标题： 滤分的Tannakian群的一阶部分 显示英文标题

标题： Depth one part of Tannakian groups of filtrations

摘要： 设$(F_r M)_{r\in\mathbb{Z}}$是一个在零特征的中性 Tannakian 范畴$\mathbf{T}$中的对象$M$上的有限滤链。 设$u(M)=u^F(M)$是 Tannakian 基本群$M$的子群的李代数，该子群在相关的分次$Gr^FM$上作用平凡。 该滤链$F_\bullet M$在内部 Hom$\underline{Hom}(M,M)$上诱导一个滤链，进而在$u(M)$上诱导一个滤链$F_\bullet u(M)$。 该滤链在$u(M)$上集中在负次数上。 在本文中，我们给出了分次项$Gr^F_{-1}u( M)$的描述，该描述基于扩张$F_{r+1}M/F_{r-1}M\in Ext^1(Gr^F_{r+1}M, Gr^F_{r}M)$。 特别是，这些扩展确定了$Gr^F_{-1}u(M)$。 请注意，此处我们既不假设过滤是函子性的，也不假设$Gr^FM$是半单的。 在这一普遍性下研究$u(M)$的问题是出于理解与混合动机及其表示相关的Tannakian群的动机，包括那些纯动机表示的半单性尚未可知的表示，以及缺乏有趣函子性权重过滤的表示。 我们还给出了两个相关应用。 第一个是在上述普遍性下，当$u(M)$与其平凡上界$F_{-1}\underline{Hom}(M,M)$相同时的等价条件。 这个结果推广了之前在特殊情境或限制条件下由不同作者获得的关于$u(M)$最大性的早期标准。 在第二个应用中，我们将结果应用于具有函子性权重过滤的中性Tannakian范畴$W_\bullet$的情形。 结合[arXiv:2307.15487]中的构造与我们的普遍最大性准则，我们证明了一个关于对象$M$的同构类集合的结构结果，其中$Gr^WM$与给定的分次对象$A$和$u^W(M)=W_{-1}\underline{Hom}(M,M)$同构。 显示英文摘要

摘要： Let $(F_r M)_{r\in\mathbb{Z}}$ be a finite filtration on an object $M$ of a neutral Tannakian category $\mathbf{T}$ in characteristic zero. Let $u(M)=u^F(M)$ be the Lie algebra of the subgroup of the Tannakian fundamental group of $M$ that acts trivially on the associated graded $Gr^FM$. The filtration $F_\bullet M$ induces a filtration on the internal Hom $\underline{Hom}(M,M)$, which in turn induces a filtration $F_\bullet u(M)$ on $u(M)$. This filtration on $u(M)$ is concentrated in negative degrees. In this paper, we give a description of the graded piece $Gr^F_{-1}u( M)$ in terms of the extensions $F_{r+1}M/F_{r-1}M\in Ext^1(Gr^F_{r+1}M, Gr^F_{r}M)$. In particular, these extensions determine $Gr^F_{-1}u(M)$. Note that here we neither assume the filtration is functorial, nor we assume that $Gr^FM$ is semisimple. The problem of studying $u(M)$ in this generality is motivated by the desire to understand Tannakian groups associated to a mixed motive and its realizations, including realizations for which semisimplicity of realizations of pure motives is not known and realizations that lack an interesting functorial weight filtration. We also give two related applications. The first is an equivalent condition in the generality described above for when $u(M)$ coincides with its trivial upper bound $F_{-1}\underline{Hom}(M,M)$. This result generalizes earlier criteria for maximality of $u(M)$ obtained by various authors in special contexts or under limiting conditions. In the second application, we apply the results to the setting of a neutral Tannakian category with a functorial weight filtration $W_\bullet$. Combining the constructions of [arXiv:2307.15487] with our general maximality criterion we prove a result about the structure of the set of isomorphism classes of objects $M$ for which $Gr^WM$ is isomorphic to a given graded object $A$ and $u^W(M)=W_{-1}\underline{Hom}(M,M)$.