Generalized Castelnuovo-Mumford Regularity for Representations of the Virasoro Algebra

We compute the generalized noncommutative Castelnuovo-Mumford regularity of irreducible highest-weight modules over the Virasoro algebra Vir. Working over the graded negative universal enveloping algebra A=U(n−)⊂U(Vir) equipped with the level grading (degL−n=n), we analyze three structural classes of representations: generic Verma modules, irreducible quotients with a single singular generator, and rational minimal models. In particular, for the irreducible quotients L(cp,q,hr,s) in minimal models (1≤r≤p−1, 1≤s≤q−1), we show, using the Feigin–Fuchs structure theorem for Verma modules, that the BGG-type resolution of Feigin–Fuchs and Rocha-Caridi–Wallach restricts to a minimal graded free resolution over A with exactly two free summands in each positive homological degree, and we determine all generator degrees separately for odd and even homological degrees. The degree bounds are Ti=(ipq+ηi)2−(rq−sp)2/(4pq), where ηi=|rq−sp| for even i, and ηi=|pq−rq−sp| for odd i. Consequently, the asymptotic regularity exponent is e=2, and the regularity rate is r=pq/4, giving regAL(cp,q,hr,s)=(2,pq/4). The asymptotic regularity rate depends exclusively on the minimal model parameters (p,q) (and thus on the central charge cp,q), remaining invariant under the choice of highest weight hr,s, whereas the linear correction term does depend on hr,s. For (p,q)=(2,3) the result recovers Goncharova’s theorem on the homology of the Lie algebra of vector fields on the line.