Monomial ideals whose powers have a linear resolution

By Jürgen Herzog, Takayuki Hibi, Xinxian Zheng. 2003

181 citations recorded by the source corpus.

In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. Recall that a graded module M is said to have a linear resolution if all entries in the matrices representing the differentials in a graded minimal free resolution of M are linear forms. If an ideal I has a linear resolution, then necessarily all generators of I have the same degree, say t . In that case, one also says that I has a t-linear resolution. It is known [7] that polymatroidal ideals have linear resolutions and that powers of polymatroidal ideals are again polymatroidal (see [2] and [8]). In particular they have again linear resolutions. In general however, powers of ideals with linear resolution need not to have linear resolutions. The first example of such an ideal was given by Terai. He showed that over a base field of characteristic = 2 the Stanley Reisner ideal I = (abd, abf, ace, adc, aef, bde, bcf, bce, cdf, def ) of the minimal triangulation of the projective plane has a linear resolution, while I 2 has no linear resolution. The example depends on the characteristic of the base field. If the base field has characteristic 2, then I itself has no linear resolution. Another example, namely I = (def, cef, cdf, cde, bef, bcd, acf, ade) is given by Sturmfels [13]. Again I has a linear resolution, while I 2 has no linear resolution. The example of Sturmfels is interesting because of two reasons: 1. it does not depend on the characteristic of the base field, and 2. it is a linear quotient ideal. Recall that an equigenerated ideal I is said to have linear quotients if there exists an order f1, . . . , fm of the generators of I such that for all i = 1, . . . , m the colon ideals (f1, . . . , fi−1) : fi are generated by linear forms. It is quite easy to see that such an ideal has a linear resolution (independent on the characteristic of the base field). However the example of Sturmfels also shows that powers of a linear quotient ideal need not to be again

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/254f383e4a36ad58e1ccc104f12b4301b17554cb

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Citations in the corpus

The depth of powers of an ideal (2004, unknown): “A C ] 1 0 Se p”
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers (2006, background): “Subsequently, many people, including [1, 12, 13, 15, 16, 17, 18, 27, 29, 28, 30, 32], have been working on a program to build a dictionary between the algebraic properties of I(G) and the combinatorial structure of G.”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, background): “To learn more about the properties of edge ideals, one should see [1, 2, 6, 16, 17, 25, 26, 27, 36, 38, 39, 40, 41, 44, 46]; further properties of facet ideals can be found in the sequels [12, 13, 14] to Faridi’s paper cited above.”
Monomial and toric ideals associated to Ferrers graphs (2006, methodology): “By a result of Herzog-Hibi-Zheng [32], all the powers Ik λ have a linear resolution so that the Castelnuovo-Mumford regularity of Ik λ , for any integer k ≥ 1, is reg(I k λ) = 2k.”
Powers of edge ideals of regularity three bipartite graphs (2014, background): “Herzog, Hibi and Zheng showed in [5] that powers of edge ideals with linear resolution have linear resolution themselves.”
Edge ideals: algebraic and combinatorial properties (2010, background): “If I is the edge ideal of a graph, then I has linear quotients if and only if I has a linear resolution if and only if each power of I has a linear resolution [64].”
Regularity of powers of forests and cycles (2014, background): “This question is inspired by previous work of Herzog, Hibi and Zheng [24], of Nevo and Peeva [31], and by our main results, Theorems 4.”
Symbolic powers of cover ideal of very well-covered and bipartite graphs (2016, unknown): “However, Herzog, Hibi and Zheng [17] prove that a monomial ideal I generated in degree 2 has linear resolution if and only if every power of I has linear resolution.”
Powers of sums and their homological invariants (2016, background): “2. A catalog. We provide in this section more classes of ideals which are of small or doubly small type. For some classes of ideals satisfying the hypothesis of the next result, see, e.g., [8], [29], [30]. POWERS OF SUMS 13 Proposition 4.6. Assume that all the powers of Iare Koszul. Then: (i) Iis of”