论文标题
整体封闭理想的单体中分解
Factorization in the Monoid of Integrally Closed Ideals
论文作者
论文摘要
给定一个Noetherian Ring $ a $，$ a $中的所有正式封闭理想的集合包含一个非二元驱动器，表示为$ ic（a）$，在操作$ i*j = j = \ overline {ij} $的操作下形成了一种取消词，该产品的整体关闭产品的整体封闭。 MONOID是无扭力和原子的 - 每个$*$ - $*$*$*$*$*$*$ - 不可约定的整体封闭理想的$ A $中的每个封闭的理想都可以考虑到$ a $。限制了$ a $是多项式环，而所讨论的理想是单一的，我们表明，在他们的牛顿多面体的翻译中，整体多层群体中有一个从整体多层组组成的圆形同态。值得注意的是，在Minkowski的加法和翻译不变性下，整体多层组是具有整数顶点的多面体组，具有显式的基础，可以在MONOID中进行显式分解。
Given a Noetherian ring $A$, the collection of all integrally closed ideals in $A$ which contain a nonzerodivisor, denoted $ic(A)$, forms a cancellative monoid under the operation $I*J=\overline{IJ}$, the integral closure of the product. The monoid is torsion-free and atomic -- every integrally closed ideal in $A$ containing a nonzerodivisor can be factored in this $*$-product into $*$-irreducible integrally closed ideals. Restricting to the case where $A$ is a polynomial ring and the ideals in question are monomial, we show that there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. Notably, the Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid.
