where the subscript j on the right of a graded module denotes, throughout the paper, the degree j component of that module. There are two monomial ideals canonically attached to I: the generic initial ideal Gin(I) with respect to the revlex order and the lex-segment ideal Lex(I). They play a fundamental role in the investigation of many algebraic, homological, combinatorial and geometric proper...

The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that J m = m. Vasconcelos conjectured that r(R/I) ≤ r(R/ in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture. 1. Reduction ...

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger. Let R = k[x1, x2, . . . , xr] be a polynomial ring over a field. We will say that an ideal I ⊆ R has the saturated chain p...

We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra R over an algebraically closed field k. Building on work Römer and Schmitz, we give formula for each ideal, express associated quasivaluations in terms certain I-adic filtrations. As corollary, show that case is domain, every ideal coming from codimension 1 skeleton variety prime, so “generic presentation...

We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute the symmetric algebraic shifted complex of two disjoint simplicial complexes as was conjectured by Kalai. This result is the symmetric analogue of a theorem of ...

Abstract In this paper, we survey the theory of Cartwright–Sturmfels ideals. These are $\mathbb {Z}^{n}$ ℤ n -graded ideals, whose multigraded generic initial ideal is radical. ideals have surprising properties, mostly stemming from fact that thei...

Recently, Nevo introduced the notion of strongly edge decomposable spheres. In this paper, we characterize the algebraic shifted complex of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley–Reisner ideal of Gorenstein* complexes having the strong Lefschetz property in characteristic 0.

We study generic toric rings. We prove that they are Golod rings, so the Poincaré series of the residue field is rational. We classify when such a ring is Koszul, and compute its rate. Also resolutions related to the initial ideal of the toric ideal with respect to reverse lexicographic order are described.