of cohomology in small deformations, and an index formula which turns out to be equivalent to the Hirzebruch-Riemann-Roch relations.