×

zbMATH — the first resource for mathematics

Square-free Gröbner degenerations. (English) Zbl 07233317
Let \(K\) denote a field and let \(S = K[x_1,\ldots,x_n]\) denote the polynomial ring equipped with the standard grading. For a finitely generated graded \(S\)-module \(M\) let \(h^{ij}(M)\) denote the \(K\)-dimension of the degree \(j\) component of the \(i\)-th local cohomology module \(H^i_{\mathfrak{m}}(M)\) supported in the maximal ideal \(\mathfrak{m} = (x_1,\ldots,x_n)\). Let \(I \subset S\) be a homogeneous ideal with \(J\) an initial ideal with respect to a term order. Jürgen Herzog conjectured that the extremal Betti numbers of \(S/I\) and \(S/J\) coincide provided \(J\) is square-free. In their main result the authors prove that \(h^{ij}(S/I) = h^{ij}(S/J)\) provided \(J\) is square-free. Among others this solves Herzog’s Conjecture in the affirmative and has various interesting further consequences. Moreover, they discuss properties of ideals admitting square-free initial ideals and consequences of the main result about \(\operatorname{depth},\operatorname{reg}\) and the cohomological dimension of \(S/I\) and \(S/J\) resp. It is shown that ideals defining ASLs, Cartwright-Sturm ideals and Knutson ideals are families with square-free initial ideals. Finally further developments and questions are discussed. The main technical tool for the authors’ proof is the notion of cohomological full singularity as introduced by A. De Stefani [“Cohomologically full rings”, Preprint, arXiv:1806.00536].

MSC:
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D45 Local cohomology and commutative rings
Software:
CoCoA; Macaulay2
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abbott, J., Bigatti, A.M., Robbiano, L.: CoCoA: a system for doing computations in commutative algebra. available at http://cocoa.dima.unige.it
[2] Adiprasito, K.; Benedetti, B., The Hirsch conjecture holds for normal flag complexes, Math. Oper. Res., 39, 1340-1348 (2014) · Zbl 1319.52017
[3] Baclawski, K., Rings with lexicographic straightening law, Adv. Math., 39, 185-213 (1981) · Zbl 0466.13004
[4] Bayer, D.; Charalambous, H.; Popescu, S., Extremal Betti numbers and applications to monomial ideals, J. Algebra, 221, 497-512 (1999) · Zbl 0946.13008
[5] Bayer, D.; Stillman, M., A criterion for detecting \(m\)-regularity, Invent. Math., 87, 1-12 (1987) · Zbl 0625.13003
[6] Benedetti, B.; Varbaro, M., On the dual graph of Cohen-Macaulay algebras, Int. Math. Res. Not., 17, 8085-8115 (2015) · Zbl 1342.13015
[7] Brion, M., Multiplicity-free subvarieties of flag varieties, Contemp. Math., 331, 13-23 (2003) · Zbl 1052.14055
[8] Bruns, W.; Herzog, J., Cohen-Macaulay Rings (1993), Cambridge: Cambridge University Press, Cambridge
[9] Bruns, W.; Schwänzl, R., The number of equations defining a determinantal variety, Bull. Lond. Math. Soc., 22, 439-445 (1990) · Zbl 0725.14039
[10] Bruns, W.; Vetter, U., Determinantal Rings (1988), Berlin: Springer, Berlin
[11] Caviglia, G.; Constantinescu, A.; Varbaro, M., On a conjecture by Kalai, Isr. J. Math., 204, 469-475 (2014) · Zbl 1312.13022
[12] Cartwright, D.; Sturmfels, B., The Hilbert scheme of the diagonal in a product of projective spaces, Int. Math. Res. Not., 9, 1741-1771 (2010) · Zbl 1213.14010
[13] Chardin, M., Some results and questions on Castelnuovo-Mumford regularity, Syzygies and Hilbert functions, Lect. Not. Pure Appl. Math., 254, 1-40 (2007) · Zbl 1127.13014
[14] Conca, A., Linear spaces, transversal polymatroids and ASL domains, J. Alg. Combin., 25, 25-41 (2007) · Zbl 1108.13022
[15] Conca, A.; De Negri, E.; Gorla, E., Universal Gröbner bases for maximal minors, Int. Math. Res. Not., 11, 3245-3262 (2015) · Zbl 1325.13028
[16] Conca, A.; De Negri, E.; Gorla, E.; Conca, A.; Gubeladze, J.; Römer, T., Multigraded generic initial ideals of determinantal ideals, Homological and Computational Methods in Commutative Algebra, 81-96 (2017), Heidelberg: Springer, Heidelberg · Zbl 1406.13009
[17] Conca, A., De Negri, E., Gorla, E.: Universal Gröbner bases and Cartwright-Sturmfels ideals, to appear in Int. Math. Res. Not. (2016). arXiv:1608.08942 · Zbl 1439.13075
[18] Conca, A.; De Negri, E.; Gorla, E., Cartwright-Sturmfels ideals associated to graphs and linear spaces, J. Comb. Algebra, 2, 231-257 (2018) · Zbl 1400.13016
[19] Constantinescu, A., De Negri, E., Varbaro, M.: Singularities and radical initial ideals (2019). arXiv:1906.03192
[20] Dao, H., De Stefani, A., Ma, L.: Cohomologically full rings, To appear in IMRN (2018). arXiv:1806.00536
[21] De Concini, D.; Eisenbud, D.; Procesi, C., Hodge algebras, Astérisque, 91, 510 (1982)
[22] Di Marca, M.; Varbaro, M., On the diameter of an ideal, J. Algebra, 511, 471-485 (2018) · Zbl 1400.13009
[23] Eisenbud, D., Introduction to algebras with straightening laws, Lect. Not. Pure Appl. Math., 55, 243-268 (1980)
[24] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry (1994), Berlin: Springer, Berlin
[25] Eisenbud, D.; Green, M.; Harris, J., Higher castelnuovo theory, Astérisque, 218, 187-202 (1993) · Zbl 0819.14001
[26] Herzog, J., Sbarra, E.: Sequentially Cohen-Macaulay modules and local cohomology Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res., pp. 327-340 (2002) · Zbl 1072.13504
[27] Herzog, J., Rinaldo, G.: On the extremal Betti numbers of binomial edge ideals of block graphs (2018). arXiv:1802.06020 · Zbl 1395.13010
[28] Holmes, B.: On the diameter of dual graphs of Stanley-Reisner rings with Serre \((S_2)\) property and Hirsch type bounds on abstractions of polytopes. Electron. J. Combin. 25 (2018) · Zbl 1391.13022
[29] Hochster, M.; Roberts, JL, The purity of the Frobenius and local cohomology, Adv. Math., 21, 117-172 (1976) · Zbl 0348.13007
[30] Knutson, A.: Frobenius splitting, point-counting, and degeneration (2009). arXiv:0911.4941
[31] Kollár, J., Kovács, S.J.: Deformations of log canonical singularities (2018). arXiv:1803.03325
[32] Kollár, J., Kovács, S.J.: Deformations of log canonical and F-pure singularities (2018). arXiv:1807.07417
[33] Lyubeznik, G., On the Local Cohomology Modules \(H_a^i(R)\) for Ideals \(a\) Generated by Monomials in an \(R\)-Sequence (1983), Berlin: Springer, Berlin
[34] Ma, L.; Schwede, K.; Shimomoto, K., Local cohomology of Du Bois singularities and applications to families, Compos. Math., 153, 2147-2170 (2017) · Zbl 1387.14064
[35] Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
[36] Miyazaki, M., On the discrete counterparts of algebras with straightening laws, J. Commut. Algebra, 2, 79-89 (2010) · Zbl 1237.13041
[37] Peskine, C.; Szpiro, L., Dimension projective finie et cohomologie locale, IHES Publ. Math., 42, 47-119 (1973) · Zbl 0268.13008
[38] Schenzel, P., Zur lokalen Kohomologie des kanonischen Moduls, Math. Z., 165, 223-230 (1979) · Zbl 0377.13004
[39] Schenzel, P., Applications of Dualiziang Complexes to Buchsbaum Rings, Adv. Math., 44, 61-77 (1982) · Zbl 0492.13011
[40] Schwede, K., \(F\)-injective singularities are Du Bois, Am. J. Math., 131, 445-473 (2009) · Zbl 1164.14001
[41] Singh, AK, \(F\)-regularity does not deform, Am. J. Math., 121, 919-929 (1999) · Zbl 0946.13002
[42] Sturmfels, B., Gröbner bases and Stanley decompositions of determinantal rings, Math. Z., 205, 137-144 (1990) · Zbl 0685.13005
[43] Sturmfels, B., Gröbner Bases and Convex Polytopes (1995), Providence: American Mathematical Society, Providence
[44] Stacks Project Authors, Stacks project. Available at http://stacks.math.columbia.edu
[45] Varbaro, M., Gröbner deformations, connectedness and cohomological dimension, J. Algebra, 322, 2492-2507 (2009) · Zbl 1200.13029
[46] Varbaro, M.: Connectivity of hyperplane sections of domains. Commun. Algebra 47, 2540-2547 (2019). arXiv:1802.09445 · Zbl 1439.13063
[47] Yanagawa, K., Dualizing complex of the face ring of a simplicial poset, J. Pure Appl. Algebra, 215, 2231-2241 (2011) · Zbl 1219.13014
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.