Picard groups and the K-theory of curves with cuspidal singularities

TL;DR

This paper computes the algebraic K-theory of cuspidal curve coordinate rings over regular F_p-algebras, confirming Hesselholt's conjecture, and also determines the Picard group of certain p-complete spectra.

Contribution

It provides the first explicit calculation of the K-theory for these singular curves and computes the Picard group of the associated homotopy category, advancing understanding in algebraic K-theory and stable homotopy.

Findings

Verified Hesselholt's conjecture for cuspidal curves.

Calculated the algebraic K-theory of the coordinate ring of cuspidal curves.

Determined the Picard group of p-complete genuine C_{p^n}-spectra.

Abstract

We calculate the algebraic $K$-theory of the coordinate ring of a planar cuspidal curve over a regular $\mathbb{F}_p$-algebra, thereby verifying a conjecture due to Hesselholt. In the course of the proof we compute the Picard group of the homotopy category of $p$-complete genuine $C_{p^n}$-spectra.