A Structure Sheaf for Kirch Topology

TL;DR

This paper introduces a sheaf of locally Lipschitz functions on Kirch topology, exploring its cohomological properties and extending the analogy with complex-analytic sheaves.

Contribution

It constructs a natural sheaf of rings on Kirch topology and investigates its basic cohomological properties, filling a gap in the analogy with complex analysis.

Findings

Analyzed zeroth and first cohomology groups of the sheaf

Studied Cech cohomology with respect to basic open covers

Established foundational properties of the sheaf on Kirch topology

Abstract

Kirch topology on $\mathbb N$ goes back to 1969, and is remarkable for being Hausdorff, connected, and locally connected. In this sense, it is analogous to the usual topology on $\mathbb C,$ yet, to the author's knowledge, there have been no Kirch topology analogs of the sheaf of complex-analytic functions until very recently. In our latest paper we constructed such natural sheaf of rings, the sheaf of locally LIP functions. In this paper we investigate some of its basic properties, primarily regarding zeroth and first cohomology and Cech cohomology with respect to covers by basic open sets.