标题： 有限交换群的线性化说明 显示英文标题

标题： Note on the linearisation of finite abelian groups

摘要： 如果$K$是一个包含足够多单位根的域，且$V$是一个阿贝尔群，那么群$V$的$K$-代数$K[V]$是分裂半单的，因此规范同态$K[V]\to K^{V^\sharp}$，其中$V^\sharp$表示$V$的对偶群（可视为 Hom$(V,K^\times)$），是$K$-代数之间的同构。 如果去掉假设$K$具有足够的单位根，可以通过使用基变换和 Krull-Schmidt 容易地推导出，当限制到被固定非零整数消去的有限群$V$时，它仍然是一个在群$V$上自然的$K$-线性同构$K[V]\xrightarrow{\simeq} K^{V^\sharp}$。 这样的同构，在阿贝尔群$V$上是自然的，是否在仅要求$V$是有限的且其阶在$K$中是可逆的情况下仍然存在，这个问题并不明显；我们通过使用高斯和，在某种更一般的情况下积极地解决了它（$K$为任何交换环）。 我们还探讨了一些相关的函子性问题。 显示英文摘要

摘要： If $K$ is a field with enough roots of unity and $V$ an abelian group, the $K$-algebra $K[V]$ of the group $V$ is split semisimple, so that the canonical morphism $K[V]\to K^{V^\sharp}$, where $V^\sharp$ denotes the dual group of $V$ (which may be seen as Hom$(V,K^\times)$), is an isomorphism of $K$-algebras. If one removes the assumption that $K$ has enough roots of unity, one can easily deduce from it (by using a base change and Krull-Schmidt) that it remains a $K$-linear isomorphism $K[V]\xrightarrow{\simeq} K^{V^\sharp}$ natural in the group $V$ if one restricts to finite groups $V$ canceled by a fixed nonzero integer. The question of whether such an isomorphism, natural in the abelian group $V$, still exists without any other restriction than $V$ is finite and its order is invertible in $K$, is less obvious; we solve it positively, in a somewhat more general setting ($K$ being any commutative ring), by using Gauss sums. We also explore some related functorial questions.