The minimal free resolution of a class of square-free monomial ideals.
In this very well written paper one finds a thorough study of monomial ideals generated by minors of the direct sum of several vectors of indeterminates. The authors call such an ideal transversal and denote it by \(I_t\) if it is generated by degree \(t\) monomials. A transversal ideal is square-free, so it may be associated to a simplicial complex in the usual way. A first result asserts that a transversal ideal is Gorenstein exactly when it is principal. Although the simplicial complex naturally associated to a transversal ideal is pure and shellable if and only if all vectors of indeterminates have length one (cf. proposition 1.1), its Alexander dual is always pure and shellable (cf. proposition 2.1). Therefore, \(I_t\) has a linear resolution and the quotient ring \(S/I_t\) is Golod. Moreover, the Buchsbaum property of a quadratic transversal ideal is equivalent to the purity of the associated simplicial complex. The Hilbert series of a transversal ideal is obtained explicitly in theorem~2.4 from an exact sequence which is usually considered for shellable complexes. The minimal free resolution of the Stanley-Reisner ring associated to a transversal ideal is computed in section 3. The construction relies on the Eagon-Northcott complex associated to maximal minors of matrices of indeterminates and on the minimal free resolution of determinantal rings defined by generic diagonal matrices. The final sections contains applications to the study of generic multiple points in algebraic geometry. It is shown that transversal ideals generalize the ideals of minors of a generic pluri-circulant matrix over a field with convenient properties.
scientific article; zbMATH DE number 2078862