Divisorial Linear Algebra of Normal Semigroup Rings
Divisorial Linear Algebra of Normal Semigroup Rings
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正规半群环的除数线性代数
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
J. Gubeladze
中科院分区：
文献类型：
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作者：
W. Bruns;J. Gubeladze
研究了正规半群环上生成元的最小个数μ和除理想的深度.这样的理想由与半群的支撑超平面相关联的线性不等式的非齐次系统定义。主要结果是：对于每个有界C，直到同构，只存在n个除理想I使得μ（I）≤C.因此，只存在1000个Cohen-Macaulay因子类。此外，我们还确定了除子理想的最小深度以及除子类群中“算术级数”中μ和深度的性质，并将结果推广到更一般的线性不等式组，其齐次形式定义了半群，但不一定是无冗余的.这样产生的理想也可以看作是由非齐次线性丢番图方程组的非负解所定义的.我们还对除子理想的生成元的最小个数定理给出了一个更环论的方法：它是多级Hilbert函数增长定理的一个特例.
We investigate the minimal number of generators μ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that μ(I)≤C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of μ and depth in ‘arithmetic progressions’ in the divisor class group.The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations.We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.