• 제목/요약/키워드: Syzygy

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Minimal Generators of Syzygy Modules Via Matrices

  • Haohao Wang;Peter Oman
    • Kyungpook Mathematical Journal
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    • 제64권2호
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    • pp.197-204
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    • 2024
  • Let R = 𝕂[x] be a univariate polynomial ring over an algebraically closed field 𝕂 of characteristic zero. Let A ∈ Mm,m(R) be an m×m matrix over R with non-zero determinate det(A) ∈ R. In this paper, utilizing linear-algebraic techniques, we investigate the relationship between a basis for the syzygy module of f1, . . . , fm and a basis for the syzygy module of g1, . . . , gm, where [g1, . . . , gm] = [f1, . . . , fm]A.

THE PROJECTIVE MODULE P(2) OVER THE AFFINE COORDINATE RING OF THE 2-SPHERE S2

  • Kim, Sanghee
    • 호남수학학술지
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    • 제43권3호
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    • pp.403-416
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    • 2021
  • It is known that the rank 2 stably free syzygy module P(2) is not free. This algebraic fact was proved analytically, but this remarkable fact still lacks of a simple algebraic proof. The main purpose of this paper is to give a partially algebraic proof by making use of a theorem whose proof is quite topological, and the further properties of the module will be discussed.

SOME HILBERT FUNCTIONS FROM k-CONFIGURATIONS ONLY

  • SHIN DONG-SOO
    • Journal of applied mathematics & informatics
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    • 제18권1_2호
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    • pp.685-689
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    • 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.

FREE AND NEARLY FREE CURVES FROM CONIC PENCILS

  • Dimca, Alexandru
    • 대한수학회지
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    • 제55권3호
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    • pp.705-717
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    • 2018
  • We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

  • WANG, FANGGUI;QIAO, LEI
    • 대한수학회보
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    • 제52권4호
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    • pp.1327-1338
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    • 2015
  • In this paper, we introduce and study the w-weak global dimension w-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a $Pr\ddot{u}fer$ v-multiplication domain if and only if w-w.gl.dim(R) ${\leq}1$. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the w-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is w-coherent, then w-w.gl.dim(R[x]) = w-w.gl.dim(R).

깊이의 식과 토르 게임에 대하여 (On Depth Formula and Tor Game)

  • 최상기
    • 한국수학사학회지
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    • 제17권4호
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    • pp.37-44
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    • 2004
  • 힐베르트 시지지 정리와 그에 따른 분해에서 호몰로지 대수의 흔적을 찾을 수 있다. 1950년대에 이르러 대수 위상의 발달과 함께 그의 이론적인 도구로서 호몰로지 대수의 실질적인 시작을 볼 수 있다. 1956년 쎄레는 정칙 국소환의 대역적 차원이 유한하다는 것을 증명하였는데, 이 정리는 호모로지 대수적인 문제 풀이에서 근본적인 도구를 제공하고 있다. 호모로지 대수에서 토르를 구하고 그의 깊이를 계산하는 것은 어려운 문제인데, 이 논문에서는 1961년 오슬랜더가 제시한 토르 가군의 깊이에 관한 문제와 그에 따른 토르 게임(Tor game)에 대하여 논하고자 한다.

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PROJECTIONS OF ALGEBRAIC VARIETIES WITH ALMOST LINEAR PRESENTATION I

  • Ahn, Jeaman
    • 충청수학회지
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    • 제32권1호
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    • pp.15-21
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    • 2019
  • Let X be a reduced closed subscheme in ${\mathbb{P}}^n$ and $${\pi}_q:X{\rightarrow}Y={\pi}_q(X){\subset}{\mathbb{P}}^{n-1}$$ be an isomorphic projection from the center $q{\in}{\mathbb{P}}^n{\backslash}X$. Suppose that the minimal free presentation of $I_X$ is of the following form $$R(-3)^{{\beta}2,1}{\oplus}R(-4){\rightarrow}R(-2)^{{\beta}1,1}{\rightarrow}I_X{\rightarrow}0$$. In this paper, we prove that $H^1(I_X(k))=H^1(I_Y(k))$ for all $k{\geq}3$. This implies that Y is k-normal if and only if X is k-normal for $k{\geq}3$. Moreover, we also prove that reg(Y) ${\leq}$ max{reg(X), 4} and that $I_Y$ is generated by homogeneous polynomials of degree ${\leq}4$.