Codimension and projective dimension up to symmetry. (English) Zbl 1509.13012
This article concerns the asymptotic behaviour of codimension and projective dimension of an ascending chain of ideals invariant under actions of symmetric groups.
For an integer \(c\in \mathbb{N}\) and for each \(n\in \mathbb{N}\), let \( R_n=K[x_{k,j}\ | \ 1\le k\le c,1\le j\le n] \) and let \(R= \bigcup_{n\ge1}R_n\). A sequence of ideals \((I_n)_{n\ge1}\) where \(I_n\subseteq R_n\) is \(\mathrm{Sym}\)-invariant if \(\mathrm{Sym}(n)(I_m)\subseteq I_n\quad\text{for all }\ m\le n, \) where \(\mathrm{Sym}(n)\) is the \(n\)-th symmetric group.
Consider the monoid of increasing functions on \(\mathbb{N}\): \(\mathrm{Inc }= \{\pi \colon\mathbb{N} \to \mathbb{N} \ |\ \pi(j)<\pi(j+1)\ \text{ for all }\ j\ge 1\},\) and submonoids \(\mathrm{Inc}^i\) fixing the initial segment of \(\mathbb{N}\) up to \(i\). Let moreover \(\mathrm{Inc}^i_{m,n}=\{\pi \in\mathrm{Inc}^i \ | \ \pi(m)\le n\}.\) A chain \(I_n\) of ideals in \(R_n\) is \(\mathrm{Inc}^i\)-invariant if \(\mathrm{Inc}^ii_{m,n}(I_m)\subseteq I_n\quad \text{for all }\ m\le n.\) (The aim of such notions is to have an action which has, for an invariant chain of ideals, also invariant chain of initial ideals, with a suitably chosen order).
The main result concerning codimension of ideals in \(\mathrm{Inc}^i\)-invariant chain \(J\) of monomial ideals requires more notions i.e. of \(\gamma_i(J)\) and \(\Gamma_i(J).\) Both notions are defined in the paper for (proper) monomial ideals, with an efficient way to compute \(\gamma_i(J)\). As the definitions are a bit lenghty, we refer the reader to the paper.
The main result concerning the codimension is now as follows. Let \(J=(I_n)_{n\ge 1}\) be an \(\mathrm{Inc}^i\)-invariant chain of monomial ideals. Then there exists an integer \(D(J)\) such that \(\mathrm{codim} I_n=\gamma_i(J)n + D(J) \) for \( n\gg 0\).
The main results concerning the projective dimension give the following two lower bounds:
Let \(J=(I_n)_{n\ge 1}\) be an \(\mathrm{Inc}^i\)-invariant chain of proper monomial ideals. Then there exists an integer \(\tilde{D}(J)\) such that \(\mathrm{pd}(I_n)\ge \Gamma_i(J)n+\tilde{D} (J) \) for \( n\gg 0.\)
Let \(J=(I_n)_{n\ge 1}\) be an \(\mathrm{Inc}^i\)-invariant chain of monomial ideals. For a certain \(r\in \mathbb{N}\) (depending on \((I_n)_{n\ge 1}\)) and for a suitably chosen set of monomials \( M\in\mathcal{M}\) and certain monomials \(\nu_M\), there exists an integer \(\tilde{D} (J)\) such that \(\mathrm{pd}(I_n)\geq \max\{\gamma_i(J:v_M)\ | \ M\in \mathcal{M}\}n+\tilde{D}(J) \)for \( n\gg0\), where for the precise definitions of \(r\),\(\mathcal{M}\) and \(\nu_M\) we again refer thr reader to the paper under review.
For an integer \(c\in \mathbb{N}\) and for each \(n\in \mathbb{N}\), let \( R_n=K[x_{k,j}\ | \ 1\le k\le c,1\le j\le n] \) and let \(R= \bigcup_{n\ge1}R_n\). A sequence of ideals \((I_n)_{n\ge1}\) where \(I_n\subseteq R_n\) is \(\mathrm{Sym}\)-invariant if \(\mathrm{Sym}(n)(I_m)\subseteq I_n\quad\text{for all }\ m\le n, \) where \(\mathrm{Sym}(n)\) is the \(n\)-th symmetric group.
Consider the monoid of increasing functions on \(\mathbb{N}\): \(\mathrm{Inc }= \{\pi \colon\mathbb{N} \to \mathbb{N} \ |\ \pi(j)<\pi(j+1)\ \text{ for all }\ j\ge 1\},\) and submonoids \(\mathrm{Inc}^i\) fixing the initial segment of \(\mathbb{N}\) up to \(i\). Let moreover \(\mathrm{Inc}^i_{m,n}=\{\pi \in\mathrm{Inc}^i \ | \ \pi(m)\le n\}.\) A chain \(I_n\) of ideals in \(R_n\) is \(\mathrm{Inc}^i\)-invariant if \(\mathrm{Inc}^ii_{m,n}(I_m)\subseteq I_n\quad \text{for all }\ m\le n.\) (The aim of such notions is to have an action which has, for an invariant chain of ideals, also invariant chain of initial ideals, with a suitably chosen order).
The main result concerning codimension of ideals in \(\mathrm{Inc}^i\)-invariant chain \(J\) of monomial ideals requires more notions i.e. of \(\gamma_i(J)\) and \(\Gamma_i(J).\) Both notions are defined in the paper for (proper) monomial ideals, with an efficient way to compute \(\gamma_i(J)\). As the definitions are a bit lenghty, we refer the reader to the paper.
The main result concerning the codimension is now as follows. Let \(J=(I_n)_{n\ge 1}\) be an \(\mathrm{Inc}^i\)-invariant chain of monomial ideals. Then there exists an integer \(D(J)\) such that \(\mathrm{codim} I_n=\gamma_i(J)n + D(J) \) for \( n\gg 0\).
The main results concerning the projective dimension give the following two lower bounds:
Let \(J=(I_n)_{n\ge 1}\) be an \(\mathrm{Inc}^i\)-invariant chain of proper monomial ideals. Then there exists an integer \(\tilde{D}(J)\) such that \(\mathrm{pd}(I_n)\ge \Gamma_i(J)n+\tilde{D} (J) \) for \( n\gg 0.\)
Let \(J=(I_n)_{n\ge 1}\) be an \(\mathrm{Inc}^i\)-invariant chain of monomial ideals. For a certain \(r\in \mathbb{N}\) (depending on \((I_n)_{n\ge 1}\)) and for a suitably chosen set of monomials \( M\in\mathcal{M}\) and certain monomials \(\nu_M\), there exists an integer \(\tilde{D} (J)\) such that \(\mathrm{pd}(I_n)\geq \max\{\gamma_i(J:v_M)\ | \ M\in \mathcal{M}\}n+\tilde{D}(J) \)for \( n\gg0\), where for the precise definitions of \(r\),\(\mathcal{M}\) and \(\nu_M\) we again refer thr reader to the paper under review.
Reviewer: Halszka Tutaj-Gasinska (Kraków)
MSC:
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |
| 16W22 | Actions of groups and semigroups; invariant theory (associative rings and algebras) |
References:
| [1] | M.Aschenbrenner and C. J.Hillar, Finite generation of symmetric ideals, Trans. Amer. Math. Soc.359 (2007), no. 11, 5171-5192. · Zbl 1129.13008 |
| [2] | A. E.Brouwer and J.Draisma, Equivariant Gröbner bases and the Gaussian two‐factor model, Math. Comp.80 (2011), no. 274, 1123-1133. · Zbl 1211.13018 |
| [3] | W.Bruns and A.Conca, Gröbner bases, initial ideals and initial algebras, Homological Methods in Commutative Algebra (L. L. Avramov et al., eds.), IPM Proceedings, Teheran, 2004. |
| [4] | W.Bruns and J.Herzog, Cohen-Macaulay rings. Revised edition, Cambridge Stud. Adv. Math., vol. 39, Cambridge University Press, 1998. · Zbl 0909.13005 |
| [5] | W.Bruns and U.Vetter, Determinantal rings, Lecture Notes in Math., vol. 1327, Springer‐Verlag, 1988. · Zbl 0673.13006 |
| [6] | G.Caviglia, H. T.Hà, J.Herzog, M.Kummini, N.Terai, and N. V.Trung, Depth and regularity modulo a principal ideal, J. Algebraic Combin.49 (2019), no. 1, 1-20. · Zbl 1448.13028 |
| [7] | T.Church, J. S.Ellenberg, and B.Farb, FI‐modules and stability for representations of symmetric groups, Duke Math. J.164 (2015), no. 9, 1833-1910. · Zbl 1339.55004 |
| [8] | D. E.Cohen, On the laws of a metabelian variety, J. Algebra5 (1967), 267-273. · Zbl 0157.34802 |
| [9] | S. D.Cutkosky, J.Herzog, and N. V.Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compos. Math.118 (1999), no. 3, 243-261. · Zbl 0974.13015 |
| [10] | J.Draisma, Finiteness of the k‐factor model and chirality varieties, Adv. Math.223 (2010), no. 1, 243-256. · Zbl 1211.13020 |
| [11] | J.Draisma, Noetherianity up to symmetry, Combinatorial Algebraic Geometry, Lecture Notes in Math., vol. 2108, Springer, 2014, pp. 33-61. · Zbl 1328.13002 |
| [12] | J.Draisma, R.Eggermont, R.Krone, and A.Leykin, Noetherianity for infinite‐dimensional toric varieties, Algebra Number Theory9 (2015), no. 8, 1857-1880. · Zbl 1354.13025 |
| [13] | J.Draisma and J.Kuttler, Bounded‐rank tensors are defined in bounded degree, Duke Math. J.163 (2014), no. 1, 35-63. · Zbl 1314.14109 |
| [14] | D.Grayson and M.Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/. |
| [15] | S.Güntürkün and U.Nagel, Equivariant Hilbert series of monomial orbits, Proc. Amer. Math. Soc.146 (2018), no. 6, 2381-2393. · Zbl 1395.13021 |
| [16] | C.Hillar and A.Martín del Campo, Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals, J. Symbolic Comput.50 (2013), 314-334. · Zbl 1263.13031 |
| [17] | C. J.Hillar and S.Sullivant, Finite Gröbner bases in infinite dimensional polynomial rings and applications, Adv. Math.229 (2012), no. 1, 1-25. · Zbl 1233.13012 |
| [18] | S.Hoşten and S.Sullivant, A finiteness theorem for Markov bases of hierarchical models, J. Combin. Theory Ser. A114 (2007), no. 2, 311-321. · Zbl 1111.62053 |
| [19] | V.Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc.128 (2000), no. 2, 407-411. · Zbl 0929.13004 |
| [20] | R.Krone, A.Leykin, and A.Snowden, Hilbert series of symmetric ideals in infinite polynomial rings via formal languages, J. Algebra485 (2017), 353-362. · Zbl 1401.13048 |
| [21] | D. V.Le, U.Nagel, H. D.Nguyen, and T.Römer, Castelnuovo-Mumford regularity up to symmetry, (preprint, 2018) available at arXiv:1806.00457. |
| [22] | U.Nagel and T.Römer, Equivariant Hilbert series in non‐Noetherian polynomial rings, J. Algebra486 (2017), 204-245. · Zbl 1372.13016 |
| [23] | U.Nagel and T.Römer, FI‐ and OI‐modules with varying coefficients, J. Algebra535 (2019), 286-322. · Zbl 1485.13033 |
| [24] | A.Putman and S. V.Sam, Representation stability and finite linear groups, Duke Math. J.166 (2017), no. 13, 2521-2598. · Zbl 1408.18003 |
| [25] | S. V.Sam and A.Snowden, Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc.30 (2017), no. 1, 159-203. · Zbl 1347.05010 |
| [26] | S. V.Sam and A.Snowden, Representations of categories of G‐maps, J. Reine Angew. Math.750 (2019), 197-226. · Zbl 1412.18005 |
| [27] | F.Santos and B.Sturmfels, Higher Lawrence configurations, J. Combin. Theory Ser. A103 (2003), no. 1, 151-164. · Zbl 1036.52018 |
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.
