Obstruction modules of non-generic Eagon–Northcott complexes: explicit homology decompositions and spectral-sequence degeneration

For the generic p×q matrix the Eagon–Northcott complex is a minimal free resolution of the maximal-minors ideal, and its Betti table is classical. When the generic matrix is specialized to a structured pencil, the specialized complex may fail to be acyclic. Its homology modules Di = TorS i (R, M) are the obstruction modules of the title: they measure, as graded modules supported on coordinate subspaces, the failure of acyclicity of the specialized Eagon–Northcott complex. We compute Di completely for the rank-one pencil A = x1B and for the decomposable diagonal pencil A = diag(x1, . . . , xp, 0, . . . , 0); for the latter we give a complete decomposition of every Di into submodules supported on the distinct coordinate subspaces V (xt : t ∈ T). For the weighted diagonal variant with an arbitrary multiplicity map v : [p] → [s], the first obstruction module is L ∅=S⊆[p] (R/(xv(r) : r ∈ S))( q | − S| p ) (−p), which interpolates between the diagonal and rank-one families. We also treat the full-column augmentation A = diag(x1, . . . , xp, 0, 0) + xp+11e T p+1 of the diagonal pencil: here Ip(A) is the squarefree monomial ideal generated by all degree p products of the p + 1 variables, H1 is the direct sum of the R/(xi , xj )(−p), and all higher obstruction modules vanish. For the full-column pencil with several appended zero columns we reduce to the truncated complexes C [t] (1 ≤ t ≤ p + 1) and compute each Hn(C [t] ) as the middle term of a short exact sequence whose outer terms are direct sums of the cyclic modules R/(xs : s ∈ S); for r ≤ p + 1 this yields the corresponding decomposition of Hi(F(A)). The change-of-rings spectral sequence Epq 2 = TorR p (k, Dq) ⇒ TorS p+q (k, M) expresses the generic Betti table as a filtration whose graded pieces are subquotients of TorR • (k, Dq), and yields an upper bound on the regularity of R/Ip(A); this bound is sharp on the principal stratum. The q = 0 row records the exact Betti table of the specialized quotient; when the spectral sequence degenerates at E2 , as it does for the computed families, this row also identifies the corresponding associated-graded pieces of the generic Betti table. We compare these bounds with the generic linearization predictions of Cabarcas–Gaggero–Gorla [7] and with the sparse determinantal resolutions of Boocher [9]. The main results are the explicit computation of the obstruction modules Di and of the E2 -degeneration for structured non-generic pencils, together with the quantitative collapse of the free resolution in explicit parameter ranges.