# Morita theory and singularity categories

## TL;DR

This paper develops a new categorical framework for augmented ring spectra inspired by Morita theory, establishing dualities and invariants that measure regularity and coregularity in homotopy-theoretic contexts.

## Contribution

It introduces an analogue of the bounded derived category for augmented ring spectra, defines singularity and cosingularity categories, and proves their Koszul duality, extending classical algebraic concepts to homotopy theory.

## Key findings

- The new category is often independent of the normalization chosen.
- Singularity and cosingularity categories are Koszul dual.
- Applications include Koszul algebras, Ginzburg DG-algebras, and cochains in homotopy theory.

## Abstract

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.