Paul Ziegler
We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor can be split by a grading fpqc-locally on the base scheme.
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Let $X$ be a complete toric variety equipped with the action of a torus $T$ and $G$ a reductive algebraic group, defined over an algebraically closed field $K$. We introduce the notion of a compatible $Σ$--filtered algebra associated to $X$, generalizing the notion of a compatible $Σ$--filtered vector space due to Klyachko, where $Σ$ denotes the fan of $X$. We combine Klyachko's classification of $T$--equivariant vector bundles on $X$ with Nori's Tannakian approach to principal $G$--bundles, to give an equivalence of categories between $T$--equivariant principal $G$--bundles on $X$ and certain…
The principal aim of this note is to give an elementary proof of the fact that any two fiber functors of a Tannakian category are locally isomorphic. This builds on an idea of Deligne concerning scalar extensions of Tannakian categories and implements a proof strategy which Deligne attributes to Grothendieck. Besides categorical generalities, the proof merely relies on basic properties of exterior powers and the classification of finitely generated modules over a principal ideal domain. Using related ideas (but less elementary means) we also present an alternative characterization of Tannakian…
We introduce and develop the notion of scalar extension for abelian categories. Given a field extension F'/F, to every F-linear abelian category A satisfying a suitable finiteness condition we associate an F'-linear abelian category A' and an exact F-linear functor t: A --> A'. This functor is universal among F-linear right exact functors with target an F'-linear abelian category. We discuss various basic properties of this concept, among others compatibilities with multilinear endofunctors such as tensor products, and the permanence of favourable properties of the functors and categories invo…
Given a rigid tensor-triangulated category and a vector space valued homological functor for which the Künneth isomorphism holds, we construct a universal graded-Tannakian category through which the given homological functor factors. We use this to (unconditionally) construct graded-Tannakian categories of pure motives associated to a fixed Weil cohomology theory, with a fiber functor realizing the given cohomology theory. For $\ell$-adic cohomology and a ground field which is algebraic over a finite field, this category is Tannakian. In this case, we obtain in particular motivic Galois groups…
Theta-Categories and Tannakian duality
本文引入$\Theta$-范畴细化对称幺半$\infty$-范畴，借助纤维函子栈证明Tannakian对偶，在任意特征基环上连接Tannakian $\Theta$-范畴与概形同伦类型。
Filtered fiber functors over a general base
本文证明了在具有足够可对偶表示的光滑仿射群概型的可对偶表示范畴上，每个滤过纤维函子均源自分次纤维函子。
We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport-Zink functors associated to triples $(G,\{μ\},[b])$, where $G$ is a flat affine group scheme over $\mathbb{Z}_p$ and $μ$ is a cocharacter of $G$ defined over a finite unramified extension of $\mathbb{Z}_p$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport-Zink spaces given by Bültel and Papp…
This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequ…
We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R …
This paper proves the Beilinson-Soul{é} vanishing conjecture for motives attached to the moduli spaces of curves of genus 0 with n marked points. As part of the proof, it is also proved that these motives are mixed Tate. As a consequence of Levine's work, one obtains then well defined categories of mixed Tate motives over the moduli spaces of curves . It is shown that morphisms between moduli spaces forgetting marked points and embedding as boundary components induce functors between those categories and how tangential bases points fit in these functorialities. Tannakian formalism attaches gro…
We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures. In other words, every non-zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated étale motives over F with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of F.
On endomorphisms of extensions in Tannakian categories
本文证明了Tannakian范畴中扩张自同态的Schur引理类似结论，研究了稳定子对象的特定自同态诱导标量映射的条件，并在一般与过滤设定下给出了相应结果及精确性讨论。
For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties - expressible entirely within the language of tensor categories - which are necessary and suf…
A fully faithful p-adic Riemann--Hilbert functor for filtered coherent D-modules
本文证明光滑刚性解析簇上滤过凝聚 D-模的求解函子完全忠实，恢复 Scholze 水平截面函子，并引入 de Rham Zariski 可构造层以推广经典 de Rham 局部系统。
Stokes phenomenon of Kloosterman and Airy connections
本文通过Tannakian类别理论证明Kloosterman和Airy连接的物理刚性，利用Stokes过滤和分级的$G$-局部系统与辫形空间的关系。
A Comparison of Categories of Nori Motivic Sheaves
本文证明了两种Nori动机sheaves理论的自然等价性，并提供了与六操作兼容的实现实用函子。
We study the constant term functor for $\mathbb{F}_p$-sheaves on the affine Grassmannian in characteristic $p$ with respect to a Levi subgroup. Our main result is that the constant term functor induces a tensor functor between categories of equivariant perverse $\mathbb{F}_p$-sheaves. We apply this fact to get information about the Tannakian monoids of the corresponding categories of perverse sheaves. As a byproduct we also obtain geometric proofs of several results due to Herzig on the mod $p$ Satake transform and the structure of the space of mod $p$ Satake parameters.
Depth one part of Tannakian groups of filtrations
本文研究了在零特征下，Tannakian范畴中滤波对象的Tannakian基本群的Lie代数的深度一部分，通过扩展来描述其梯度部分，并探讨了混合动机及其实现在此一般情况下的应用。
We give a Tannakian description for categories of l-adic perverse sheaves on semiabelian varieties which combines a construction of Gabber and Loeser for algebraic tori with a generic vanishing theorem for the cohomology of constructible sheaves on abelian varieties. As an application we explain how the arising Tannaka groups on abelian varieties can be studied in terms of semiabelian degenerations via the functor of nearby cycles.
We use filtered log-$\mathscr{D}$-modules to represent the (dual) localization of Saito's Mixed Hodge Modules along a smooth hypersurface, and show that they also behave well under the direct image functor and the dual functor in the derived category of filtered log-$\mathscr{D}$-modules. The results of this paper can be used to generalize the result of M. Popa and C. Schnell about Kodaira dimension and zeros of holomorphic one-forms into the log setting.
We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able to consider neutral tannakian categories with a fibre functor as many-sorted first order structures. The class of diagonalizable proalgebraic groups is analyzed in detail. We show that the theory of a diagonalizable proalgebraic group $G$ is determined by the theory of the base field and the theory of the character group of $G$. Some initial steps towards a comprehensive study of types are also made.
Normed Fiber Functors
该研究借助Tannakian形式体系，将一般线性群Bruhat-Tits建筑的非阿基米德范数描述推广到非分歧约化群情形，并给出对应仿射群概形的模论刻画。
Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in this case the category of its finite dimensional representations. Lurie's result that the pseudofunctor which sends a geometric stack to its category of quasi-coherent sheaves is fully faithful provides a conceptual explanation for why this works. In this paper we prove a generalized Tannakian recognition theorem, in order to characterize a part of the imag…
A filtered Hochschild-Kostant-Rosenberg theorem for real Hochschild homology
本文在$ C_2 $-Mackey函子中引入导出对合代数，证明其实Hochschild同调存在滤过且相关分次等同对合de Rham复形，并特化为Raksit的HKR-滤过Hochschild同调。
This article provides some basic results on weight structures, weight complex functors and homotopy categories. We prove that the full subcategories K(A)^{w < n}, K(A)^{w > n}, K(A)^- and K(A)^+ (of objects isomorphic to suitably bounded complexes) of the homotopy category K(A) of an additive category A are idempotent complete, which confirms that (K(A)^{w <= 0}, K(A)^{w >= 0}) is a weight structure on K(A). We discuss weight complex functors and provide full details of an argument sketched by M. Bondarko, which shows that if w is a bounded weight structure on a triangulated category T that ha…
We define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if and only if it is a filtered category, and our construction yields a category equivalent to the category resulting from the usual construction of filtered colimits of categories. Weaker axioms suffice for this construction, and we call the corresponding notion pre 2-filtered 2-category. The full set of axioms is necessary to prove that 2-filtered bicolimits have the properties corresponding to the essenti…
Much of the homotopical and homological structure of the categories of chain complexes and topological spaces can be deduced from the existence and properties of the 'simple' functors Tot : {double chain complexes} -> {chain complexes} and geometric realization : {sSets} -> {Top}, or similarly, Tot : {simplicial chain complexes} -> {chain complexes} and | | : {sTop} -> {Top}. The purpose of this thesis is to abstract this situation, and to this end we introduce the notion of '(co)simplicial descent category'. It is inspired by Guillen-Navarros's '(cubical) descent categories'. The key ingredie…
Using the functoriality of C. McCrory and A. Parusiński's weight complex -which induces an analog of the weight filtration for complex algebraic varieties on the Borel-Moore homology with $\mathbb{Z}_2$ coefficients of real algebraic varieties-, we define a weight complex with action on the real algebraic varieties equipped with a finite group action. Emphasizing on the two elements group, we then establish a filtered version of Smith short sequence, taking into account the Nash-constructible filtration which realizes the weight complex with action. Its exactness is implied by the splitting of…
A classical result of Tannaka duality is the fact that a coalgebra over a field can be reconstructed from its category of finite dimensional representations by using the forgetful functor which sends a representation to its underlying vector space. There is also a corresponding recognition result, which characterizes those categories equipped with a functor to finite dimensional vector spaces which are equivalent to the category of finite dimensional representations of a coalgebra. In this paper we study a generalization of these questions to an arbitrary cosmos, that is, a complete and cocomp…
Hodge theory, intertwining functors, and the Orbit Method for real reductive groups
本文研究实约化群幂单表示的Hodge滤过，证明其与轨道方法预测的量子化滤过一致，从而推导出酉性与K-型描述等猜想，并给出部分旗簇上混合Hodge模的新上同调消失定理。
Filtrations in $\mathbb{C}$-motivic stable homotopy theory
本文在复动机稳定同伦范畴中利用滤子谱模型研究三类滤子，计算了多种谱的切片及有效切片谱序列，验证了相关猜想并得到新结果。
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