Abstract Given a noncommutative partial resolution A = End R ( R ⊕ M ) of a Gorenstein singularity R, we show that the relative singularity category Δ R ( A ) of Kalck–Yang is controlled by a certain connective dga A / L A e A , the derived quotient of Braun–Chuang–Lazarev. We think of A / L A e A as a kind of ‘derived exceptional locus’ of the partial resolution A, as we show that it can be thought of as the universal dga fitting into a suitable recollement. This theoretical result has… Expand