In this paper, we introduce the class of ideals with $(d_1,ldots,d_m)$-linear quotients generalizing the class of ideals with linear quotients. Under suitable conditions we control the numerical invariants of a minimal free resolution of ideals with $(d_1,ldots,d_m)$-linear quotients. In particular we show that their first module of syzygies is a componentwise linear module.

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the Stanley-Reisner correspondence, to associate to a square-free monomial...

In this paper, we study the minimal free resolution of lex-ideals over a Koszul toric ring. In particular, we study in which toric ring R all lexideals are componentwise linear. We give a certain necessity and sufficiency condition for this property, and show that lex-ideals in a strongly Koszul toric ring are componentwise linear. In addition, it is shown that, in the toric ring arising from t...

Let C ⊂ Z be an affine semigroup, R = K[C] its semigroup ring, and *modC R the category of finitely generated “C-graded” R-modules (i.e., Z -graded modules M with M = ⊕ c∈C Mc). When R is Cohen-Macaulay and simplicial, we show that information on M ∈ *modC R such as depth, CohenMacaulayness, and (Sn) condition, can be read off from numerical invariants of the minimal irreducible resolution (i.e...

We combine the theory of Pommaret bases with a (slight generalisation of a) recent construction by Sköldberg based on discrete Morse theory. This combination allows us the explicit determination of a (generally non-minimal) free resolution for a graded polynomial module with the computation of only one Pommaret basis. If only the Betti numbers are needed, one can considerably simplify the compu...

A very well–covered graph is an unmixed without isolated vertices such that the height of its edge ideal half number vertices. We study these graphs by means Betti splittings and mapping cone constructions. show cover ideals Cohen–Macaulay are splittable. As a consequence, we compute explicitly minimal graded free resolution class prove have homological linear quotients. Finally, conjecture sam...

The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P are determined whenever Z is supported at any 6 or fewer distinct points. We also handle a broad range of cases in which the points can be infinitely near, related to the classification of normal cubic surfaces. All results hold over an arbitrary algebra...

Rank 2 indecomposable arithmetically Cohen-Macaulay bundles E on a nonsingular cubic surface X in P are classified, by means of the possible forms taken by the minimal graded free resolution of E over P. The admissible values of the Chern classes of E are listed and the vanishing locus of a general section of E is studied. Properties of E such as slope (semi) stability and simplicity are invest...

Given multigraded free resolutions of two monomial ideals we construct a multigraded free resolution of the sum of the two ideals. Introduction Let K be a field, S = K[x1, . . . , xn] the polynomial ring in n variables over K, and let I and J be two monomial ideals in S. Suppose that F is a multigraded free S-resolution of S/I and G a multigraded free S-resolution of S/J . In this note we const...