  We propose an analogue of the bounded derived category for an augmented ring
spectrum, defined in terms of a notion of Noether normalization. In many cases
we show this category is independent of the chosen normalization. Based on
this, we define the singularity and cosingularity categories measuring the
failure of regularity and coregularity and prove they are Koszul dual in the
style of the BGG correspondence. Examples of interest include Koszul algebras
and Ginzburg DG-algebras, $C^*(BG)$ for finite groups (or for compact Lie
groups with orientable adjoint representation), cochains in rational homotopy
theory and various examples from chromatic homotopy theory.
