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5c0e382c408670dbc9aa1109bf241aa1c43319ed | subsection | 28 | 124 | Mapping cones | Then the iterated mapping
cone F, derived from the sequence u_1,\ldots ,u_m is a minimal graded free resolution
of S/I, and for all i > 0 the symbolsf(\sigma ; u)\ \mbox{with}\ u\in G(I),\ \sigma \subset \operatorname{set}(u),\ |\sigma | = i - 1form a homogeneous basis of the S-module F_i. Moreover \deg (f(\sigma ; u))... | {
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439d6a5c3a65b615a2d409cd5e8e41070d8f14d4 | subsection | 29 | 124 | Mapping cones | Hence0\longrightarrow F_3\stackrel{\partial _3}{\longrightarrow }F_2\stackrel{\partial _2}{\longrightarrow }F_1\stackrel{\partial _1}{\longrightarrow }S\longrightarrow S/I\longrightarrow 0.It is easy to see that:F_1 has the basis \lbrace f(\emptyset ;u)\ :\ u\in G(I)\rbrace ,F_2 has the basis \lbrace f(1;x_2x_3x_4),\ f... | {
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b50a69402d97179f1fa64093a5555ae4541cd1ee | subsection | 30 | 124 | Basic notions | Let n>0 be an integer. We denote by [n] the set \lbrace 1,\ldots ,n\rbrace .Definition 1.7.1 A simplicial complex \Delta on [n] is a collection of subsets of [n] such that the following conditions hold:\lbrace i\rbrace \in \Delta for all i\in [n];
if F\in \Delta and G\subseteq F, then G\in \Delta .The set [n] is the v... | {
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646f47b29efaaa31c7b10fb208e7ac3ced0d9623 | subsection | 31 | 124 | Basic notions | Here x_F is the square-free monomialx_F=x_{i_1}\cdots x_{i_s},where F=\lbrace i_1,\ldots ,i_s\rbrace .For the simplicial complex considered in Figure REF , the Stanley–Reisner ideal isI_{\Delta }=(x_1x_4,\ x_2x_4).The factor ringk[\Delta ]=\frac{k[x_1,\ldots ,x_n]}{I_{\Delta }}is called the Stanley–Reisner ring of \Del... | {
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2080175bbf3fb718ace1498f6bbd272757543609 | subsection | 32 | 124 | Basic notions | Then\operatorname{lk}(\lbrace 1,3\rbrace ,\Delta )=\langle \lbrace 2\rbrace \rangle .Remark 1.7.11 Let \Delta be a simplicial complex. Then \operatorname{lk}(\emptyset ,\Delta )=\Delta .A particular class of simplicial complexes is the class of pure simplicial complexes. Recall that a simplicial complex is pure if all ... | {
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e48220a01f1af7690a229fa68efa9dfe9ac1fe37 | subsection | 33 | 124 | Classes of pure simplicial complexes | We recall some known classes of pure simplicial complexes. We give examples and counter-examples of such simplicial complexes and we establish a hierachy on them.The most larger and important class of pure simplicial complexes is the class of Cohen–Macaulay simplicial complexes. Recall that a simplicial complex \Delta ... | {
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d67a730cc6211c2c7e9a54e2648c298771fff15c | subsection | 34 | 124 | Classes of pure simplicial complexes | (ii)
If \Delta _1,\ \Delta _2 are d-dimensional constructible simplicial complexes and \Delta _1\cap \Delta _2 is a constructible simplicial complex of dimension d-1, then \Delta _1\cup \Delta _2 is a d-dimensional constructible simplicial complex.The following simplicial complex is given by M. Hachimori and it is a co... | {
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ba5b0d989ba49230ac6e82292026362a15edd5d1 | subsection | 35 | 124 | Classes of pure simplicial complexes | \Delta is called vertex-decomposable if \Delta =\lbrace \emptyset \rbrace or there exists a vertex v\in \Delta such that both \operatorname{del}(v,\Delta ) and \operatorname{lk}(v,\Delta ) are vertex-decomposable.Example 1.7.24 Let \Delta be the simplicial complex from Figure REF with the facets\mathcal {F}(\Delta )=\l... | {
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d95d2fd436c641e41f9f6b59582520e9fe35a52b | subsection | 36 | 124 | Classes of pure simplicial complexes | But this simplicial complex is not shifted. Indeed, if we consider a total order on the vertex set such that v_1<v_3, then looking to the face \lbrace v_3,v_4\rbrace and replacing v_3 by v_1 we obtain \lbrace v_1,v_4\rbrace which is not a face in \Delta . If we consider a total order on the vertices such that v_3<v_1, ... | {
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fc7c735c634174c22637faea17c810249404ec97 | subsection | 37 | 124 | Simplicial complexes and Alexander duality | Many properties of a simplicial complex can be obtained by studying the Alexander dual and its Stanley–Reisner ideal.Definition 1.7.29 Let \Delta be a simplicial complex on the vertex set [n]. The Alexander dual is the simplicial complex\Delta ^{\vee }=\lbrace F^c\ \mid \ F\notin \Delta \rbrace ,where we denote F^c=[n]... | {
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7a3a5f91a2a806f32be268b9c34fe11b8b4a386d | subsection | 38 | 124 | Simplicial complexes and Alexander duality | \Delta is shellable if and only if I_{\Delta ^{\vee }} has linear quotients.Therefore, if we consider the connection with the Stanley–Reisner ideal of the Alexander dual, for a pure simplicial complex \Delta we have the following diagram:\begin{array}{ccccc}
\Delta \ \mbox{is shellable}&\Longrightarrow &\Delta \ \mbox{... | {
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acd460e93e72e3ee8b10d1a1bc74de0134774935 | subsection | 39 | 124 | Coxeter groups | In this section we recall some basic notions related to Coxeter groups, following J.E. Humphreys . They will be needed in the last chapter of our thesis. See A. Björner and F. Brenti or J.E. Humphreys for more details.Definition 1.8.1 A Coxeter system is a pair (W,S) consisting of a group W and a set of generators S\su... | {
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38fc92d5aaba77f232e1bf9d68528d774d87e032 | subsection | 40 | 124 | Coxeter groups | \ell (w^{-1})=\ell (w).The "Exchange Property" is a fundamental combinatorial property of a Coxeter system.Theorem 1.8.5 (Exchange Property ) Suppose that w=\sigma _1\sigma _2\cdots \sigma _k is a reduced expression, \sigma _i\in S for i=1,\ldots ,k and \sigma \in S a simple reflection. If \ell (\sigma w)<\ell (w), the... | {
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4bbee7259741a5d2466527077a2654eb514712f2 | subsection | 41 | 124 | Subword complexes in Coxeter groups | Subword complexes were introduced by A. Knutson and E. Miller for the study of Schubert polynomials and combinatorics of determinantal ideals. Many properties of the subword complexes in Coxeter groups are obtained by using the Demazure algebra and the Demazure product.Let (W,S) be an arbitrary Coxeter system consistin... | {
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dac71558b83228aa865bcc951bb2c2a595df0015 | subsection | 42 | 124 | Subword complexes in Coxeter groups | The set of all the subwords of Q that represent \pi is\lbrace (\sigma _1,\sigma _2,\sigma _4,\sigma _6),\ (\sigma _1,\sigma _4,\sigma _6,\sigma _7),\ (\sigma _3,\sigma _4,\sigma _6,\sigma _7),\ (\sigma _4,\sigma _5,\sigma _6,\sigma _7)\rbrace .Therefore, the subword complex \Delta =\Delta (Q,\pi ) is the simplicial com... | {
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2c28afe95db151dd40fbf5a4a5149a4d57869d56 | subsection | 43 | 124 | Subword complexes in Coxeter groups | (b)
If \delta (P)=\pi , then every subword of P containing \pi has the Demazure product \pi .
(c)
If \delta (P)\succ \pi , then P contains a word T representing an element \tau \succ \pi satisfying |T|=\ell (\tau )=\ell (\pi )+1.Lemma 1.9.8 () Let T be a word in W and let \pi be an element in W such that |T|=\ell (\pi ... | {
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35de34e87e601f09f14f9a4a3864a9f5512a5121 | subsection | 44 | 124 | Subword complexes in Coxeter groups | (c)
If T represents \tau \succ \pi , then T\setminus \sigma represents \pi for exactly one \sigma \in T.The following theorem gives a complete description of the structure of the subword complexes in Coxeter groups.Theorem 1.9.9 () The subword complex \Delta (Q,\pi ) is a sphere if \delta (Q)=\pi and a ball otherwise.R... | {
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bc470bf8c4b24c659eff8e9e753525913a95db13 | subsection | 45 | 124 | Subword complexes in Coxeter groups | Provan ), subword complexes in Coxeter groups are shellable.In this chapter, we get the following facts which will be published in our paper :We prove directly that subword complexes in Coxeter groups are shellable, by using the Alexander duality.
As a consequence, we get a shelling on the facets of the subword comple... | {
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0.0073274876922369,
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0.0... | |
8b34aecf7496b2d67dd59aa5d0dd403c6541ecd3 | subsection | 46 | 124 | Completely lexsegment ideals with a linear resolution | Let S=k[x_1,\ldots , x_n] be the polynomial ring in n variables over a field k. We order lexicographically the monomials of S such that x_1>x_2>\ldots > x_n. Let d\ge 2 be an integer and \mathcal {M}_d be the set of all the monomials of S of degree d.Theorem 2.1.1
Let u=x_1^{a_1}\cdots x_n^{a_n}, with a_1>0, and v=x_1... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0... | |
d12521a0306c3fc4791feaeb8c47a735d0cfab16 | subsection | 47 | 124 | Completely lexsegment ideals with a linear resolution | Hence I has linear quotients with respect to \prec since it is obvious that the extension in the ring k[x_1,\ldots , x_n] of a monomial ideal with linear quotients in k[x_2,\ldots ,x_n] has linear quotients, too.Next we assume that u,v satisfy either the condition (b) or the condition (c).By definition, I has linear qu... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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f47f5808be2e2dd67426d3b8959b5f81ad517a62 | subsection | 48 | 124 | Completely lexsegment ideals with a linear resolution | This will imply that \bar{w}\ge _{lex}v, hence \bar{w}\in \mathcal {L}(u,v).The inequality \bar{w}\ge _{lex}w_j is obviously fulfilled if \alpha _1-1>\beta _1 or if \alpha _1-1=\beta _1 and at least one of the inequalities \alpha _s\ge \beta _s for 2\le s<l is strict. If \alpha _1-1=\beta _1 and \alpha _s=\beta _s for ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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... | |
55adf6350459c97082c9871c5158a9d1f494b0d7 | subsection | 49 | 124 | Completely lexsegment ideals with a linear resolution | If m\le _{lex}m^{\prime } then\frac{m}{x_{\max (m)}}\le _{lex}\frac{m^{\prime }}{x_{\max (m^{\prime })}}.Let m <_{lex}m^{\prime }. Then there exists s\ge 1 such that a_1=b_1,\ldots , a_{s-1}=b_{s-1} and a_s<b_s. It is clear that \max (m^{\prime })\ge s. Comparing the degrees of m and m^{\prime }, we get \max (m)>s.If \... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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... | |
86dfcc159bd666ec6d6583d7559dffffba621d18 | subsection | 50 | 124 | Completely lexsegment ideals with a linear resolution | On the other hand,\frac{x_1w^{\prime \prime }}{x_{\max (w^{\prime \prime })}}=\frac{x_1x_lw_i}{x_1x_{\max (w_i)}}=\frac{x_lw_i}{x_{\max (w_i)}}=w^{\prime }.Therefore, it results w^{\prime }\le _{lex}u, which contradicts our assumption on w^{\prime }.Consequently, we have w^{\prime }\le _{lex}u or w^{\prime \prime }\ge ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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... | |
27260f104edb00f81463e4d3222181d59d990a78 | subsection | 51 | 124 | Completely lexsegment ideals with a linear resolution | We label the monomials from G(I) as follows: u_1=x_2^3,\ u_2=x_2^2x_3,\ u_3=x_2x_3^2,\ u_4=x_1x_2x_3,\ u_5=x_1x_3^2, so u_1\prec u_2\prec \ldots \prec u_5. Then (u_1):u_2=(x_2),\ (u_1,u_2):u_3=(x_2),\ (u_1,u_2,u_3):u_4=(x_2,x_3), (u_1,u_2,u_3,u_4):u_5=(x_2).We further study the decomposition function of a completely le... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 456,
"openalex_id": "",
"raw": "J. Herzog, Y. Takayama, Resolutions by mapping cones, Homology, Homotopy and Applications, 4(2)(2002), 277–294",
"source_ref_id": "53152b6700a96cb6adc14b4f13c499fc18c4738c",
"start": 361... | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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789cc9f0ac15a9171a6179c8e70a28d26f57f242 | subsection | 52 | 124 | Completely lexsegment ideals with a linear resolution | \end{array}\right.Let u=x_1^{a_1}\cdots x_n^{a_n},\ v=x_1^{b_1}\cdots x_n^{b_n},\ a_1>0, and w=x_1^{\alpha _1}\cdots x_n^{\alpha _n}.In the first place we considerx_sw\ge _{lex}x_1v.Since, by Lemma REF , we have s\ge 2, the above inequality shows that \nu _1(w)\ge 1. We have to show that g(x_sw)=x_s w/x_1, that is\frac... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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9c6bd25f359756ce5526ea5023574cbc6e49b8a0 | subsection | 53 | 124 | Completely lexsegment ideals with a linear resolution | Since w\le _{lex} u, by using Lemma REF , we have\frac{x_sw}{x_{\max (w)}}\le _{lex}\frac{x_su}{x_{\max (u)}}=\frac{x_su}{x_2}\le _{lex} u,the last inequality being true by Lemma REF . Therefore, x_s w/x_{\max (w)}\in \mathcal {L}(u,v).If a_1=b_1+1 then the condition (c) in Theorem REF holds. Let z be the greatest mono... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0.0074... | |
744dd38fc5186677ba163ccd20d7351e79f843c2 | subsection | 54 | 124 | Completely lexsegment ideals with a linear resolution | It follows that\tilde{w}=\frac{x_sw}{x_t}\le _{lex}\frac{x_sw}{x_{\max (w)}}.If t=1 we have x_1\tilde{w}=x_sw <_{lex}x_1 v, which implies that \tilde{w}<_{lex}v, contradiction. Therefore t\ne 1 and, moreover, \tilde{w}\succeq x_sw/x_{\max (w)}, the inequality being true by the definition of the ordering \prec .
This yi... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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... | |
f8302184b36964c891512a32fde0b2fd8a42a81d | subsection | 55 | 124 | Completely lexsegment ideals with a linear resolution | The following inequalities hold:v<_{lex}\frac{x_1w^{\prime }}{x_s}<_{lex}w\le _{lex}u,the first one being true since v\le _{lex}w^{\prime }, so \nu _1(v)\le \nu _1(w^{\prime }). These inequalities show that x_1w^{\prime }/x_s\in \mathcal {L}(u,v). But we also have x_1w^{\prime }/x_s\prec w, hence x_1w^{\prime }/x_s\in ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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-... | |
b7f27fd5ba2d0ae02cba1eaa2eeae8eb2ec19b83 | subsection | 56 | 124 | Completely lexsegment ideals with a linear resolution | Hence, let\frac{x_1w^{\prime }}{x_s}>_{lex}u\ \mbox{and}\ \frac{x_jw^{\prime }}{x_s}<_{lex}v.We will show that this inequalities cannot hold simultaneously. Comparing the exponents of x_1 in the monomials involved in the above inequalities, we obtain \nu _1(w^{\prime })=b_1\ge a_1-1. Since, by hypothesis, x_sw>_{lex} x... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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15005cbad44ef6680525106e57d052d106419744 | subsection | 57 | 124 | Completely lexsegment ideals with a linear resolution | Since \nu _1(x_sw/x_{\max (w)})=\nu _1(w), the above inequality implies that\frac{x_sw}{x_{\max (w)}}>_{lex}w,that is x_s>_{lex}x_{\max (w)} which means that s<\max (w).As t\in \operatorname{set}(g(x_sw)), there exist w^{\prime }\prec \ x_sw/x_{\max (w)}, w^{\prime }\in \mathcal {L}(u,v), and a variable x_j, such that\... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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... | |
34a79db46d4bc7af383133533eb683092f95cf92 | subsection | 58 | 124 | Completely lexsegment ideals with a linear resolution | Since \nu _1(x_1w^{\prime }/x_s)=\nu _1(w), by using the definition of the ordering \prec we get x_1w^{\prime }/x_s\in I_{\prec w}. Then we may writex_tw=x_{\max (w)}\frac{x_1w^{\prime }}{x_s}\in I_{\prec w}.It remains to consider that
x_{\max (w)}w^{\prime }/x_s<_{lex}v\ \mbox{and}\ x_1w^{\prime }/x_s>_{lex}u.
Procee... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0.00... | |
4d236121801d1df5b14cb2400b25506f0863f6d2 | subsection | 59 | 124 | Completely lexsegment ideals with a linear resolution | Then the chain map of \mathbb {F}_{\bullet } is given by\partial (f(\sigma ; w)) = -\sum _{s\in \sigma }(-1)^{\alpha (\sigma ;s)}x_sf(\sigma \setminus s;w)+\sum _{\stackrel{s\in \sigma :}{x_sw\ge _{lex}}x_1v}(-1)^{\alpha (\sigma ;s)}x_1f\left(\sigma \setminus s;\frac{x_sw}{x_1}\right)++\sum _{\stackrel{s\in \sigma :}{x... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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dd5d6c9143f35dacc3300d36ef17caf8e45f6955 | subsection | 60 | 124 | Completely lexsegment ideals with a linear resolution | Let \mathbb {F}_{\bullet } be the minimal graded free resolution of S/I.Since \max \lbrace |\operatorname{set}(w)|\mid w\in \mathcal {L}(u,v)\rbrace =2, we have F_i=0, for all i\ge 4.A basis for the S-module F_1 is \lbrace f(\emptyset ;u_1),\ f(\emptyset ;u_2),\ f(\emptyset ;u_3),\ f(\emptyset ;u_4),\ f(\emptyset ;u_5)... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0.0... | |
27c0da7d1502c6cceab7d81541e5e68d705307a2 | subsection | 61 | 124 | Non-completely lexsegment ideals with a linear resolution | Theorem 2.2.1
Let u=x_1^{a_1}\cdots x_n^{a_n},\ v=x_2^{b_2}\cdots x_n^{b_n} be monomials of degree d in k[x_1,\ldots ,x_n],with a_1\ne 0. Suppose that the ideal I=({\mathcal {L}}(u,v)) is not a completely lexsegment ideal. Then I has a linear resolution if and only if I has linear quotients.We only have to proof that ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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91f57ca6b77f99aae2bed247811465b985e3f8c9 | subsection | 62 | 124 | Non-completely lexsegment ideals with a linear resolution | Since J
has linear quotients with respect to \prec , it follows that J_{\prec g}:g is generated by variables. Now it is enough
to show that, for any generator h of K, the colon ideal I_{\prec h}:h is generated by variables.
We note thatI_{\prec h}:h=J:h + K_{\prec h}:h.Since K is with linear quotients, we already know ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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c0a40eb6514cd13d13e694107d822566e108a497 | subsection | 63 | 124 | Non-completely lexsegment ideals with a linear resolution | By Theorem REF , I has a linear resolution and by the proof of Theorem REF , I has linear quotients with respect to the following ordering of its minimal monomial generators:x_2^3,\ x_2^2x_3,\ x_2^2x_4,\ x_2x_3^2,\ x_2x_3x_4,\ x_2x_4^2,\ x_1x_4^2,\ x_1x_3x_4.We note that \operatorname{set}(x_1x_4^2)=\lbrace 2\rbrace an... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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d346c922f56811ef37a53c73a5ca8309bc275eec | subsection | 64 | 124 | Cohen-Macaulay lexsegment ideals | In this section we study the dimension and the depth of arbitrary lexsegment ideals. These results are applied to describe the lexsegments ideals which are Cohen-Macaulay. We begin with the study of the dimension. As in the previous sections, let d\ge 2 be an integer. We denote {m}=(x_1,\ldots ,x_n). It is clear that i... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0.000022231308321352117... | |
c4301d4d82e2c70bf15584ab810b82206795d558 | subsection | 65 | 124 | Cohen-Macaulay lexsegment ideals | The lexsegment \mathcal {L}(u/x_1,x_n^{d-1}) contains the lexsegment \mathcal {L}(x_2^{d-1},x_n^{d-1}). Let p be a monomial prime ideal which contains I and such that x_1\notin p. Then p\supset \mathcal {L}(x_2^{d-1},x_n^{d-1}) which implies that (x_2,\ldots ,x_n)\subset p. Obviously we also have I\subset (x_2,\ldots ,... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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... | |
123a1413153913f5f286ba380aec51ae9b803dd6 | subsection | 66 | 124 | Cohen-Macaulay lexsegment ideals | Then l<n and, with similar arguments as above, we obtain \dim (S/I)=n-q.In order to study the depth of arbitrary lexsegment ideals, we note that one can restrict to those lexsegments defined by monomials of the form u=x_1^{a_1}\cdots x_n^{a_n},\ v=x_1^{b_1}\cdots x_n^{b_n} of degree d with a_1>0 and b_1=0.Indeed, if a_... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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26251401b879dc65cc778e01c1991f5ad03b5aa9 | subsection | 67 | 124 | Cohen-Macaulay lexsegment ideals | Therefore, u is of the form u=x_1x_l^{a_l}\cdots x_n^{a_n}, l\ge 2,\ a_l>0. Moreover, we have l\ge q.Let us suppose that x_1-x_n is not regular on S/I, that is there exists at least a polynomial f\notin I such that f(x_1-x_n)\in I. One may assume that all monomials of \operatorname{supp}(f) do not belong to I. Let us c... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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830ead68dec07996ebb3b440d4f553e79b8345c7 | subsection | 68 | 124 | Cohen-Macaulay lexsegment ideals | If \deg \delta \ge d there exists a variable x_j, with j\ge 2, such that (x_j\beta /x_n )| \delta , thus (x_j\beta /x_n) | w. It is obvious that x_2^d\ge _{lex} x_j\beta /x_n \ge _{lex} \beta \ge _{lex} v, hence x_j\beta /x_n is a minimal generator of I which divides w, contradiction. This implies that \delta has the d... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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11b73e008e20d6a13cbacc4bd8e12d1ae5976e85 | subsection | 69 | 124 | Cohen-Macaulay lexsegment ideals | Therefore\operatorname{depth}(S/I)=\operatorname{depth}(S^{\prime }/I^{\prime })+1,where S^{\prime }=k[x_2,\ldots ,x_n] and I^{\prime } is the ideal of S^{\prime } whose minimal monomial generating set is G(I^{\prime })=x_n\mathcal {L}(u\,^{\prime },x_n^{d-1})\cup \mathcal {L}^i(v).Lemma 2.3.6 In the above notations an... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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b29bb1cfd60c1306d70a97fe8ae38ad7f24260c6 | subsection | 70 | 124 | Cohen-Macaulay lexsegment ideals | Hence I^{\prime } is minimally generated by the following set of monomials\lbrace x_n\gamma \ |\ \gamma \ \mbox{is a monomial of degree}\ d-1\ \mbox{such that}\ x_n^{d-1}\le _{lex}\gamma \le _{lex}u\,^{\prime } \rbrace \cup\cup \lbrace x_2^d,\ x_2^{d-1}x_3,\ldots ,x_2^{d-1}x_j\rbrace .Then \lbrace x_{j+1},\ldots ,x_{l-... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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086d9db4913714d3a58c22e3cfd3f3d247c25f34 | subsection | 71 | 124 | Cohen-Macaulay lexsegment ideals | Indeed, x_2^d\nmid w and if we assume that there exists x_n^{d-1}\le _{lex}\gamma \le _{lex}u\,^{\prime }, \deg \gamma =d-1, such that x_n\gamma |w, we obtain x_n\gamma |x_n^{d-1} which is impossible.We show that x_jw\in I^{\prime } for all 2\le j\le n. Indeed, x_2w=x_2^dx_n^{d-1}\in I^{\prime }. Let 3\le j\le n. Then ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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06d902198010ca797526b02a6645bb26842a5fc2 | subsection | 72 | 124 | Cohen-Macaulay lexsegment ideals | Namely, we consider the monomial w^{\prime }=x_2x_n^{d-2} which does not belong to G(I^{\prime }) since it has degree d-1.If a_2\ge 2, for any j such that 2\le j\le n, we have
x_n^{d-1}<_{lex}x_jw^{\prime }/x_n=x_2x_jx_n^{d-3}<_{lex}x_l^{a_l}\cdots x_n^{a_n}=u\,^{\prime }.
This shows that x_jw^{\prime }\in I^{\prime }... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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71cea93b2c251fa740affa0386dcdf7b1cd1a155 | subsection | 73 | 124 | Cohen-Macaulay lexsegment ideals | The conclusion follows applying Lemma REF .By using the Auslander–Buchsbaum Theorem, we get the following corollary:Corollary 2.3.8 Let I=(\mathcal {L}(u,v)) be a lexsegment ideal defined by the monomials u=x_1x_l^{a_l}\cdots x_n^{a_n},\ v=x_q^{b_q}\cdots x_n^{b_n} where a_l,\ b_q>0, l,q\ge 2 and x_nu/x_1<_{lex} v. The... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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ea6f711f57706163bc3f6dd315213584be0ddd25 | subsection | 74 | 124 | Cohen-Macaulay lexsegment ideals | By Proposition REF , since \operatorname{depth}(S/I)>0, we must have x_nu/x_1<_{lex}v, thus we get (b).Finally, let \dim (S/I)\ge 2, that is q\le n-2. By using Proposition REF , we obtain q=2. Therefore \dim (S/I)=\operatorname{depth}(S/I)=n-2. Using again Proposition REF (a),(b), it follows that u=x_1x_n^{d-1} and v=x... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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37b12ecc3ad84a76df3e969583b88a5ef363a685 | subsection | 75 | 124 | Constructible ideals | We define a new class of monomial ideals, namely constructible ideals. We prove that a square-free constructible ideal is closely related to the notion of constructible simplicial complexes. Next we will determine some properties for constructible ideals and we give a formula for computing the Betti numbers. | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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88b29eed5dd4a3d1982a67c46ec2290e65014f41 | subsection | 76 | 124 | Constructible ideals and constructible simplicial complexes | Let S=k[x_1,\ldots ,x_n] be a polynomial ring in n variables over a field, k. For a monomial ideal I of S, we will denote by G(I) the minimal monomial set of generators of I.Definition 3.1.1 A monomial ideal I of S generated in degree q is a constructible ideal if it can be obtained by the following recursive procedure... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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74f11e8aedbef8f949b46ad5bbbc35b2af7adc25 | subsection | 77 | 124 | Constructible ideals and constructible simplicial complexes | I is an \mathbf {\underline{a}}–constructible ideal\ if it can be obtained by the following recursive procedure:(i)
If u\in \mathcal {M}_{\mathbf {\underline{a}}} and I=(u), then I is an \mathbf {\underline{a}}–constructible ideal;
(ii)
If I_1,\ I_2\in \mathcal {J}_{\mathbf {\underline{a}}} are \mathbf {\underline{a}}–... | {
"cite_spans": [
{
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"end": 2093,
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"raw": "M. Hachimori, Decomposition of two dimensional simplicial complexes, To appear in Discrete Mathematics.",
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"start": 1987
... | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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b8bdd2eb9bf8797952cc53a631c5deb5840473b8 | subsection | 78 | 124 | Constructible ideals and constructible simplicial complexes | On the other hand, the Stanley–Reisner ideal of the Alexander dual of the Dunce Hat is an example of ideal with a linear resolution which is not constructible (see also Example REF ).In the following lemma we will prove some properties of the simplicial complexes which are often used and which we will need in the proof... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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ad5d636b46d02ccab528e55d1b5218013bb71bfa | subsection | 79 | 124 | Constructible ideals and constructible simplicial complexes | In other words, x_{F^c} is in I_{\Delta _1^{\vee }} or x_{F^c} is in I_{\Delta _2^{\vee }}. But this is equivalent with the fact that F is in \Delta _1 or F is a face in \Delta _{2}, that is F is a face of \Delta _1\cup \Delta _2.(c) "\Leftarrow " F is a face in \Delta if and only if x_F\notin I_{\Delta }=I_{\Delta _1}... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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f3d4fc9e2607b9df751d7083315ba250af375075 | subsection | 80 | 124 | Constructible ideals and constructible simplicial complexes | We will prove in Proposition REF that every monomial ideal with linear quotients generated in one degree is a constructible ideal. Hence I_{\Delta ^{\vee }} is a \underline{\mathbf {1}}–constructible ideal.Assume that for all constructible simplicial complexes \Delta of dimension at most d-1, I_{\Delta ^{\vee }} is a \... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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842bee01d5c6508243a2b8ab3df3a90822b2c16e | subsection | 81 | 124 | Constructible ideals and constructible simplicial complexes | Hence I_{\Delta ^{\vee }}=I_{\Delta _1^{\vee }}+I_{\Delta _2^{\vee }} is a \mathbf {\underline{1}}–constructible ideal."(b)\Rightarrow (a)" We use descending induction on the degree of the monomials from the minimal system of generators of the monomial ideal I.If I=(x_1\cdots x_n), then I=I_{\Gamma } and \Gamma ^{\vee ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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c1511ca0e03b142f49e3d31ebfcb516bca7e827a | subsection | 82 | 124 | Constructible ideals and constructible simplicial complexes | We have to prove that \Gamma ^{\vee } is a constructible simplicial complex.By induction hypothesis, \Gamma _1^{\vee },\ \Gamma _2^{\vee } are constructible simplicial complex of dimension n-q-1.We have that I_1\cap I_2=I_{\Gamma _1}\cap I_{\Gamma _2}=I_{(\Gamma _1^{\vee }\cap \Gamma _2^{\vee })^{\vee }}. So \Gamma ^{\... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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01de94b2d7ac6dbbd7739b7bdf5ab24526e9157c | subsection | 83 | 124 | Properties of constructible ideals | From the above diagram, one may note that for square-free monomial ideals generated in one degree we have the following implications:\begin{array}{ccccc}
\end{array}\mbox{ square-free monomial}&&\mbox{square-free }&&\mbox{ square-free monomial}\\
\mbox{ideals with linear}&\Longrightarrow &\mbox{}&\Longrightarrow &\mbox... | {
"cite_spans": [
{
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"end": 768,
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"raw": "T. Römer, Generalized Alexander duality and applications, Osaka J. Math., 38(2001), 469–485.",
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"start": 674
},
{
... | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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dd2558a67716f5fb7fe05f4ec7947df4183d118d | subsection | 84 | 124 | Properties of constructible ideals | In the same way, \operatorname{Tor}_{i}^{R}(M^{\prime },k)_{i+j}=0 for all j\ne q+1.For j=q we obtain0\rightarrow \operatorname{Tor}_{i}^{R}(M,k)_{i+q}\rightarrow \operatorname{Tor}_{i}^{R}(M^{\prime \prime },k)_{i+q}\rightarrow \operatorname{Tor}_{i-1}^{R}(M^{\prime },k)_{i+q}\rightarrow 0.If j=q+1 we have \operatorna... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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93d5de171665a313204da4cc05818f219d4c5a38 | subsection | 85 | 124 | Properties of constructible ideals | From the exact sequence0\rightarrow I_1\rightarrow I_1\oplus I_2\rightarrow I_2\rightarrow 0we have, by Lemma REF , that I_1\oplus I_2 has a q–linear resolution and from the exact sequence:0\rightarrow I_1\cap I_2\rightarrow I_1\oplus I_2\rightarrow I_1+I_2\rightarrow 0,I_1+I_2 has a q–linear resolution, by Lemma REF .... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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] | 2,008 | en | Mathematics | [
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d1da04617505fcda718da0d24d07193c059328da | subsection | 86 | 124 | Polarization of constructible ideals | We prove that the property of constructibility is preserved in the polarization process.The polarization of a monomial ideal was introduced by R. Fröberg in . In the polarization process, homological properties of a monomial ideal are preserved. Since the polarization of a monomial ideal is a square-free monomial ideal... | {
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{
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"raw": "R. Fröberg, A study of graded extremal rings and of monomial rings, Math. Scand. 51(1982), 22–34.",
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"start": 88
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... | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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84fb401ed78f4edbc58a69270932e45f3b0a16cf | subsection | 87 | 124 | Polarization of constructible ideals | We use induction on the number of monomials from the minimal system of generators of the ideal I.If u\in \mathcal {M}_{\mathbf {\underline{a}}}, \deg (u)=q and I=(u), the statement is obvious.Let I\in \mathcal {J}_{\mathbf {\underline{a}}} be an \mathbf {\underline{a}}–constructible ideal generated in degree q with |G(... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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] | 2,008 | en | Mathematics | [
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94e2ba208ba3c0009fcc489db5e30d22efe5d001 | subsection | 88 | 124 | Ideals with linear quotients | In this section we describe the relation between monomial ideals with linear quotients and constructible ideals.Proposition 3.4.1
Let I be a monomial ideal of S with linear quotients generated in degree q. Then I is a constructible ideal.We prove by induction on the number of monomials in the minimal system of generat... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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67c94210b5208a5d5f91808b82051610c1b86ef5 | subsection | 89 | 124 | Ideals with linear quotients | Therefore|G(I_{k-1}\cap (u_k))|=|G(I_{k-1}:(u_k))|=d_k.Since I_k has linear quotients, the ideal I_{k-1}:(u_k) is generated by a regular sequence of length d_k, and then\beta _{i-1}(I_{k-1}:(u_k))=\left(\begin{array}{c}
d_k\\
i
\end{array}
\right).Summing in (*) for k=2,3,\ldots ,m, we get\beta _{i}(I)=\sum _{k=2}^{m}\... | {
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"raw": "A. Soleyman Jahan, Prime filtrations of monomial ideals and polarizations. to appear in J. Alg, Arxiv:math.AC/0605119.",
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"Anda Olteanu"
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e7387829ce863003d7194d53fcdd43d5df361ecb | subsection | 90 | 124 | Examples | In the sequel, we analise some examples. First two examples arise from the Stanley–Reisner ideal of suitable simplicial complexes.Example 3.5.1 The following example of constructible and non-shellable simplicial complex is due to M. | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
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"math.AC",
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c1a6103d2063f42f9239d6b95c30ea27efa32875 | subsection | 91 | 124 | Examples | Hachimori .
[Figure: NO_CAPTION]The simplicial complex is constructible because we can split it by the bold line and we obtain two shellable simplicial complexes \Delta _1,\ \Delta _2 of dimension 2 whose intersection is a shellable 1-dimensional simplicial complex.The shelling order of the facets for the simplicial co... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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823cf2b4f42b87c6ede5607766553a9f65019b57 | subsection | 92 | 124 | Examples | SinceI_{\Delta _1^{\vee }}\cap I_{\Delta _2^{\vee }}=I_{(\Delta _1\cap \Delta _2)^{\vee }}=(x_1x_2 x_3 x_5 x_6x_7 x_8 x_9,\ x_0x_1x_2 x_3 x_6 x_7x_8 x_9,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_0x_1x_2 x_3 x_4x_7 x_8 x_9,\ x_0x_1x_3 x_4 x_5x_7 x_8 x_9),the ideal I_{\Delta _1^{\vee }}\... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
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9a5840c5b180d04da4f809704eef171f49c6e679 | subsection | 93 | 124 | Examples | Hachimori ).13213212365487The Stanley–Reisner ideal for the Alexander dual of \Delta is:I_{\Delta ^{\vee }}=(x_3x_5x_6x_7x_8,\ x_3x_4x_5x_6x_8,\ x_3x_4x_5x_6x_7,\ x_2x_5x_6x_7x_8,\ \ \ \ \ \ \ \ \ \ \ x_2x_4x_6x_7x_8,\ x_2x_4x_5x_7x_8,\ x_2x_3x_4x_7x_8,\ x_2x_3x_4x_5x_6,\ \ \ \ \ \ \ \ \ \ \ x_1x_4x_6x_7x_8,\ x_1x_4x_5... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
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fed8723654b1a1966a93168f9ec90b082cdf4f3f | subsection | 94 | 124 | Examples | We argue this using the fact that constructibility is "preserved" during the polarization process.Example 3.5.3 Let I\in k[x_1,\ldots ,x_8] be the monomial idealI=(x_1x_2x_5x_6x_7x_8,\ x_2x_3x_5x_6x_7x_8,\ x_2^2x_3x_5x_6x_7,\ x_2^2x_3x_4x_6x_7,\ x_1x_2^2x_3x_6x_7,\ \ \ \ \ \ x_2x_3x_4x_5x_7x_8,\ x_2^2x_3x_4x_7x_8,\ x_1... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
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"math.AC",
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073b34faed7efde19a597b9f4125c3762896fbbd | subsection | 95 | 124 | Examples | Ziegler, with 10 vertices and 21 facets (see G.M. Ziegler ):\Delta _1=\langle \lbrace 1,2,3,4\rbrace ,\ \lbrace 1,2,4,9\rbrace ,\ \lbrace 1,4,8,9\rbrace ,\ \lbrace 1,5,8,9\rbrace ,\ \lbrace 1,4,5,8\rbrace ,\ \lbrace 1,2,6,9\rbrace ,\ \ \ \ \ \ \\ \ \ \ \ \ \ \ \lbrace 1,5,6,9\rbrace ,\ \lbrace 1,2,5,6\rbrace ,\ \lbrace... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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] | 2,008 | en | Mathematics | [
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90a822a00b1c4ade3c587d5e32be6af4ff794069 | subsection | 96 | 124 | Subword complexes in Coxeter groups and Alexander duality | Let (W,S) be a Coxeter system, Q=(\sigma _1,\ldots ,\sigma _n) be a word in W, with \sigma _i\in S for all 1\le i\le n, and \pi be an element in W. Let k[x_1,\ldots ,x_n] be the polynomial ring in n variables over a field k, where n is the size of the word Q and \Delta (Q,\pi ) be the subword complex.
We aim to determi... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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f4c61df06b1d9f3b2bffdba05ea8f87f5d992ee9 | subsection | 97 | 124 | Subword complexes in Coxeter groups and Alexander duality | Note that P_i\subseteq Q and P_i represents \pi for all 1\le i\le r.We have to prove that I_{\Delta ^{\vee }} has linear quotients with respect to the sequence of monomials \mathbf {x}_{P_1},\ldots ,\mathbf {x}_{P_r}, that is, for all i\ge 2 and for all j<i there exists an integer l\in [n] and an integer k, 1\le k<i, s... | {
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} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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cf33c8ef231456dbcd9a634c959cca505c2cf8d2 | subsection | 98 | 124 | Subword complexes in Coxeter groups and Alexander duality | Since P_i,\ P_j represent \pi , we have that\pi =\sigma _{i_1} \cdots \sigma _{i_{l-1}}\sigma _{i_l}\cdots \sigma _{i_{\ell (\pi )}}=\sigma _{j_1} \cdots \sigma _{j_{l-1}}\sigma _{j_l}\cdots \sigma _{j_{\ell (\pi )}}are reduced expressions for \pi and, by the choice of l, we obtain\sigma _{i_l}\cdots \sigma _{i_{\ell (... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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f967567a0a7f1b33861a5e6996b92c5589be14ee | subsection | 99 | 124 | Subword complexes in Coxeter groups and Alexander duality | Since both P_i and P^{\prime } represent \pi , we have that\pi =\sigma _{i_1}\cdots \sigma _{i_t}\cdots \sigma _{i_{l-1}}\sigma _{i_l}\cdots \sigma _{i_{\ell (\pi )}}=\sigma _{i_1}\cdots \sigma _{i_{t-1}}\sigma _{i_{t+1}}\cdots \sigma _{i_{l-1}}\sigma _{j_l}\sigma _{i_l}\cdots \sigma _{i_{\ell (\pi )}}are two reduced e... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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3e535a00ae02e5cc9b8f3280798233489a708305 | subsection | 100 | 124 | Subword complexes in Coxeter groups and Alexander duality | Thus, there exists a monomial \mathbf {x}_{P^{\prime }} and an integer j_l\in [n] such that \mathbf {x}_{P^{\prime }}/\gcd (\mathbf {x}_{P^{\prime }},\mathbf {x}_{P_i})=x_{j_l} and x_{j_l} divides \mathbf {x}_{P_{j}}/\gcd (\mathbf {x}_{P_i},\mathbf {x}_{P_j}) which ends our proof.Example 4.1.2 \ Let (S_4,S) be the Coxe... | {
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} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
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a40e473206a155fbfdc4da2267035f4aeed52fd3 | subsection | 101 | 124 | Subword complexes in Coxeter groups and Alexander duality | Since(\mathbf {x}_{P_1})\colon \mathbf {x}_{P_2}=(x_2),\ (\mathbf {x}_{P_1},\mathbf {x}_{P_2})\colon \mathbf {x}_{P_3}=(x_1)\ \mbox{and}\ (\mathbf {x}_{P_1},\mathbf {x}_{P_2},\mathbf {x}_{P_3})\colon \mathbf {x}_{P_4}=(x_1,x_3),I_{\Delta ^{\vee }} has linear quotients with respect to this order of the monomials from G(... | {
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{
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"raw": "A. Björner, M.L. Wachs, Shellable non-pure complexes and posets II, Trans. Amer. Math. Soc., 349(1997), 3945–3975.",
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"Anda Olteanu"
] | [
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9e02492a064121d26eddc2236fc7dec510d01f8a | subsection | 102 | 124 | Subword complexes in Coxeter groups and Alexander duality | We obtained that F_1=\lbrace \sigma _3,\sigma _5,\sigma _7,\sigma _8\rbrace ,\ F_2=\lbrace \sigma _2,\sigma _3,\sigma _5,\sigma _8\rbrace ,\ F_3=\lbrace \sigma _1,\sigma _2,\sigma _5,\sigma _8\rbrace ,\ F_4=\lbrace \sigma _1,\sigma _2,\sigma _3,\sigma _8\rbrace , is a shelling on the facets of \Delta (see Example REF )... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
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3111b5a89fd70e524a5920bcac614515988f9ed4 | subsection | 103 | 124 | Subword complexes in Coxeter groups and Alexander duality | Hence,\operatorname{lk}(\sigma _3,\Delta _1^{\prime })=\Delta (Q^{\prime \prime \prime },\pi )=\langle \lbrace \sigma _8\rbrace \rangleand\operatorname{del}(\sigma _3,\Delta _1^{\prime })=\Delta (Q^{\prime \prime \prime },\sigma _3\pi )=\Delta (Q^{\prime \prime \prime },s_2s_3s_2)=\langle \lbrace \sigma _5,\sigma _8\rb... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
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... | |
565e04a03a729b4a0a592627faa257ab5cc420cf | subsection | 104 | 124 | Subword complexes in Coxeter groups and Alexander duality | The set of all the reduced expressions of \pi is\lbrace s_1s_2s_3s_2,\ s_1s_3s_2s_3,\ s_3s_1s_2s_3\rbrace .We denoteQ=(\sigma _1,\ \sigma _2,\ \sigma _3,\ \sigma _4,\ \sigma _5,\ \sigma _6).The set of all the subwords of Q that represent \pi is\lbrace (\sigma _1,\sigma _2,\sigma _5,\sigma _6),\ (\sigma _1,\sigma _3,\si... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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7adf261c31c7eee4a066e0693cceed385954346f | subsection | 105 | 124 | Subword complexes in Coxeter groups and Alexander duality | We will need the following lemma.Lemma 4.1.9 Let I be a square-free monomial ideal with G(I)=\lbrace w_1,\ldots ,w_r\rbrace and w_1>_{lex}\ldots >_{lex}w_r such that I has linear quotients with respect to this order of the generators. Then\operatorname{set}(w_i)\subseteq [\max (w_i)]\setminus \operatorname{supp}(w_i),f... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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fc3d0f83244214a1e4dea5cb2da13c82c479d79f | subsection | 106 | 124 | Subword complexes in Coxeter groups and Alexander duality | For any 1\le i\le r, we have\operatorname{set}(\mathbf {x}_{P_i})=\lbrace \min (P_j\setminus P_i)\ \mid \ 1\le j<i\rbrace .We will use I instead of I_{\Delta ^{\vee }} to simplify the notation.Let s\in \operatorname{set}(\mathbf {x}_{P_i}). We have to prove that there exists a monomial \mathbf {x}_{P_j} such that s=\mi... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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162385fc68c51b776b333ec757bdfa9f40189ff3 | subsection | 107 | 124 | Subword complexes in Coxeter groups and Alexander duality | One may note that w_1>_{lex}w_2>_{lex}w_3 and I has linear quotients with respect to this order of generators. We have that \operatorname{set}(w_2)=\lbrace 1\rbrace and \operatorname{set}(w_3)=\lbrace 3\rbrace . If we denote F_i=\operatorname{supp}(w_i), 1\le i\le 3, we have that \min (F_1\setminus F_3)=\lbrace 1\rbrac... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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7f54646168cb3e5312ff14c22c591ca620cf9d21 | subsection | 108 | 124 | Subword complexes in Coxeter groups and Alexander duality | The Stanley–Reisner ideal of the Alexander dual of \Delta has the minimal monomial set of generatorsG(I_{\Delta ^{\vee }})=\lbrace x_1x_4x_6x_7,\ x_2x_4x_6x_7,\ x_3x_4x_6x_7,\ x_4x_5x_6x_7\rbrace .Then\mathbf {x}_{P_1}=x_1x_4x_6x_7>_{lex} \mathbf {x}_{P_2}=x_2x_4x_6x_7>_{lex} \mathbf {x}_{P_3}=x_3x_4x_6x_7>_{lex} \math... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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92925cf28c52268c7bf690f4426ed0bedfb45f1c | subsection | 109 | 124 | Subword complexes in Coxeter groups and Alexander duality | Let us assume by contradiction that\operatorname{proj\,dim}(I_{\Delta ^{\vee }})> n-\ell (\pi ).Hence, there exists 1\le k\le r such that\operatorname{proj\,dim}(I_{\Delta ^{\vee }})=d_k> n-\ell (\pi ).By Lemma REF , we have\operatorname{set}(\mathbf {x}_{P_k})\cap \operatorname{supp}(\mathbf {x}_{P_k})=\emptyset .Sinc... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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eff912d167239ba51e468c22ba85039accef8b88 | subsection | 110 | 124 | Subword complexes in Coxeter groups and Alexander duality | Assume that u,v>_{lex}w and \min (u/\gcd (u,w))\ne \min (v/\gcd (v,w)). Then u>_{lex} v if and only if \min (u/\gcd (u,w)) <\min (v/\gcd (v,w)).In the following, for a monomial m=x_1^{\alpha _1}\cdots x_n^{\alpha _n}, we denote by \nu _i(m) the exponent of the variable x_i in m, that is \nu _i(m)=\alpha _i, i=1,\ldots ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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ea6bf4ab003df386a898203de861af6da44bbbd5 | subsection | 111 | 124 | Subword complexes in Coxeter groups and Alexander duality | Similar, if l^{\prime \prime }<l^{\prime }, we have l>l^{\prime \prime } and we get \nu _{l^{\prime \prime }}(w)<\nu _{l^{\prime \prime }}(v)=\nu _{l^{\prime \prime }}(u)=\nu _{l^{\prime \prime }}(w), contradiction.Therefore, we must have l=\min (l^{\prime },l^{\prime \prime }). Let us assume that l^{\prime \prime }<l^... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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bb3d31c02a7b499d42299b75cc97ecff5e0f8a49 | subsection | 112 | 124 | Subword complexes in Coxeter groups and Alexander duality | This will imply that, for all 1\le k,s<j with k\ne s, \min (P_k\setminus P_j)\ne \min (P_s\setminus P_j) and hence d_j=j-1.Since k<j, by Lemma REF ,\min (P_k\setminus P_i)<\min (P_j\setminus P_i).On the other hand, since k<i, we have that \mathbf {x}_{P_k}>_{lex}\mathbf {x}_{P_i} that is for all t<\min (P_k\setminus P_... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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f310c9f66f0de3784f9125496632366297bfb330 | subsection | 113 | 124 | Subword complexes in Coxeter groups and Alexander duality | Let us assume by contradiction that there exist i_t,i_{t^{\prime }}\in \operatorname{supp}(\mathbf {x}_{P_i}), i_t\ne i_{t^{\prime }} such that\mathbf {x}_{P_j}=x_{\min (P_j\setminus P_i)}\frac{\mathbf {x}_{P_i}}{x_{i_t}},\ \mathbf {x}_{P_k}=x_{\min (P_k\setminus P_i)}\frac{\mathbf {x}_{P_i}}{x_{i_{t^{\prime }}}}.By th... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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4bb80f3daf0051616cf4bc024d96e8b126ecf9cb | subsection | 114 | 124 | Subword complexes in Coxeter groups and Alexander duality | Since \mathbf {x}_{P_j}\ne \mathbf {x}_{P_k} for all 1\le j,k\le i-1, j\ne k, we get that d_i=i-1. | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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78fafff669b10eb4f40fbfac6703fc00ff889562 | subsection | 115 | 124 | A special class of subword complexes | In this section we consider only subword complexes \Delta =\Delta (Q,\pi ) such that the minimal monomial generating system of I_{\Delta ^{\vee }} has r\le n-\ell (\pi )+1 elements, where n is the size of Q, and for which d_r=r-1. In the following proposition, we construct classes of such subword complexes.Proposition ... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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3d45b2da0e1ea8564b74b0ca9fa8503a85337838 | subsection | 116 | 124 | A special class of subword complexes | The set of all the reduced expressions of \pi is\lbrace s_2s_3s_2s_1s_2s_3,\ s_3s_2s_3s_1s_2s_3,\ s_3s_2s_1s_3s_2s_3,\ s_3s_2s_1s_2s_3s_2\rbrace .Let us denoteQ=(\sigma _1,\sigma _2,\sigma _3,\sigma _4,\sigma _5,\sigma _6,\sigma _7,\sigma _8,\sigma _9)All the subwords of Q that represent \pi are(\sigma _1,\sigma _2,\si... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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7351aba52a2928f40622a1b9cf0022190e0b965b | subsection | 117 | 124 | A special class of subword complexes | Then there exists a unique l\in \operatorname{supp}(\mathbf {x}_{P_r}) such thatI_{\Delta ^{\vee }}=\frac{\mathbf {x}_{P_r}}{x_l}(x_{\min (P_1\setminus P_r)},\ldots ,x_{\min (P_{r-1}\setminus P_r)},x_l).Since d_r=r-1, by Lemma REF we have that there exists a unique l such that \mathbf {x}_{P_j}=x_{\min (P_j\setminus P_... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0.0... | |
d79d0ebb5d34dc886e233da2c94a011b3563c6da | subsection | 118 | 124 | A special class of subword complexes | Then there exists a unique integer l\in [n] such that the Koszul complex associated to the sequencex_{\min (P_1\setminus P_r)},\ldots ,x_{\min (P_{r-1}\setminus P_r)},x_lis isomorphic to the minimal graded free resolution of k[\Delta ^{\vee }].By Lemma REF there exists a unique l\in [n] such thatI_{\Delta ^{\vee }}=\fr... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0d3bac64fb1d48096f817a52600f520aa065a1c3 | subsection | 119 | 124 | A special class of subword complexes | Then the Hilbert numerator of the Hilbert series is\mathcal {K}_{I_{\Delta ^{\vee }}}(t)=\sum _{i=0}^{r-1}(-1)^i\left({\genfrac{}{}{0.0pt}1{\scriptstyle {r}}{\scriptstyle {i+1}}} \right)t^{i+\ell (\pi )}.Since I_{\Delta ^{\vee }} has a \ell (\pi )–linear resolution, \beta _{ij}(I_{\Delta ^{\vee }})=0 for all j\ne i+\el... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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0.036... | |
13eeb9e79c1adb3ae999490429b3c079b821d151 | subsection | 120 | 124 | A special class of subword complexes | In \mathbb {Z}–grading, we have\mathcal {K}_{I_{\Delta ^\vee }}(t) = \sum _{\genfrac{}{}{0.0pt}1{\scriptstyle {P \subseteq Q}}{\scriptstyle {\delta (P)=\pi }}} (-1)^{|P|-\ell (\pi )} t^{|P|}=\sum _{\genfrac{}{}{0.0pt}1{\scriptstyle {P \subseteq Q}}{\scriptstyle {\delta (P)=\pi }}}\sum ^n_{\genfrac{}{}{0.0pt}1{\scriptst... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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621bf31e7f73a46f938faf30d382444f92011f7c | subsection | 121 | 124 | A special class of subword complexes | Hence m_n\ne 0 if and only if r=n-\ell (\pi )+1.Proposition 4.2.10 Let \Delta be the subword complex \Delta (Q,\pi ) and let n be the size of Q. Assume that G(I_{\Delta ^{\vee }})=\lbrace \mathbf {x}_{P_1},\ldots ,\mathbf {x}_{P_r}\rbrace with \mathbf {x}_{P_1}>_{lex}\ldots >_{lex}\mathbf {x}_{P_r}, r \le n-\ell (\pi )... | {
"cite_spans": []
} | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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83a931e114a47a24f2074f3af2358d44345630f4 | subsection | 122 | 124 | A special class of subword complexes | Then there exists a unique integer l such that the Koszul complex associated to the sequence x_{\min (P_1\setminus P_r)}\ldots x_{\min (P_{r-1}\setminus P_r)}x_l,\ x_i\ :i\in \operatorname{supp}(\mathbf {x}_{P_r}/x_l) is a minimal graded free resolution of k[{\Delta }].By Proposition REF , we have thatG(I_{\Delta })=\l... | {
"cite_spans": [
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"raw": "J.A. Eagon, V. Reiner, Resolutions of Stanley–Reisner rings and Alexander duality, J.Pure and Appl. Algebra, 130(1998), 265–275.",
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"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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a998e2c59d25b73c9f9fd9fab1fbc4838def7818 | subsection | 123 | 124 | Ideas for future | Alexander duality has become an important tool in the study of the square-free monomial ideals, due to the remarkable result of J.A. Eagon and V.Reiner and extensions by N. Terai , relating data of the resolution of the Stanley–Reisner ring of a simplicial complex to that of its Alexander dual. The Eagon–Reiner theorem... | {
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... | 0807.1675 | Classes of Monomial Ideals | [
"Anda Olteanu"
] | [
"math.AC",
"math.CO"
] | 2,008 | en | Mathematics | [
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-0.012202371843159199,
0.03162390738725662,
0.018452368676662445,
0.0... | |
bb6dd5c10906b21241f684898ca8b28bb4c55ae5 | abstract | 0 | 20 | Abstract | We consider a type of long-range percolation problem on the positive
integers, motivated by earlier work of others on the appearance of (in)finite
words within a site percolation model. The main issue is whether a given
infinite binary word appears within an iid Bernoulli sequence at locations that
satisfy certain cons... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.07190394401550293,
0.010414780117571354,
0.003723379224538803,
-0.00614967942237854,
-0.021729884669184685,
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-0.018708454445004463,
0.008606499060988426,
... | |
5ea200d65f5d6df2ff4addcbccef788d7de349f9 | subsection | 1 | 20 | Introduction | Let W=(w_1,w_2,\dots )\in \lbrace 0,1\rbrace ^\mathbb {N} be an infinite binary word,
and X = (X_1,X_2,\dots ) and Y=(Y_1, Y_2,\dots ) be independent Bernoulli sequences (i.e., iid sequences of Bernoulli random variables)
with parameters p_X=P(X_i=1),p_Y=P(Y_j=1) \in (0,1).
Let M be a positive integer. An admissible (M... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 865,
"openalex_id": "",
"raw": "Benjamini, I., Kesten, H., Percolation of arbitrary words in \\lbrace 0,1\\rbrace ^{\\mathbb {N}}, Ann. Probab. 23 (1995) 1024–1060.",
"source_ref_id": "e1ae41de79ddb706d11731778e15cd6143f7daa... | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.018626682460308075,
0.006384329870343208,
0.003731820033863187,
-0.001576059963554144,
0.018931787461042404,
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0.021265843883156776,
-0.055986836552619934,
-0.040548503398895264,
0.048420220613479614,
... | |
7f8769927d3071a64f091bfccb1545cd745f94a4 | subsection | 2 | 20 | Introduction | In Section 5, we will show that the truth of the statements
“for every M, P(X\text{ is $M$-seen\ in } Y)=0" and
“for every M, P(W\text{ is $M$-seen\ in } Y)=0"
(for an arbitrary infinite word W) is
independent of the parameters p_X,p_Y of the Bernoulli sequences
X,Y; see Theorem REF . Therefore, except in that section,... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.017452746629714966,
0.020793788135051727,
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0.004347930662333965,
-0.000955877301748842,
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0.021434536203742027,
-0.03524112328886986,
-0.05101571977138519,
0.0022826637141406536,
0.0... | |
a03c50908e8c1e67d7ea0c8e97d973acf666dc75 | subsection | 3 | 20 | Introduction | (An example illustrating this is given
in Section 3.)
However, they do so for the constant and alternating words.Theorem 3 (a) The constant word of length n is M-seen in Y if and only if \tau _k\le M
for all 1\le k\le n.(b) The alternating word A_n of length n is M-seen in Y
if and only ifT_k\le kM\text{ for all }1\le ... | {
"cite_spans": []
} | 0807.1676 | Percolation of arbitrary words in one dimension | [
"Geoffrey R. Grimmett",
"Thomas M. Liggett",
"Thomas Richthammer"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
-0.003206247230991721,
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0.04165641963481903,
-0.022659260779619217,
-0.02702326700091362,
-0.009239180013537407,
... |
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