Homological support of big objects in tensor-triangulated categories
Paul Balmer
本文借助同调剩余域为张量三角范畴中的大对象引入支撑概念，并建立相应的张量积公式。
利用同调剩余域，我们定义了张量三角范畴中大对象的支撑，并证明了一个张量积公式。
Using homological residue fields, we define supports for big objects in tensor-triangulated categories and prove a tensor-product formula.
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Finiteness of homological dimensions in triangulated categories
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The theory of derivators enhances and simplifies the theory of triangulated categories. In this article a notion of fibered (multi-)derivator is developed, which similarly enhances fibrations of (monoidal) triangulated categories. We present a theory of cohomological as well as homological descent in this language. The main motivation is a descent theory for Grothendieck's six operations.
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本文研究张量三角几何中的几何点，通过高阶扎里斯基几何中的自由构造构造反例，讨论构造谱，并利用这些谱构建几何 incarnations of 原子素。
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本文研究弱近似可三角化范畴的定位现象，证明在弱假设下，此类范畴的Recollement诱导出若干自然子范畴的短正合列，应用于环的导出范畴、DG代数和概形。
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…
$t$-Structures with Grothendieck hearts via functor categories
研究带余积三角范畴中 $t$-结构的心成为 AB5 或 Grothendieck 范畴的条件，通过函子范畴将心表示为 Serre 商并分类保余积同调函子，证明标准良生成三角范畴中纯内射对象总余生成 $t$-结构。
Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building blocks. The manifestation of this general pattern, in the study of derived and triangulated categories, dates back almost to the beginnings of the subject -- more precisely to articles by Illusie in SGA6, way back in the early 1970s. What's new, at least new in the world of derived and triangulated categories, is that one gets extra mileage from analysing mor…
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Bounding The Orlov Spectrum For A Completion Of Discrete Cluster Categories
本文通过引入同调连通对象及其分解，对 Dynkin 型 $A_{\infty}$ 离散簇范畴的 Paquette-Yıldırım 完备化中的厚子范畴与经典生成子进行分类，并利用同调长度界定了该范畴的 Orlov 谱上界与 Rouquier 维度。
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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…
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