• Title/Summary/Keyword: Betti numbers

Search Result 21, Processing Time 0.019 seconds

SIGNED A-POLYNOMIALS OF GRAPHS AND POINCARÉ POLYNOMIALS OF REAL TORIC MANIFOLDS

  • Seo, Seunghyun;Shin, Heesung
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.2
    • /
    • pp.467-481
    • /
    • 2015
  • Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph G is related to the $Poincar\acute{e}$ polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multipartite graphs G.

Cohomology of flat vector bundles

  • Kim, Hong-Jong
    • Communications of the Korean Mathematical Society
    • /
    • v.11 no.2
    • /
    • pp.391-405
    • /
    • 1996
  • In this article, we calculate the cohomology groups of flat vector bundles on some manifolds.

  • PDF

SOME HILBERT FUNCTIONS FROM k-CONFIGURATIONS ONLY

  • SHIN DONG-SOO
    • Journal of applied mathematics & informatics
    • /
    • v.18 no.1_2
    • /
    • pp.685-689
    • /
    • 2005
  • We find some Hilbert functions of codimension 4, which are obtained from only k-configurations in $\mathbb{P}^{3}$ and support the $3^{rd}$ linear syzygy.

ON THE LOCAL COHOMOLOGY OF MINIMAX MODULES

  • Mafi, Amir
    • Bulletin of the Korean Mathematical Society
    • /
    • v.48 no.6
    • /
    • pp.1125-1128
    • /
    • 2011
  • Let R be a commutative Noetherian ring, a an ideal of R, and M a minimax R-module. We prove that the local cohomology modules $H^j_a(M)$ are a-cominimax; that is, $Ext^i_R$(R/a, $H^j_a(M)$) is minimax for all i and j in the following cases: (a) dim R/a = 1; (b) cd(a) = 1, where cd is the cohomological dimension of a in R; (c) dim $R{\leq}2$. In these cases we also prove that the Bass numbers and the Betti numbers of $H^j_a(M)$ are finite.

COMINIMAXNESS OF LOCAL COHOMOLOGY MODULES WITH RESPECT TO IDEALS OF DIMENSION ONE

  • Roshan-Shekalgourabi, Hajar
    • Honam Mathematical Journal
    • /
    • v.40 no.2
    • /
    • pp.211-218
    • /
    • 2018
  • Let R be a commutative Noetherian ring, a be an ideal of R and M be an R-module. It is shown that if $Ext^i_R(R/a,M)$ is minimax for all $i{\leq}{\dim}\;M$, then the R-module $Ext^i_R(N,M)$ is minimax for all $i{\geq}0$ and for any finitely generated R-module N with $Supp_R(N){\subseteq}V(a)$ and dim $N{\leq}1$. As a consequence of this result we obtain that for any a-torsion R-module M that $Ext^i_R(R/a,M)$ is minimax for all $i{\leq}dim$ M, all Bass numbers and all Betti numbers of M are finite. This generalizes [8, Corollary 2.7]. Also, some equivalent conditions for the cominimaxness of local cohomology modules with respect to ideals of dimension at most one are given.

ON THE MINIMAL FREE RESOLUTION OF CURVES OF MAXIMAL REGULARITY

  • Lee, Wanseok;Park, Euisung
    • Bulletin of the Korean Mathematical Society
    • /
    • v.53 no.6
    • /
    • pp.1707-1714
    • /
    • 2016
  • Let $C{\subset}{\mathbb{P}}^r$ be a nondegenerate projective curve of degree d > r + 1 and of maximal regularity. Such curves are always contained in the threefold scroll S(0, 0, r - 2). Also some of such curves are even contained in a rational normal surface scroll. In this paper we study the minimal free resolution of the homogeneous coordinate ring of C in the case where $d{\leq}2r-2$ and C is contained in a rational normal surface scroll. Our main result provides all the graded Betti numbers of C explicitly.

NON-EXISTENCE OF SOME ARTINIAN LEVEL O-SEQUENCES OF CODIMENSION 3

  • Shin, Dong-Soo
    • Journal of applied mathematics & informatics
    • /
    • v.23 no.1_2
    • /
    • pp.517-523
    • /
    • 2007
  • Let R/I be an Artinian algebra of codimension 3 with Hilbert function H such that $h_{d-1}>h_d=h_{d+1}$. Ahn and Shin showed that A cannot be level if ${\beta}_{1,d+2}(Gin(I))={\beta}_{2,d+2}(Gin(I))$ where Gin(I) is a generic initial ideal of I. We prove that some certain graded Artinian algebra R/I cannot be level if either ${\beta}_{1,d}(I^{lex})={\beta}_{2,d}(I^{lex})+1\;or\;{\beta}_{1,d+1}(I^{lex})={\beta}_{2,d+1}(I^{lex})\;where\;I^{lex}$ is a lex-segment ideal associated to I.

CONSECUTIVE CANCELLATIONS IN FILTERED FREE RESOLUTIONS

  • Sharifan, Leila
    • Bulletin of the Korean Mathematical Society
    • /
    • v.56 no.4
    • /
    • pp.1077-1097
    • /
    • 2019
  • Let M be a finitely generated module over a regular local ring (R, n). We will fix an n-stable filtration for M and show that the minimal free resolution of M can be obtained from any filtered free resolution of M by zero and negative consecutive cancellations. This result is analogous to [10, Theorem 3.1] in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order ${\sigma}$, Betti numbers of I are obtained from those of $LT_{\sigma}(I)$ by so-called zero ${\sigma}$-consecutive cancellations. This provides a stronger version of the well-known cancellation "cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

REALIZING A FAKE PROJECTIVE PLANE AS A DEGREE 25 SURFACE IN ℙ5

  • Lev Borisov;Zachary Lihn
    • Journal of the Korean Mathematical Society
    • /
    • v.61 no.4
    • /
    • pp.683-692
    • /
    • 2024
  • Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in ℙ9. In this paper, we study Keum's fake projective plane (a = 7, p = 2, {7}, D327) and use the equations of [1] to construct an embedding of fake projective plane in ℙ5. We also simplify the 84 cubic equations defining the fake projective plane in ℙ9.