Minisymposium: Algebraic models arising from phenomenals in statistic, transportation problems, manufacturing and other ¯elds Gotzmann ideals and applications to graphs Vittoria BONANZINGA University of Reggio Calabria-DIMET Faculty of Engineering, via Graziella (Feo di Vito), 89100 Reggio Calabria (ITALIA). bonanzin@ing.unirc.it, vittoria.bonanzinga@tin.it Loredana SORRENTI University of Messina Department of Mathematics, C.da Papardo, Salita Sperone 31, 98166 Messina (ITALIA). sorrenti@dipmat.unime.it, sorrenti.loredana@tiscali.it 1 Introduction Let k be an in¯nite ¯eld and S = k[x1; : : : ; xn] the polynomial ring over k with each deg xi = 1 and m = (x1; : : : ; xn) the graded maximal ideal of S: If I is a graded ideal of S, then we write Ihki for the ideal generated by all homogeneous polynomials of degree k belonging to I: Given a graded ideal I ½ S; there exists a unique lexsegment ideal Ilex such that I and Ilex have the same Hilbert function. A graded ideal I generated in degree d is called a Gotzmann ideal if the number of generators of mI is the smallest possible, namely, equal to the number of generators of (mI)lex: By Gotzmann's persistence theorem [9] a graded ideal I generated in degree d is a Gotzmann ideal if, and only if, I and (Ilex)hdi have the same Hilbert function. Thus, in particular, a Gotzmann ideal has a linear resolution with the same Betti numbers as the corresponding lexsegment ideal. Knowing which mono- mial ideals are Gotzmann is still an open problem. The general classi¯cation of Gotzmann ideals is unknown at the moment. By de¯nition, all lexsegment ideals are Gotzmann. In the theory of monomial ideals, there is the following hierarchy of ideals: lexsegment monomial ideals ) strongly stable monomial ideals ) stable monomial ideals. When the base ¯eld k is of characteristic 0, the Borel-¯xed ideals coincide with the strongly stable ideals. Strongly stable ideals play a special role in the study of Hilbert functions. The reason is that one can often apply GrÄobner basis techniques to reduce the general case to the study of Hilbert functions of strongly stable ideals (see e.g.[1], [2], [4], [3], [7], [8], [11], [12], [15]). Among the Borel ideals the principal ones are the most simple. Let u be a monomial of S, then hui denotes the smallest Borel ideal which contains u. The ideal hui is called principal Borel with Borel generator u. In [5] all principal Borel ideals with Borel generators up to degree 4 which are Gotzmann are characterized. From a combinatorial point of view, it is particularly interesting to consider ideals generated by squarefree monomials, since 1 they appear as the de¯ning ideals of Stanley-Reisner rings. Let G = ([n];E(G)) be a ¯nite graph on the vertex set [n] = f1; 2; : : : ; ng with the edge set E(G) without loops, multiple edges and isolated vertices. Write I(G) for the ideal generated by all squarefree monomials xixj with fi; jg 2 E(G). A graph G is called Gotzmann if the edge ideal I(G) is Gotzmann. The graph G is called Cohen-Macaulay over a ¯eld K, if K[x1; : : : ; xn]=I(G) is a Cohen-Macaulay ring, and is called Cohen-Macaulay if it is Cohen-Macaulay over any ¯eld. We will give some properties of Gotzmann graphs when the edge ideal is a principal Borel ideal. We also characterize all Cohen- Macaulay graphs whose edge ideal is principal Borel and Gotzmann. We will show some classes of graphs which are Gotzmann. 2 Main results We will characterize all Gotzmann graphs whose edge ideal is a principal Borel ideal. Proposition 2.1 Let I(G) = hui be a principal Borel ideal. The following condi- tions are equivalent: (a) I(G) is a Gotzmann ideal. (b) I(G) is a lexsegment ideal. If the equivalent conditions hold, then u = x1xi2 or u = xi1xn: Remark 2.2 In general (b) implies (a), but not the converse. In fact if I(G) is not a principal Borel ideal the following example shows that I(G) = (x1x2; x1x4; x2x3; x3x4) ½ k[x1; x2; x3; x4] is Gotzmann, but it is not lexsegment. We need some preliminaries. Let G be a graph and (fr1; r2g; : : : ; frs; rs+1g) a cycle in G, where rs+1 = r1: An edge fi; jg 2 E(G) is called a chord of the cycle (fr1; r2g; : : : ; frs; rs+1g) if i; j 2 fr1; r2; : : : ; rsg and fi; jg 6= fri; ri+1g for all i. We call G chordal if every cycle in G of lenght > 3 has a chord. A stable subset or clique of G is a subset F of [n] such that fi; jg 2 E(G) for all i; j 2 F with i 6= j: We write ¢(G) for the simplicial complex on [n] whose faces are the stable subset of G. A graph G is called shifted if, for any edge fi; jg 2 E(G) and for any integers i0 · i and j0 · j one has fi0; j0g 2 E(G). The shifting operation of graphs is an operation which associates a shifted graph ¢(G) with a graph G . Shifting operations are ¯rst considered by ErdÄos, Ko and Rado [6]. Their shifting operation G ¡! ¢c(G) is called combinatorial shifting. Instead, algebraic shifting was introduced by Kalai, [13]. The main variants of algebraic shifting are exterior algebraic shifting G ¡! ¢e(G) and symmetric algebraic shifting G ¡! ¢s(G), see Kalai [14] and [10]. Shifting operations are de¯ned for simplicial complexes, however, we here only consider ¯nite graphs. Let R = K[x1; : : : ; xn] the polynomial ring in n variables over a ¯eld K with each deg xi = 1. The graded Betti numbers 2 ¯ij(I) of a homogeneos ideal I ½ R are the integers ¯ij(I) = dimK(Tori(I;K)j): In other words, ¯ij(I) appear in the minimal graded free resolution 0 ¡! M j A(¡j)¯hj ¡! : : : ¡! M j A(¡j)¯1j ¡! M j A(¡j)¯0j ¡! I ¡! 0 (1) of I over R. Corollary 2.3 If I(G) = hui is Gotmann then (a) G is a chordal graph and ¢e(G) = ¢s(G) (b) ¯ii+2(I(G)) = ¯ii+2(I(¢e(G))) = ¯ii+2(I(¢s(G))) for all i ¸ 0. We will characterize all Cohen-Macaulay graphs whose edge ideal is principal Borel and Gotzmann. Theorem 2.4 Let K be a ¯eld, let G be a graph on the vertex set [n] and let I(G) = hui be a principal Borel ideal which is Gotzmann. Let F1; : : : ; Fm be the facets of ¢(G) which admit a free vertex. The following conditions are equivalent: (a) G is Cohen-Macaulay; (b) G is Cohen-Macaulay over K; (c) G is unmixed; (d) [n] is the disjoint union of F1; : : : ; Fm. The complement G of a graph G has vertex set V (G) and two vertices are adjacent in G if and only if they are not adjacent in G: Let G¤ be the graph obtained by the complement G by removing the set fx1; : : : ; xrg of all its isolated vertexes: G¤ = G n fx1; : : : ; xrg: We will show some classes of graphs which are Gotzmann. Theorem 2.5 Let R = k[x1; : : : ; xn]. Let G be a connected graph on the vertex set V (G) = fx1; : : : ; xng and I(G) ½ R its edge ideal. Then G is Gotzmann in the following cases: 1. G is complete; 2. G¤ is complete; 3. jE(G¤)j · 2; 4. jE(G)j · 2; 3 5. G = Ka;b is a bipartite complete graph with a · 2 and b · 3: Remark 2.6 We can remark that the bipartite complete graph K3;3 is not Gotz- mann and a cycle G of lenght l · 4 is Gotzmann, but it is not Gotzmann if l ¸ 5. REFERENCES 1. A. Aramova, J. Herzog, T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra, 191 (1997), 174-211. 2. D. Bayer, The division algorithm and the Hilbert scheme, Ph.D. Thesis, Harvard University, Cambridge, MA, 1982. 3. D. Bayer, M. Stillman, A criterion for detecting m-regularity, Invent. Math. 87 (1987), 1-11. 4. A. Bigatti, Aspetti Combinatorici e Computazionali dell'Algebra Commutativa, Gen- ova, Dissertation 1995. 5. V. Bonanzinga, Principal Borel ideals and Gotzmann ideals, Arch. Math. , 81 (2003), 385-396. 6. P. ErdÄos, C. Ko and R. Rado, Intersection theorems for systems of ¯nite sets, Quart. J. Math. Oxford Ser., (2), 12 (1961), 313-320. 7. A. Gallico, A propos du Th¶eorem de Pr¶eparation de Weierstrass, L. N., Math. 409 (1974) 543-579. 8. V. Gasharov, Extremal properties of Hilbert functions, Illinois J. Math. 41 (1997), 4, 612-629. 9. G. Gotzmann, Eine Bedingung fÄur die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z, 158 (1978), 61-70. 10. J. Herzog, Generic initial ideals and graded Betti numbers, in "Computational Com- mutative Algebra and Combinatorics" (T. Hibi, Ed.), Advanced Studies in Pure Math., 33, (2002), 75-120. 11. J. Herzog, D. Popescu, Hilbert functions and homogeneous generic forms, Compos. Math. 116 (1999), 167-172. 12. H. Hulett, A generalization of Macaulay's theorem, Comm. Algebra 23 (1995), 1249-1263. 13. G. Kalai, Hyperconnectivity of graphs, Graphs combin., 1, (1985), 65-79. 14. G. Kalai, Algebraic shifting, in "Computational Commutative Algebra and Combi- natorics" (T. Hibi, Ed.), Advanced Studies in Pure Math., 33, (2002), 121-163. 15. K. Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), 4, 564-585. 4
Gotzmann ideals and applications to graphs
BONANZINGA, Vittoria;
2006-01-01
Abstract
Minisymposium: Algebraic models arising from phenomenals in statistic, transportation problems, manufacturing and other ¯elds Gotzmann ideals and applications to graphs Vittoria BONANZINGA University of Reggio Calabria-DIMET Faculty of Engineering, via Graziella (Feo di Vito), 89100 Reggio Calabria (ITALIA). bonanzin@ing.unirc.it, vittoria.bonanzinga@tin.it Loredana SORRENTI University of Messina Department of Mathematics, C.da Papardo, Salita Sperone 31, 98166 Messina (ITALIA). sorrenti@dipmat.unime.it, sorrenti.loredana@tiscali.it 1 Introduction Let k be an in¯nite ¯eld and S = k[x1; : : : ; xn] the polynomial ring over k with each deg xi = 1 and m = (x1; : : : ; xn) the graded maximal ideal of S: If I is a graded ideal of S, then we write Ihki for the ideal generated by all homogeneous polynomials of degree k belonging to I: Given a graded ideal I ½ S; there exists a unique lexsegment ideal Ilex such that I and Ilex have the same Hilbert function. A graded ideal I generated in degree d is called a Gotzmann ideal if the number of generators of mI is the smallest possible, namely, equal to the number of generators of (mI)lex: By Gotzmann's persistence theorem [9] a graded ideal I generated in degree d is a Gotzmann ideal if, and only if, I and (Ilex)hdi have the same Hilbert function. Thus, in particular, a Gotzmann ideal has a linear resolution with the same Betti numbers as the corresponding lexsegment ideal. Knowing which mono- mial ideals are Gotzmann is still an open problem. The general classi¯cation of Gotzmann ideals is unknown at the moment. By de¯nition, all lexsegment ideals are Gotzmann. In the theory of monomial ideals, there is the following hierarchy of ideals: lexsegment monomial ideals ) strongly stable monomial ideals ) stable monomial ideals. When the base ¯eld k is of characteristic 0, the Borel-¯xed ideals coincide with the strongly stable ideals. Strongly stable ideals play a special role in the study of Hilbert functions. The reason is that one can often apply GrÄobner basis techniques to reduce the general case to the study of Hilbert functions of strongly stable ideals (see e.g.[1], [2], [4], [3], [7], [8], [11], [12], [15]). Among the Borel ideals the principal ones are the most simple. Let u be a monomial of S, then hui denotes the smallest Borel ideal which contains u. The ideal hui is called principal Borel with Borel generator u. In [5] all principal Borel ideals with Borel generators up to degree 4 which are Gotzmann are characterized. From a combinatorial point of view, it is particularly interesting to consider ideals generated by squarefree monomials, since 1 they appear as the de¯ning ideals of Stanley-Reisner rings. Let G = ([n];E(G)) be a ¯nite graph on the vertex set [n] = f1; 2; : : : ; ng with the edge set E(G) without loops, multiple edges and isolated vertices. Write I(G) for the ideal generated by all squarefree monomials xixj with fi; jg 2 E(G). A graph G is called Gotzmann if the edge ideal I(G) is Gotzmann. The graph G is called Cohen-Macaulay over a ¯eld K, if K[x1; : : : ; xn]=I(G) is a Cohen-Macaulay ring, and is called Cohen-Macaulay if it is Cohen-Macaulay over any ¯eld. We will give some properties of Gotzmann graphs when the edge ideal is a principal Borel ideal. We also characterize all Cohen- Macaulay graphs whose edge ideal is principal Borel and Gotzmann. We will show some classes of graphs which are Gotzmann. 2 Main results We will characterize all Gotzmann graphs whose edge ideal is a principal Borel ideal. Proposition 2.1 Let I(G) = hui be a principal Borel ideal. The following condi- tions are equivalent: (a) I(G) is a Gotzmann ideal. (b) I(G) is a lexsegment ideal. If the equivalent conditions hold, then u = x1xi2 or u = xi1xn: Remark 2.2 In general (b) implies (a), but not the converse. In fact if I(G) is not a principal Borel ideal the following example shows that I(G) = (x1x2; x1x4; x2x3; x3x4) ½ k[x1; x2; x3; x4] is Gotzmann, but it is not lexsegment. We need some preliminaries. Let G be a graph and (fr1; r2g; : : : ; frs; rs+1g) a cycle in G, where rs+1 = r1: An edge fi; jg 2 E(G) is called a chord of the cycle (fr1; r2g; : : : ; frs; rs+1g) if i; j 2 fr1; r2; : : : ; rsg and fi; jg 6= fri; ri+1g for all i. We call G chordal if every cycle in G of lenght > 3 has a chord. A stable subset or clique of G is a subset F of [n] such that fi; jg 2 E(G) for all i; j 2 F with i 6= j: We write ¢(G) for the simplicial complex on [n] whose faces are the stable subset of G. A graph G is called shifted if, for any edge fi; jg 2 E(G) and for any integers i0 · i and j0 · j one has fi0; j0g 2 E(G). The shifting operation of graphs is an operation which associates a shifted graph ¢(G) with a graph G . Shifting operations are ¯rst considered by ErdÄos, Ko and Rado [6]. Their shifting operation G ¡! ¢c(G) is called combinatorial shifting. Instead, algebraic shifting was introduced by Kalai, [13]. The main variants of algebraic shifting are exterior algebraic shifting G ¡! ¢e(G) and symmetric algebraic shifting G ¡! ¢s(G), see Kalai [14] and [10]. Shifting operations are de¯ned for simplicial complexes, however, we here only consider ¯nite graphs. Let R = K[x1; : : : ; xn] the polynomial ring in n variables over a ¯eld K with each deg xi = 1. The graded Betti numbers 2 ¯ij(I) of a homogeneos ideal I ½ R are the integers ¯ij(I) = dimK(Tori(I;K)j): In other words, ¯ij(I) appear in the minimal graded free resolution 0 ¡! M j A(¡j)¯hj ¡! : : : ¡! M j A(¡j)¯1j ¡! M j A(¡j)¯0j ¡! I ¡! 0 (1) of I over R. Corollary 2.3 If I(G) = hui is Gotmann then (a) G is a chordal graph and ¢e(G) = ¢s(G) (b) ¯ii+2(I(G)) = ¯ii+2(I(¢e(G))) = ¯ii+2(I(¢s(G))) for all i ¸ 0. We will characterize all Cohen-Macaulay graphs whose edge ideal is principal Borel and Gotzmann. Theorem 2.4 Let K be a ¯eld, let G be a graph on the vertex set [n] and let I(G) = hui be a principal Borel ideal which is Gotzmann. Let F1; : : : ; Fm be the facets of ¢(G) which admit a free vertex. The following conditions are equivalent: (a) G is Cohen-Macaulay; (b) G is Cohen-Macaulay over K; (c) G is unmixed; (d) [n] is the disjoint union of F1; : : : ; Fm. The complement G of a graph G has vertex set V (G) and two vertices are adjacent in G if and only if they are not adjacent in G: Let G¤ be the graph obtained by the complement G by removing the set fx1; : : : ; xrg of all its isolated vertexes: G¤ = G n fx1; : : : ; xrg: We will show some classes of graphs which are Gotzmann. Theorem 2.5 Let R = k[x1; : : : ; xn]. Let G be a connected graph on the vertex set V (G) = fx1; : : : ; xng and I(G) ½ R its edge ideal. Then G is Gotzmann in the following cases: 1. G is complete; 2. G¤ is complete; 3. jE(G¤)j · 2; 4. jE(G)j · 2; 3 5. G = Ka;b is a bipartite complete graph with a · 2 and b · 3: Remark 2.6 We can remark that the bipartite complete graph K3;3 is not Gotz- mann and a cycle G of lenght l · 4 is Gotzmann, but it is not Gotzmann if l ¸ 5. REFERENCES 1. A. Aramova, J. Herzog, T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra, 191 (1997), 174-211. 2. D. Bayer, The division algorithm and the Hilbert scheme, Ph.D. Thesis, Harvard University, Cambridge, MA, 1982. 3. D. Bayer, M. Stillman, A criterion for detecting m-regularity, Invent. Math. 87 (1987), 1-11. 4. A. Bigatti, Aspetti Combinatorici e Computazionali dell'Algebra Commutativa, Gen- ova, Dissertation 1995. 5. V. Bonanzinga, Principal Borel ideals and Gotzmann ideals, Arch. Math. , 81 (2003), 385-396. 6. P. ErdÄos, C. Ko and R. Rado, Intersection theorems for systems of ¯nite sets, Quart. J. Math. Oxford Ser., (2), 12 (1961), 313-320. 7. A. Gallico, A propos du Th¶eorem de Pr¶eparation de Weierstrass, L. N., Math. 409 (1974) 543-579. 8. V. Gasharov, Extremal properties of Hilbert functions, Illinois J. Math. 41 (1997), 4, 612-629. 9. G. Gotzmann, Eine Bedingung fÄur die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z, 158 (1978), 61-70. 10. J. Herzog, Generic initial ideals and graded Betti numbers, in "Computational Com- mutative Algebra and Combinatorics" (T. Hibi, Ed.), Advanced Studies in Pure Math., 33, (2002), 75-120. 11. J. Herzog, D. Popescu, Hilbert functions and homogeneous generic forms, Compos. Math. 116 (1999), 167-172. 12. H. Hulett, A generalization of Macaulay's theorem, Comm. Algebra 23 (1995), 1249-1263. 13. G. Kalai, Hyperconnectivity of graphs, Graphs combin., 1, (1985), 65-79. 14. G. Kalai, Algebraic shifting, in "Computational Commutative Algebra and Combi- natorics" (T. Hibi, Ed.), Advanced Studies in Pure Math., 33, (2002), 121-163. 15. K. Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), 4, 564-585. 4I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
