Paul Balmer, Giordano Favi
We discuss gluing of objects and gluing of morphisms in tensor triangulated categories. We illustrate the results by producing, among other things, a Mayer-Vietoris exact sequence involving Picard groups.
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Residual regularity in tensor triangular geometry
本文提出张量三角范畴的剩余正则性概念，证明其在有限可分扩张下可传递，并分类了所有置换模导出范畴为剩余正则的有限群。
Gluing of Fourier-Mukai partners in a triangular spectrum and birational geometry
本文在三角谱中构造Fourier-Mukai轨迹以粘合FM伙伴的张量三角谱，结合实例揭示其与代数簇几何、双有理性质的关联，并对比相关轨迹提出关于Serre不变轨迹的精确猜想。
We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable homological tensor-functors to Grothendieck categories.
We compare the homological support and tensor triangular support for `big' objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support.
This is a chapter of the Handbook of Homotopy Theory, that surveys the classifications of thick tensor-ideals.
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during t…
Geometric Points in Tensor Triangular Geometry
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。
Tensor triangular geometry -- Notes for an Oberwolfach Seminar
本文旨在发展三角范畴的支持概念，并将其应用于分类在有限群模表示论中出现的厚和局部化张量理想。
We develop a theory of cosupport and costratification in tensor triangular geometry. We study the geometric relationship between support and cosupport, provide a conceptual foundation for cosupport as categorically dual to support, and discover surprising relations between the theory of costratification and the theory of stratification. We prove that many categories in algebra, topology and geometry are costratified by developing and applying descent techniques. An overarching theme is that cosupport is relevant for diverse questions in tensor triangular geometry and that a full understanding …
Some notes on tensor triangular geometry
本文通过格理论视角，介绍张量三角几何的基本概念与方法，涵盖小范畴和可紧生成范畴，并展示了Thomason定理的证明及新思路。
We recall P. Balmer's definition of tensor triangular Chow group for a tensor triangulated category $\mathcal{K}$ and explore some of its properties. We give a proof that for a suitably nice scheme $X$ it recovers the usual notion of Chow group from algebraic geometry when we put $\mathcal{K} = \mathrm{D^{perf}}(X)$. Furthermore, we identify a class of functors for which tensor triangular Chow groups behave functorially and show that (for suitably nice schemes) proper push-forward and flat pull-back of algebraic cycles can be interpreted as being induced by the derived inverse and direct image…
We apply the machinery of relative tensor triangular Chow groups to the action of the derived category of quasi-coherent sheaves on a noetherian scheme $X$ on the derived category of quasi-coherent $\mathcal{A}$-modules, where $\mathcal{A}$ is a (not necessarily commutative) quasi-coherent $\mathcal{O}_X$-algebra. When $\mathcal{A}$ is commutative and coherent, we recover the tensor triangular Chow groups of the relative Spec of $\mathcal{A}$. We also obtain concrete descriptions for integral group algebras and hereditary orders over curves, and we investigate the relation of these invariants …
Separable extensions in tensor-triangular geometry and generalized Quillen stratification
本文分析张量三角范畴可分扩张诱导的谱映射，确定其像与纤维基数，并建立直至幂零的弱下降，从而将 Quillen 分层定理推广至等变导出范畴。
On functoriality and the tensor product property in noncommutative tensor-triangular geometry
本文研究幺半三角范畴支撑理论的函子性与张量积性质，引入完全素谱作为万有支撑数据，推广Balmer重构定理至非交换情形，并给出诱导映射的判据及交叉积范畴的谱描述。
A characterization of finite étale morphisms in tensor triangular geometry
本文在张量三角几何中刻画有限 étale 态射，证明其等价于具有保守右伴随、满足 Grothendieck–Neeman 对偶且相对对偶化对象通过典范映射平凡的函子。
We extend the scope of Balmer's tensor triangular Chow groups to compactly generated triangulated categories $\mathcal{K}$ that only admit an action by a compactly-rigidly generated tensor triangulated category $\mathcal{T}$ as opposed to having a compatible monoidal structure themselves. The additional flexibility allows us to recover the Chow groups of a possibly singular algebraic variety $X$ from the homotopy category of quasi-coherent injective sheaves on $X$. We are also able to construct localization sequences associated to restricting to an open subset of $\mathrm{Spc}(\mathcal{T}^c)$,…
We initiate the theory of graded commutative 2-rings, a categorification of graded commutative rings. The goal is to provide a systematic generalization of Paul Balmer's comparison maps between the spectrum of tensor-triangulated categories and the Zariski spectra of their central rings. By applying our constructions, we compute the spectrum of the derived category of perfect complexes over any graded commutative ring, and we associate to every scheme with an ample family of line bundles an embedding into the spectrum of an associated graded commutative 2-ring.
We show that under favorable circumstances, one can construct an intersection product on the Chow groups of a tensor triangulated category $\mathcal{T}$ (as defined by Balmer) which generalizes the usual intersection product on a non-singular algebraic variety. Our construction depends on the choice of an algebraic model for $\mathcal{T}$ (a tensor Frobenius pair), which has to satisfy a $\mathrm{K}$-theoretic regularity condition analogous to the Gersten conjecture from algebraic geometry. In this situation, we are able to prove an analogue of the Bloch formula and use it to define an interse…
Noncommutative tensor triangular geometry: classification via noetherian spectra
该研究针对具有诺特谱的幺半三角范畴，建立了非交换谱的Thomason子集与厚双边半素理想间的保序双射，证明诺特谱为谱空间且具分类相关理想的泛性，为现有分类方法提供了替代方案。
Convexity in tensor triangular geometry
本文对上同调分层的刚性紧生成tt-$\infty$-范畴中的对偶化局部化理想进行分类，证明其对应于Balmer谱的凸子集，推广了Efimov关于交换noetherian环导出范畴的定理。
Triangular tensor networks, pencils of matrices and beyond
本文研究三角图相关的张量网络变种，通过矩阵族的克罗内克不变量进行表征，并探讨了其维度特性及几何条件。
Tensor triangular geometry of filtered objects and sheaves
本文利用∞-范畴与无点方法，通过分配格和半格等工具计算了滤过或分次对象所成张量三角范畴中紧对象的 Balmer 谱，并应用于概形滤过导出范畴与滤过谱同伦范畴。
Triangular spectra and their applications to derived categories of noetherian schemes
本文引入三角谱，从诺特概形的完美导出范畴重构该概形，证明特殊情形下的 Bondal-Orlov-Ballard 重构定理，并比较三角谱与 Balmer 谱。
On the tensor-triangular geometry of isotropic motives
本文研究柔性域上各向同性Tate、Artin及Artin--Tate动机范畴的张量三角几何，证明其Balmer谱和同调谱均为单点集，验证Balmer猜想并研究分层与生成性质。
In this article we initiate the study of the tensor triangular geometry of the categories Mot(k)_a and Mot(k)_l of non-commutative motives (over a base ring k). Since the full computation of the spectrum of Mot(k)_a and Mot(k)_l seems completely out of reach, we provide some information about the spectrum of certain subcategories. More precisely, we show that when k is a finite field (or its algebraic closure) the spectrum of the monogenic cores Core(k)_a and Core(k)_l (i.e. the thick triangulated subcategories generated by the tensor unit) is closely related to the Zariski spectrum of the int…
Stratification in tensor triangular geometry with applications to spectral Mackey functors
本文系统发展张量三角范畴的弱 Noether 分层理论，并借助 Balmer 谱与色度层分析，完全分类有限群上谱 Mackey 函子范畴的局部化张量理想。
We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including the field of algebraic numbers and the algebraic closure of a finite field, we arrive at a complete description of the tensor triangular spectrum and a classification of thick tensor ideals.
The Spectrum of Stable Infinity Categories with Actions
本文引入相对Matsui谱，用于分类厚子模块，结合了Balmer的张量三角谱和Matsui的三角谱，扩展了张量三角几何的应用范围。
In this paper, we compute triangular spectrum (as defined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of $G$-equivariant sheaves on a smooth scheme $X$ for a finite group $G$ which acts on $X$ with some smoothness condition on the quotient. The other class is the derived category of split superschemes.
The tensor triangular geometry of fully faithful functors
该研究证明全忠实几何函子诱导的Balmer谱映射是纤维连通的商映射，引入tt-范畴的集中与幺元化概念，计算相关范畴的Balmer谱并研究比较映射性质，给出理解tt-范畴谱的分层方法。
On the surjectivity of the map of spectra associated to a tensor-triangulated functor
本文研究张量三角函子诱导的谱映射 Spc(F)，刻画其在闭点及整个谱上满射分别与函子保守性、检测特定态射幂零性之间的等价关系。
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