On the Lex-plus-powers conjecture. (English) Zbl 1408.13032
Let \(I \subset S = \Bbbk[x_1,\ldots,x_n]\) be a homogeneous ideal containing a regular sequence of degrees \(d_1 \leq \ldots \leq d_c\). The Eisenbud-Green-Harris conjecture (see [D. Eisenbud et al., Astérisque 218, 187–202 (1993; Zbl 0819.14001)] and [D. Eisenbud et al., Bull. Am. Math. Soc., New Ser. 33, No. 3, 295–324 (1996; Zbl 0871.14024)]) states that there exists a lex ideal \(L \subseteq S\) such that \(I\) has the same Hilbert function as \(L + (x_1^{d_1}, \ldots,x_c^{d_c})\). A positive answer has various consequences about the growth of Hilbert functions. In particular, if \(L + (x_1^{d_1}, \ldots,x_c^{d_c})\) exists it has the largest number of generators among all ideals with the same Hilbert function containing a complete intersection of degrees \(d_1 \leq \ldots \leq d_c\).
This leads to the Lex-plus-powers Conjecture (attributed to Charalambous and Evans (see [C. A. Francisco and B. P. Richert, Lect. Notes Pure Appl. Math. 254, 113–144 (2007; Zbl 1125.13008)]): If there exists a lex ideal \(L \subseteq S\) such that \(I\) has the same Hilbert function as \(L + (x_1^{d_1}, \ldots,x_c^{d_c})\), then \(\beta^S_{i,j}(I) \leq \beta^S_{i,j}(L + (x_1^{d_1}, \ldots,x_c^{d_c})\) for the corresponding graded Betti numbers \(\beta^S_{i,j}\). As a main result of the paper this is shown to be true if \(\text{char} \Bbbk = 0\) and the degrees satisfy \(d_i \geq \sum_{j=1}^{i-1} (d_j-1) +1\) for all \(i \geq 3\).
This leads to the Lex-plus-powers Conjecture (attributed to Charalambous and Evans (see [C. A. Francisco and B. P. Richert, Lect. Notes Pure Appl. Math. 254, 113–144 (2007; Zbl 1125.13008)]): If there exists a lex ideal \(L \subseteq S\) such that \(I\) has the same Hilbert function as \(L + (x_1^{d_1}, \ldots,x_c^{d_c})\), then \(\beta^S_{i,j}(I) \leq \beta^S_{i,j}(L + (x_1^{d_1}, \ldots,x_c^{d_c})\) for the corresponding graded Betti numbers \(\beta^S_{i,j}\). As a main result of the paper this is shown to be true if \(\text{char} \Bbbk = 0\) and the degrees satisfy \(d_i \geq \sum_{j=1}^{i-1} (d_j-1) +1\) for all \(i \geq 3\).
Reviewer: Peter Schenzel (Halle)
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13C40 | Linkage, complete intersections and determinantal ideals |
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 14M06 | Linkage |
| 14M10 | Complete intersections |
Keywords:
Eisenbud-Green-Harris conjecture; Hilbert function; Syzygies; Betti numbers; complete intersection; linkageSoftware:
LexIdealsReferences:
| [1] | Abedelfatah, A., On the eisenbud-Green-harris conjecture, Proc. Amer. Math. Soc., 143, 105-115, (2015) · Zbl 1317.13002 |
| [2] | Bigatti, A., Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra, 21, 2317-2334, (1993) · Zbl 0817.13007 |
| [3] | Caviglia, G.; Constantinescu, A.; Varbaro, M., On a conjecture by Kalai, Israel J. Math., 204, 469-475, (2014) · Zbl 1312.13022 |
| [4] | Caviglia, G.; Kummini, M., Poset embeddings of Hilbert functions, Math. Z., 274, 805-819, (2013) · Zbl 1273.13022 |
| [5] | Caviglia, G.; Kummini, M., Poset embeddings of Hilbert functions and Betti numbers, J. Algebra, 410, 244-257, (2014) · Zbl 1348.13023 |
| [6] | Caviglia, G.; Maclagan, D., Some cases of the eisenbud-Green-harris conjecture, Math. Res. Lett., 15, 427-433, (2008) · Zbl 1154.13001 |
| [7] | Caviglia, G.; Sbarra, E., The lex-plus-powers inequality for local cohomology modules, Math. Ann., 364, 225-241, (2016) · Zbl 1410.13009 |
| [8] | Chong, K. F.E., An application of liaison theory to the eisenbud-Green-harris conjecture, J. Algebra, 445, 221-231, (2016) · Zbl 1393.13033 |
| [9] | Clements, G. F.; Lindström, B., A generalization of a combinatorial theorem of Macaulay, J. Combin. Theory, 7, 230-238, (1969) · Zbl 0186.01704 |
| [10] | Cooper, S. M., Subsets of complete intersections and the EGH conjecture, (Progress in Commutative Algebra 1, (2012), de Gruyter Berlin), 167-198 · Zbl 1250.13021 |
| [11] | Eisenbud, D., Commutative algebra: with a view toward algebraic geometry, vol. 150, (1995), Springer Science & Business Media · Zbl 0819.13001 |
| [12] | Eisenbud, D.; Green, M.; Harris, J., Higher Castelnuovo theory, Astérisque, 218, 187-202, (1993) · Zbl 0819.14001 |
| [13] | Eisenbud, D.; Green, M.; Harris, J., Cayley-bacharach theorems and conjectures, Bull. Amer. Math. Soc., 33, 295-324, (1996) · Zbl 0871.14024 |
| [14] | Eliahou, S.; Kervaire, M., Minimal resolutions of some monomial ideals, J. Algebra, 129, 1-25, (1990) · Zbl 0701.13006 |
| [15] | Francisco, C., Almost complete intersections and the lex-plus-powers conjecture, J. Algebra, 276, 737-760, (2004) · Zbl 1102.13009 |
| [16] | Francisco, C.; Richert, B., Lex-plus-powers ideals, (Peeva, I., Syzygies and Hilbert Functions, Lect. Notes Pure Appl. Math., vol. 254, (2007)) · Zbl 1125.13008 |
| [17] | Gasharov, V., Hilbert functions and homogeneous generic forms II, Compos. Math., 116, 167-172, (1999) · Zbl 0946.13011 |
| [18] | Green, M. L., Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, (Algebraic Curves and Projective Geometry, Trento, 1988, Lecture Notes in Math., vol. 1389, (1989), Springer Berlin), 76-86 · Zbl 0717.14002 |
| [19] | Green, M. L., Generic initial ideals, (Six Lectures on Commutative Algebra, (1998), Birkhäuser Basel), 119-186 · Zbl 0933.13002 |
| [20] | Herzog, J.; Popescu, D., Hilbert functions and generic forms, Compos. Math., 113, 1-22, (1998) · Zbl 0908.13008 |
| [21] | Hulett, H. A., Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra, 21, 2335-2350, (1993) · Zbl 0817.13006 |
| [22] | Macaulay, F. S., Some properties of enumeration in the theory of modular systems, Proc. Lond. Math. Soc., 2, 531-555, (1927) · JFM 53.0104.01 |
| [23] | Mermin, J.; Murai, S., The lex-plus-powers conjecture holds for pure powers, Adv. Math., 226, 3511-3539, (2011) · Zbl 1213.13033 |
| [24] | Mermin, J.; Peeva, I.; Stillman, M., Ideals containing the squares of the variables, Adv. Math., 217, 2206-2230, (2008) · Zbl 1141.13016 |
| [25] | Migliore, J. C., Introduction to liaison theory and deficiency modules, vol. 165, (2012), Springer Science & Business Media |
| [26] | Murai, S., Borel-plus-powers monomial ideals, J. Pure Appl. Algebra, 212, 1321-1336, (2008) · Zbl 1138.13008 |
| [27] | Otwinowska, A., Sur la fonction de Hilbert des algèbres graduées de dimension 0, J. Reine Angew. Math., 545, 97-119, (2002) · Zbl 0988.13010 |
| [28] | Pardue, K., Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math., 40, 564-585, (1996) · Zbl 0903.13004 |
| [29] | Richert, B. P., A study of the lex plus powers conjecture, J. Pure Appl. Algebra, 186, 169-183, (2004) · Zbl 1052.13008 |
| [30] | Richert, B. P.; Sabourin, S., The residuals of lex plus powers ideals and the eisenbud-Green-harris conjecture, Illinois J. Math., 52, 1355-1384, (2008) · Zbl 1197.13020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
