Shifting operations and completely t–spread lexsegment ideals

Linear resolutions of t-spread lexsegment ideals via Betti splittings

“…In general, proving that a graded ideal of a standard graded polynomial ring has linear resolution can be difficult. However, some classes of graded ideals with linear resolution may be found in [2,3,8,9,11,14,21,24,25] and the reference therein. Let S = K[x 1 , .…”

“…In [23], a general lexsegment ideal was introduced. Such a definition has been recently generalized in [14] by the notion of an arbitrary t-spread lexsegment ideal. Let t ≥ 1 and let M n,d,t be the set of all t-spread monomials of S of degree d. Assume that M n,d,t is endowed with the squarefree lexicographic order ≥ slex .…”

“…The classification is achieved by analyzing the behaviour of the incompletely t-spread lexsegment ideals. Such a study began in [14] using classical techniques and all completely t-spread lexsegment ideals with a linear resolution were characterized. The case t = 0 were considered in [3,11], whereas the case t = 1 was analyzed in [4].…”

“…We state a formula that gives the Betti numbers of an incompletely t-spread lexsegment ideal I of S with a linear resolution (Theorem 3.1). The case when I is a completely t-spread lexsegment ideal was faced and solved in [14]. Furthermore, we characterize all incompletely t-spread lexsegment ideals with linear quotients, for t ≥ 1.…”

Vector-spread monomial ideals and Eliahou-Kervaire type resolutions

“…The t-spread monomial ideals have been introduced in 2019 by Ene, Herzog and Qureshi [18]. The homological and combinatorial properties of these and some related classes of ideals are the subject of a large body of research [1,2,3,4,5,6,7,8,13,14,15,16,18,20,21,24,28,29,30].…”