Pure free resolutions over arbitrary Veronese subalgebras

Determine whether the Eisenbud–Fløystad–Weyman-style construction of pure free resolutions using skew Schur functors and Pieri-type maps extends to all Veronese subalgebras S^{(d)} of the polynomial ring S^•(V). Specifically, ascertain whether the identical construction described for the degree-d rational normal curve yields pure free resolutions over S^{(d)} for arbitrary d, and clarify the obstacles related to the non-uniqueness of Pieri maps for skew Schur functors and the lack of Schur’s lemma in this setting.

Background

In the paper, the authors generalize the Eisenbud–Fløystad–Weyman (EFW) construction to quadric hypersurface rings by introducing quadric Schur functors via Jacobi–Trudi structures, and they also produce resolutions over the coordinate ring of the rational normal curve S^{(d)} when dim V = 2. These constructions rely on functorial techniques (Zelevinsky’s functor) and on specific representation-theoretic properties in low rank cases.

However, extending the same approach to arbitrary Veronese subalgebras presents difficulties. In particular, Pieri maps are not uniquely defined for skew Schur functors, and Schur’s lemma cannot be applied to guarantee that the sequences of maps form complexes. The authors explicitly state uncertainty about the general case, highlighting the need to determine whether analogous pure resolutions exist for all Veronese subalgebras.