关于正则半局部环的点状谱的维特群

IF 1 2区 数学 Q1 MATHEMATICS
Stefan Gille, Ivan Panin
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We show that the Witt ring <span></span><math>\n <semantics>\n <mrow>\n <mi>W</mi>\n <mo>(</mo>\n <mi>U</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{W}(U)$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$U$</annotation>\n </semantics></math> has <span></span><math>\n <semantics>\n <mi>l</mi>\n <annotation>$l$</annotation>\n </semantics></math> non-trivial generators <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>E</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mtext>…</mtext>\n <mo>,</mo>\n <msub>\n <mi>E</mi>\n <mi>l</mi>\n </msub>\n </mrow>\n <annotation>$\\mathfrak {E}_{1},\\ldots ,\\mathfrak {E}_{l}$</annotation>\n </semantics></math> as <span></span><math>\n <semantics>\n <mrow>\n <mi>W</mi>\n <mo>(</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{W}(R)$</annotation>\n </semantics></math>-algebra, which satisfy <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>E</mi>\n <mi>i</mi>\n </msub>\n <msub>\n <mi>E</mi>\n <mi>j</mi>\n </msub>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\mathfrak {E}_{i}\\mathfrak {E}_{j}=0$</annotation>\n </semantics></math> for all <span></span><math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>⩽</mo>\n <mi>i</mi>\n <mo>,</mo>\n <mi>j</mi>\n <mo>⩽</mo>\n <mi>l</mi>\n </mrow>\n <annotation>$1\\leqslant i,j\\leqslant l$</annotation>\n </semantics></math>. If <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> is integral, then these generators become trivial over the fraction field <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>. 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引用次数: 0

摘要

设R $R$是一个维度为4 q + 1或5 $4q+1\geqslant 5$的正则半局部环,它包含1 2$\frac{1}{2}$, l小于或等于$l\geqslant 1$最大理想R $R$的个数假设它们都是相同的高度,U $U$为R $R$的刺穿谱,即没有极大理想的Spec R $\operatorname{Spec}R$。我们证明了U $U$的Witt环W (U) $\mathrm{W}(U)$有l $l$个非平凡发生器E1,…,E l $\mathfrak {E}_{1},\ldots ,\mathfrak {E}_{l}$作为W (R) $\mathrm{W}(R)$ -代数,满足E i E j = 0 $\mathfrak {E}_{i}\mathfrak {E}_{j}=0$,J≥1 $1\leqslant i,j\leqslant l$。如果R $R$是积分,那么这些生成器在R $R$的分数域K $K$上变得平凡。特别地,自然态射W (U) W (K) $\mathrm{W}(U)\longrightarrow \mathrm{W}(K)$不是内射。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

On the Witt group of the punctured spectrum of a regular semilocal ring

On the Witt group of the punctured spectrum of a regular semilocal ring

Let  R $R$ be a regular semilocal ring of dimension  4 q + 1 5 $4q+1\geqslant 5$ which contains  1 2 $\frac{1}{2}$ , l 1 $l\geqslant 1$ the number of maximal ideals of  R $R$ which are assumed to be all of the same height, and  U $U$ the punctured spectrum of  R $R$ , that is, Spec R $\operatorname{Spec}R$ without the maximal ideals. We show that the Witt ring  W ( U ) $\mathrm{W}(U)$ of  U $U$ has  l $l$ non-trivial generators E 1 , , E l $\mathfrak {E}_{1},\ldots ,\mathfrak {E}_{l}$ as  W ( R ) $\mathrm{W}(R)$ -algebra, which satisfy E i E j = 0 $\mathfrak {E}_{i}\mathfrak {E}_{j}=0$ for all 1 i , j l $1\leqslant i,j\leqslant l$ . If  R $R$ is integral, then these generators become trivial over the fraction field  K $K$ of  R $R$ . In particular, the natural morphism W ( U ) W ( K ) $\mathrm{W}(U)\longrightarrow \mathrm{W}(K)$ is not injective.

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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
186
审稿时长
6-12 weeks
期刊介绍: The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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