  The homotopy groups of a commutative algebra in spectra form a commutative
algebra in the symmetric monoidal category of graded abelian groups. The
grading and the Koszul sign rule are remnants of the structure encoded by anima
as opposed to sets. The purpose of this paper and its sequel is to develop the
geometry built from such algebras. We name this geometry Dirac geometry, since
the grading exhibits the hallmarks of spin. Indeed, it is a reflection of the
internal structure encoded by anima, and it distinguishes symmetric and
anti-symmetric behavior, as does spin. Moreover, the coherent cohomology, which
we develop in the sequel admits half-integer Serre twists.
