论文标题
在动机galois群体的一项自由基上
On unipotent radicals of motivic Galois groups
论文作者
论文摘要
让$ \ mathbf {t} $在特征零的字段上是中性的坦纳基人类别，其单位对象$ \ mathbf {1} $，并配备了过滤$ w_ \ cdot $，类似于混合动机上的重量过滤。令$ m $为$ \ mathbf {t} $的对象，$ \ usewfrak {\ mathfrak {u}}（m）\ subset w _ { - 1} \下划线{ - hom}（m，m，m）$ $ m $ m $ $ m $ $ $ $ m $ $ $ m $ $ $ m $ $ m $ nie代数的对象。 DELIGNE的结果给出了$ \下划线{\ Mathfrak {u}}（m）$的$ 0 \ longrightArrow w_pm \ longrightArrow m \ longrightArrow m \ longrightArrow m/w_pm \ longrightArrow m/w_pm \ longrightArrow 0 $：该$：该$ usperline $ \ usperline inlline inlline usperlline {smill ob} $ w _ { - 1} \下划线{hom}（m，m）$，以至于上述扩展的总和被视为$ \ mathbf {1} $ by $ w _ { - 1} \ { - 1} \ suespline {hom}（hom}（m，m）$，是$ by $ by $ by mathbf的$ w _ { - 1} \ suespline { - 1} \ $ \ useverline {\ mathfrak {u}}（m）$。在本文中，我们根据$ \ usewsline {\ mathfrak {u}}（m）$分别研究上述扩展。除其他事项外，我们还获得了DELIGNE的结果的改进，在该结果中，我们为个人提供了足够的条件。在本文的后半部分中，我们将动机的一支乐趣的GALOIS组尽可能大（即用$ \ $ \下划线{\ Mathfrak {u}}（m）= w _ { - 1} \ useverline {hom}（m，m，m）$）提供了应用。使用Grothedieck的形式主义\ textIt {延长panachées}，我们证明了这种动机的分类结果。专门针对混合泰特动机的类别，我们获得了三维混合泰特动机的分类结果。
Let $\mathbf{T}$ be a neutral Tannakian category over a field of characteristic zero with unit object $\mathbf{1}$, and equipped with a filtration $W_\cdot$ similar to the weight filtration on mixed motives. Let $M$ be an object of $\mathbf{T}$, and $\underline{\mathfrak{u}}(M)\subset W_{-1}\underline{Hom}(M,M)$ the Lie algebra of the kernel of the natural surjection from the fundamental group of $M$ to the fundamental group of $Gr^WM$. A result of Deligne gives a characterization of $\underline{\mathfrak{u}}(M)$ in terms of the extensions $0\longrightarrow W_pM \longrightarrow M \longrightarrow M/W_pM \longrightarrow 0$: it states that $\underline{\mathfrak{u}}(M)$ is the smallest subobject of $W_{-1}\underline{Hom}(M,M)$ such that the sum of the aforementioned extensions, considered as extensions of $\mathbf{1}$ by $W_{-1}\underline{Hom}(M,M)$, is the pushforward of an extension of $\mathbf{1}$ by $\underline{\mathfrak{u}}(M)$. In this article, we study each of the above-mentioned extensions individually in relation to $\underline{\mathfrak{u}}(M)$. Among other things, we obtain a refinement of Deligne's result, where we give a sufficient condition for when an individual extension $0\longrightarrow W_pM \longrightarrow M \longrightarrow M/W_pM \longrightarrow 0$ is the pushforward of an extension of $\mathbf{1}$ by $\underline{\mathfrak{u}}(M)$. In the second half of the paper, we give an application to mixed motives whose unipotent radical of the motivic Galois group is as large as possible (i.e. with $\underline{\mathfrak{u}}(M)= W_{-1}\underline{Hom}(M,M)$). Using Grothedieck's formalism of \textit{extensions panachées} we prove a classification result for such motives. Specializing to the category of mixed Tate motives we obtain a classification result for 3-dimensional mixed Tate motives over $\mathbb{Q}$ with three weights and large unipotent radicals.
