论文标题
关于非交流性欧拉系统
On non-commutative Euler systems
论文作者
论文摘要
让$ p $为素数，$ t $ a $ p $ - 亚法代表$ k $和$ \ nathcal {k} $ $ k $的任意galois扩展名。然后，对于每个非负整数$ r $，我们定义了$ t $的“非交易性Euler系统”的自然概念，相对于扩展$ \ Mathcal {k}/k $。我们证明，如果$ p $是奇怪的，$ t $和$ \ mathcal {k}/k $满足某些温和的假设，那么存在非共同的Euler系统，可以控制$ t $的同胞组的GALOIS结构，而不是$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mathcal {k}/k $的中间领域，并依赖于$ t $ t $ t $ t $ t $ t。作为这种方法的第一个具体应用，我们（无条件地）将环环单元的经典Euler系统扩展到$ \ mathbb {q} $的任意完全真实的galois扩展，并描述此扩展的环形欧拉系统之间的显着链接，并在构建$ l $ l $ l-l $ ly-l $ l $ ly-l $ ly-l $ ly-l $ ly-l $ ly-l $ ly-l $ l $ l的值之间进行明确联系。作为这些结果的制定和证明的重要初步，我们介绍了多种标准构建体的自然非交通概括，包括较高拟合的不变性，较高的外部力量和Grothendieck-Knudsen-Mumumford n-Mumumford确定性函数，具有独立利益的完美络合物。
Let $p$ be a prime, $T$ a $p$-adic representation over a number field $K$ and $\mathcal{K}$ an arbitrary Galois extension of $K$. Then for each non-negative integer $r$ we define a natural notion of a `non-commutative Euler system of rank $r$' for $T$ relative to the extension $\mathcal{K}/K$. We prove that if $p$ is odd and $T$ and $\mathcal{K}/K$ satisfy certain mild hypotheses, then there exist non-commutative Euler systems that control the Galois structure of cohomology groups of $T$ over intermediate fields of $\mathcal{K}/K$ and have rank that depends explicitly on $T$. As a first concrete application of this approach, we (unconditionally) extend the classical Euler system of cyclotomic units to the setting of arbitrary totally real Galois extensions of $\mathbb{Q}$ and describe explicit links between this extended cyclotomic Euler system, the values at zero of derivatives of Artin $L$-series and the Galois structures of ideal class groups. As an important preliminary to the formulation and proof of these results, we introduce natural non-commutative generalizations of several standard constructions in commutative algebra including higher Fitting invariants, higher exterior powers and the Grothendieck-Knudsen-Mumford determinant functor on perfect complexes that are of independent interest.
