Comparing Dualities in the $K(n)$-local Category

TL;DR

This paper clarifies and details the duality equivalences in the $K(n)$-local category at large primes, focusing on algebraic structures related to Lubin-Tate space and invertible sheaves.

Contribution

It provides detailed proofs and clarifications of duality equivalences in the $K(n)$-local category, emphasizing algebraic aspects at large primes.

Findings

Two important invertible sheaves become isomorphic modulo p in the Picard group of Lubin-Tate space.

The duality equivalences are explicitly established and clarified for the $K(n)$-local category.

The results are algebraic and valid at large primes, extending previous accessible but less detailed results.

Abstract

In their work on the period map and the dualizing sheaf for Lubin-Tate space, Gross and the second author wrote down an equivalence between the Spanier-Whitehead and Brown-Comenetz duals of certain type $n$-complexes in the $K(n)$-local category at large primes. In the culture of the time, these results were accessible to educated readers, but this seems no longer to be the case; therefore, in this note we give the details. Because we are at large primes, the key result is algebraic: in the Picard group of Lubin-Tate space, two important invertible sheaves become isomorphic modulo $p$.