Semigroup rings and simplicial complexes
Semigroup rings and simplicial complexes
复制标题
半群环和单纯复形
DOI：
10.1016/s0022-4049(97)00051-0
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发表时间：
1997
影响因子：
0.8
通讯作者：
J. Herzog
中科院分区：
文献类型：
--
作者：
W. Bruns;J. Herzog
研究了环T=S I的最小自由分解F，其中S是域K上的正仿射半群环，I是由单项式生成的S理想.我们将主要利用这样一个事实，即T的多分次Betti数可以从单形复形的相对同调中计算出来，我们称之为无平方除数复形。在某种意义上，如果忽略不可约元素在元素表示中出现的重数，这些单纯复形就代表了S中的可除性关系。在第一节中，我们研究了自由分解对K的特征的依赖关系。第二节中，我们证明了，直到同伦等价时，每个单纯复形都可以在正规半群环上‘实现’，也可以在一维半群环上‘实现’。此外，我们还刻画了无平方因子复形中的所有图。在第三节中，我们从众所周知的半群环的自由分解的信息推导出关于某些棋盘型单纯复形的断言。
We study the minimal free resolution F of a ring T = S I where S is a positive affine semi-group ring over a field K, and I is an ideal in S generated by monomials. We will essentially use the fact that the multigraded Betti numbers of T can be computed from the relative homology of simplicial complexes that we shall call squarefree divisor complexes. In a sense, these simplicial complexes represent the divisibility relations in S if one neglects the multiplicities with which the irreducible elements appear in the representation of an element. In Section 1 we study the dependence of the free resolution on the characteristic of K. In Section 2 we show that, up to an equivalence in homotopy, every simplicial complex can be ‘realized’ in a normal semigroup ring and also in a one-dimensional semigroup ring. Furthermore, we describe all the graphs among the squarefree divisor complexes. In Section 3 we deduce assertions about certain simplicial complexes of chessboard type from information about free resolutions of well-understood semigroup rings.