Abstract
This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring A = k x 1 , … , x n . Besides, we compare graded linearity with componentwise linearity in general. For modules minimally generated by a regular sequence in a maximal ideal of A , we show that graded linear quotients imply graded linear resolution property for the colon ideals. On the other hand, we provide specific characterizations of graded linear resolution property for the Stanley–Reisner ring of broken circuit complexes and generalize the results of Van Le and Römer, related to the decomposition of matroids into the direct sum of uniform matroids. Specifically, we show that the matroid can be stratified such that each stratum has a decomposition into uniform matroids. We also present analogs of our results for the Orlik–Terao ideal of hyperplane arrangements which are translations of the corresponding results on matroids.
Highlights
We consider finitely generated graded modules over the polynomial ring, A k[x1, . . . , xn], where k is an infinite field
One can arrange the Betti numbers in a table by assigning βi,i+j in the (i, j) entry, called the Betti table. e graded A-module M is said to have a linear resolution if its nonzero Betti numbers in minimal A-resolution appear just in one line of entries, successively in the table
We show that graded linearity is a weaker property than componentwise linearity
Summary
We consider finitely generated graded modules over the polynomial ring, A k[x1, . . . , xn], where k is an infinite field. We define an extension of Koszulness, namely, graded Koszulness, and characterize this concept via the decomposition of the associated matroid Another important concept is the entire intersection property of Orlik–Solomon algebra and the Orlik–Terao ideal of hyperplane arrangements [3, 5–7]. E main theorem that is proven in that work is that of characterization for the decomposition of the broken circuit complex of a matroid when the associated Stanley–Reiner ring has a linear resolution. E characterization of this property for Orlik–Terao ideals of arrangement of hyperplanes can be considered as a special case of having general linear resolution mentioned above. The Orlik–Terao ideal has a 2-linear resolution [3] In this regard, the combinatorial identification of the Koszul property for algebras associated with matroids and hyperplane arrangements is one of the significant and crucial challenging problems in this area [5–7, 19, 20]
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