  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.
