Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions

Let n be a nonnegative integer. For each composition alpha of n, Berg, Bergeron, Saliola, Serrano and Zabrocki introduced a cyclic indecomposable H-n(0) module nu(alpha) with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study nu(alpha)s from the homological viewpoint. To be precise, we construct a minimal projective presentation of nu(alpha) and a minimal injective presentation of nu(alpha) as well. Using them, we compute Ext(Hn(0))(1) (nu(alpha), F-beta) and Ext(Hn(0))(1) (F-beta, nu(alpha)), where F-beta is the simple H-n(0)-module attached to a composition beta of n. We also compute Ext(Hn(0))(i) (nu(alpha), nu(beta)) when i = 0, 1 and beta <=(l) alpha, where <=(l) represents the lexicographic order on compositions.