  The nilpotence variety for extended supersymmetric quantum mechanics is a
cone over a quadric in projective space. The pure spinor correspondence, which
relates the description of off-shell supermultiplets to the classification of
modules over the corresponding hypersurface ring, reduces to a classical
problem of linear algebra. Spinor bundles, which correspond to maximal
Cohen-Macaulay modules, serve as basic building blocks. Koszul duality appears
as a deformed version of the Bernstein-Gel'fand-Gel'fand correspondence that we
make fully concrete. We illustrate in numerous examples the close relationship
between these connections and the powerful graphical technology of Adinkras. We
emphasize the role of R-symmetry for recovering higher-dimensional gauge and
gravity multiplets.
