SpecZ下的Grothendieck对偶

A Grothendieck-Witt space for stable infinity categories with duality

We construct a Grothendieck-Witt space for any stable infinity category with duality. If we apply our construction to perfect complexes over a commutative ring in which 2 is invertible we recover the classical Grothendieck-Witt space. Our Grothendieck-Witt space is a grouplike E-infinity space which is part of a genuine C_2-spectrum, the connective real K-theory spectrum.

Grothendieck-Neeman duality and the Wirthmüller isomorphism

We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk-Hu-May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: There exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothen…

Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves

Let $\mathbb{X}$ be a semiseparated Noetherian scheme with a dualizing complex $D$. We lift some well-known triangulated equivalences associated with Grothendieck duality to Quillen equivalences of model categories. In the process we are able to show that the Gorenstein flat model structure, on the category of quasi-coherent sheaves on $\mathbb{X}$, is Quillen equivalent to the Gorenstein injective model structure. Also noteworthy is that we extend the recollement of Krause to hold without the Noetherian condition. Using a set of flat generators, it holds for any quasi-compact semiseparated sc…

循环配边小圆盘代数、Grothendieck-Verdier对偶与柄体群表示

Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations

该研究刻画对称幺半双范畴中配边小圆盘operad的循环代数，证明其与两类Grothendieck-Verdier范畴等价，进而得到量子拓扑中柄体群表示推广与无半单假设的Grothendieck-Verdier对偶两项应用。

Local duality in algebra and topology

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable $\infty$-category $\mathcal{C}$ together with a collection of compact objects $\mathcal{K} \subset \mathcal{C}$ we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring $A$ and a suitable ideal in $A$, we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More…

Drinfeld 中心中作为 ribbon 对偶对象的特异可逆对象

The distinguished invertible object as ribbon dualizing object in the Drinfeld center

本文证明 pivotal 有限张量范畴的 Drinfeld 中心具有唯一 ribbon Grothendieck-Verdier 结构，其对偶对象为特异可逆对象，并由此得到 ansular 函子甚至 modular 函子。