共查询到20条相似文献,搜索用时 0 毫秒
1.
Giancarlo Rinaldo 《Archiv der Mathematik》2008,91(5):416-426
We compute the Betti numbers of the resolution of a special class of square-free monomial ideals, the ideals of mixed products.
Moreover when these ideals are Cohen-Macaulay we calculate their type.
Received: 9 March 2008 相似文献
2.
Satoshi Murai 《Journal of Algebraic Combinatorics》2008,27(3):383-398
In this paper, we will show that the color-squarefree operation does not change the graded Betti numbers of strongly color-stable
ideals. In addition, we will give an example of a nonpure balanced complex which shows that colored algebraic shifting, which
was introduced by Babson and Novik, does not always preserve the dimension of reduced homology groups of balanced simplicial
complexes.
The author is supported by JSPS Research Fellowships for Young Scientists. 相似文献
3.
Ali Akbar Yazdan Pour 《代数通讯》2017,45(4):1483-1492
Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d?1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fröberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals. 相似文献
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Giuseppe Valla 《Proceedings of the American Mathematical Society》2005,133(1):57-63
In this paper we compute the graded Betti numbers of certain monomial ideals that are not stable. As a consequence we prove a conjecture, stated by G. Fatabbi, on the graded Betti numbers of two general fat points in
7.
Saugata Basu Richard Pollack Marie-Franç oise Roy 《Proceedings of the American Mathematical Society》2005,133(4):965-974
Let be a real closed field and let and be finite subsets of such that the set has elements, the algebraic set defined by has dimension and the elements of and have degree at most . For each we denote the sum of the -th Betti numbers over the realizations of all sign conditions of on by . We prove that
This generalizes to all the higher Betti numbers the bound on . We also prove, using similar methods, that the sum of the Betti numbers of the intersection of with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form or for , is bounded by
making the bound more precise.
This generalizes to all the higher Betti numbers the bound on . We also prove, using similar methods, that the sum of the Betti numbers of the intersection of with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form or for , is bounded by
making the bound more precise.
8.
Rachelle R. Bouchat Huy Tài Hà Augustine O?Keefe 《Journal of Combinatorial Theory, Series A》2011,118(8):2411-2425
Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal It(Γ) is generated by monomials that correspond to directed paths of length (t−1) in Γ. In this paper, we study algebraic properties and invariants of It(Γ). We give a recursive formula to compute the graded Betti numbers of It(Γ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when It(Γ) has a linear resolution. 相似文献
9.
Dariush Kiani 《代数通讯》2013,41(12):5376-5394
Let R = k[x1,…, xn], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms. 相似文献
10.
F. Bihan 《Commentarii Mathematici Helvetici》2003,78(2):227-244
Let be a nonsingular real algebraic surface of degree m in the complex projective space and its real point set in . In the spirit of the sixteenth Hilbert's problem, one can ask for each degree m about the maximal possible value of the Betti number (i=0 or 1). We show that is asymptotically equivalent to for some real number and prove inequalities and .
Received: April 26, 2000 相似文献
11.
Shifting Operations and Graded Betti Numbers 总被引:2,自引:0,他引:2
The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a squarefree monomial ideal are bounded by those of the corresponding squarefree lexsegment ideal. 相似文献
12.
We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution
of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph ℋ
appears within the resolution of its edge ideal ℐ(ℋ). We discuss when recursive formulas to compute the graded Betti numbers
of ℐ(ℋ) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A.
in J. Algebra 309:405–425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally
generalizes simple graphs from the point of view that distances between intersecting edges are “well behaved.” For such a
hypergraph ℋ (and thus, for any simple graph), we give a lower bound for the regularity of ℐ(ℋ) via combinatorial information
describing ℋ and an upper bound for the regularity when ℋ=G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal
graphs. When ℋ is a triangulated hypergraph, we explicitly compute the regularity of ℐ(ℋ) and show that the graded Betti numbers
of ℐ(ℋ) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests
can now be extended to chordal graphs.
Dedicated to Anthony V. Geramita on the occasion of his 65th birthday. 相似文献
13.
Daniel Soll 《Discrete Mathematics》2009,309(9):2782-2797
For n≥3, let Ωn be the set of line segments between the vertices of a convex n-gon. For j≥2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k≥1, let Δn,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ωn not containing any (k+1)-crossing.The complex Δn,k has been the central object of many papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line segments in Ω2n which can be transformed into each other by a 180°-rotation of the 2n-gon. Let Fn be the set Ω2n after identification, then the complex Dn,k of type-B generalized triangulations is the simplicial complex of subsets of Fn not containing any (k+1)-crossing in the above sense. For k=1, we have that Dn,1 is the simplicial complex of type-B triangulations of the 2n-gon as defined in [R. Simion, A type-B associahedron, Adv. Appl. Math. 30 (2003) 2-25] and decomposes into a join of an (n−1)-simplex and the boundary of the n-dimensional cyclohedron. We demonstrate that Dn,k is a pure, k(n−k)−1+kn dimensional complex that decomposes into a kn−1-simplex and a k(n−k)−1 dimensional homology-sphere. For k=n−2 we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of Dn,k.On the algebraical side we give a term order on the monomials in the variables Xij,1≤i,j≤n, such that the corresponding initial ideal of the determinantal ideal generated by the (k+1) times (k+1) minors of the generic n×n matrix contains the Stanley-Reisner ideal of Dn,k. We show that the minors form a Gröbner-Basis whenever k∈{1,n−2,n−1} thereby proving the equality of both ideals and the unimodality of the h-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k<n. 相似文献
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We introduce the concept of t-spread monomials and t-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the t-fold stretching operator. It is shown that t-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from t-spread principal Borel ideals. 相似文献
16.
We present definitions and properties of conformal Killing forms on a Riemannian manifold and determine Tachibana numbers as analogs of the well known Betti numbers of a compact Riemannian manifold. We show some sets of conditions which characterize these numbers. Finally, we prove some results which establish relationships between Betti and Tachibana numbers. 相似文献
17.
《Mathematische Nachrichten》2018,291(1):28-40
We study the equality of the extremal Betti numbers of the binomial edge ideal and those of its initial ideal for a closed graph G. We prove that in some cases there is a unique extremal Betti number for and as a consequence there is a unique extremal Betti number for and these extremal Betti numbers are equal. 相似文献
18.
Eric Emtander 《代数通讯》2013,41(5):1545-1571
In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions. 相似文献
19.
We extend the lower bounds on the complexity of computing Betti numbers proved in [P. Bürgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic and semialgebraic sets, J. Complexity 22 (2006) 147–191] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACE-hard. Then we prove PSPACE-hardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer. 相似文献
20.
Alessandro Conflitti 《Discrete Mathematics》2009,309(4):615-621
We solve some recurrences given by E. Munarini and N. Zagaglia Salvi proving explicit formulas for Whitney numbers of the distributive lattices of order ideals of the fence poset and crown poset. Moreover, we get a method to obtain explicit formulas for Whitney numbers of lattices of order ideals of fences with higher asymmetric peaks. 相似文献