# Coalgebraic Formal Curve Spectra and the Annular Tower

## TL;DR

This paper introduces a new algebraic geometric approach to homotopy theory, producing models of the determinantal sphere and cellular decompositions in the context of Morava K-theories, and extends Snaith's theorem to Iwasawa extensions.

## Contribution

It develops a novel geometric construction within homotopy theory that yields new models and decompositions for spectra related to Morava K-theories and extends classical theorems to Iwasawa extensions.

## Key findings

- A choice-free model of the determinantal sphere.
- An efficient Picard-graded cellular decomposition of K(Z_p, d+1).
- A Snaith-type theorem for the Iwasawa extension of the K(d)-local sphere.

## Abstract

We import into homotopy theory the algebro-geometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava $K$-theory of height $d$, we show that this can be used to produce a choice-free model of the determinantal sphere as well as an efficient Picard-graded cellular decomposition of $K(\mathbb Z_p, d+1)$. Coupling these ideas to work of Westerland, we give a "Snaith's theorem" for the Iwasawa extension of the $K(d)$-local sphere.