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

Coalgebraic Formal Curve Spectra and the Annular Tower