On Monomial Ideals and Their Socles

Abstract

For a finite subset M ⊂ [x1,…, xd] of monomials, we describe how to constructively obtain a monomial ideal $I\subseteq R = K[x_{1},\ldots ,x_{d}]$ such that the set of monomials in Soc(I) ∖ I is precisely M, or such that $\overline {M}\subseteq R/I$ is a K-basis for the the socle of R/I. For a given M we obtain a natural class of monomials ideals I with this property. This is done by using solely the lattice structure of the monoid [x1,…, xd]. We then present some duality results by using anti-isomorphisms between upsets and downsets of the lattice $({\mathbb {Z}}^{d},\preceq )$ . Finally, we define and analyze zero-dimensional monomial ideals of R of type k, where type 1 are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in ${\mathbb {Z}}^{d}$ .