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Curved BGG Correspondence
The Bernstein-Gel’fand-Gel’fand Correspondence is an equivalence between the bounded derived categories of finitely generated graded modules over an exterior algebra and that of a symmetric algebra. This result was established in 1978 and has been generalized in many different directions. In particular, Avramov et al.  extend the result to include the dg-algebra structure of the exterior algebra. In this dissertation we further extend this correspondence to a Koszul Complex, i.e. an exterior algebra equipped with non-trivial differential. The corresponding category is no longer that of modules over a symmetric algebra, but instead “curved modules” over a “curved algebra”. These curved modules generalize dg modules by allowing their differentials to square to multiplication by a non-zero element of the curved algebra.
Martin, A. Amadeus, "Curved BGG Correspondence" (2021). ETD collection for University of Nebraska - Lincoln. AAI28652057.