The Grothendieck Group and the Extensional Structure of Noetherian Module Categories
The Grothendieck Group and the Extensional Structure of Noetherian Module Categories
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格洛腾迪克群和诺特模范畴的外延结构
DOI：
10.1090/conm/259/04090
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
G. Brookfield
中科院分区：
文献类型：
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作者：
G. Brookfield
对于左Notherian环R，Gothendieck群G0(R)对于从左Notherian R-模范畴到Abelian群范畴的短正合列的映射是泛的。有一个鲜为人知的么半群M(R-Noeth)，它在映射到交换么半群方面具有类似的普适性质。本文研究了这两个普遍对象之间的关系，从而对前者进行了新的、更详细的描述。有一个自然分解G0(R)∼=Zn×eG0(R)，其中n是R的极小素数理想的个数，而eG0(R)是我们证明可以嵌入到M(R-Noeth)中的群，粗略地说，就是那些可与M(R-Noeth)中的R的像相比较的元素。由此得到了降秩零模A的生成元<A>的刻画以及某些<U/U‘>=0形式的关系，其中U’⊂U是R的极小素因子环的同构一致左理想.特别地，对于Krull维1的整环，e G0(R)的生成元对应于单模，并且当x∈R不可约时，这些关系对应于模R/Rx的合成列.
For a left Noetherian ring R, the Gothendieck group G0(R) is universal for maps which respect short exact sequences from the category of left Noetherian R-modules to Abelian groups. There is a less well known monoid M(R-Noeth) which has the analogous universal property with respect to maps into commutative monoids. In this paper the relationship between these two universal objects is studied leading to a new and more detailed description of the former. There is a natural decomposition G0(R) ∼= Zn × e G0(R) where n is the number of minimal prime ideals of R and e G0(R) is a group which we show can be embedded in M(R-Noeth), roughly speaking, as those elements which are comparable to the image of R in M(R-Noeth). This leads to a description of e G0(R) in terms of generators 〈A〉 for modules A of reduced rank zero and certain relations of the form 〈U/U ′〉 = 0 where U ′ ⊂ U are isomorphic uniform left ideals of minimal prime factor rings of R. In particular, for a domain of Krull dimension 1, the generators of e G0(R) correspond to simple modules, and the relations correspond to the composition series of the modules R/Rx when x ∈ R is irreducible.