正合范畴的上同调与（非）可加层

Structured spaces: categories, sheaves, bundles, schemes and cohomology theories

In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is to define categories of structured spaces, (pre)sheaves with values in such categories, and generalised notions of vector bundles, ringed spaces and schemes. Then, we will construct (using some techniques and also the general method in Section 2 of [2]) various cohomology theories for these objects.

$\mathbb A^1$-不变平展层的驯顺与曲线驯顺上同调

Tame and curve-tame cohomology of $\mathbb A^1$-invariant étale sheaves

本文在附加假设下将非分歧曲线驯顺上同调推广至$\mathbb{A}^1$-不变平展层，定义其与苏斯林同调的配对，并证明正特征域上光滑几何连通簇的该上同调等于基域上同调。

A construction of intersection cohomology from a simplicial version of the Deligne axioms

Intersection cohomology is a way to enhance classical cohomology, allowing us to use a famous result called Poincaré duality on a large class of spaces known as stratified pseudomanifolds. There is a theoretically powerful way to arrive at intersection cohomology by classifying sheaves that satisfy what are called Deligne axioms. We stablish an abstract manifestation of the Deligne axioms, to then apply it on a simplicial complex environment, for a category of simplicial sheaves inspired on the works of D. Chataur, D. Tanré and M. Saralegi-Araguren. For a stablished topology on a triangulation…

作为谱层的微分上同调理论

Differential cohomology theories as sheaves of spectra

我们证明了，在一个稳定的（无穷大，1）-范畴（如谱或链复形）的值的光滑流形的网站上的每一个层产生一个微分上同调图和同伦公式，这是微分上同调理论的所有经典例子的共同特征。这些结构自然是从一个规范分解成一个同伦不变量的一部分和一块，其中有一个平凡的评价点的层。在经典的例子中，后者是微分形式的贡献。这种分解提出了一种自然的方案，通过确定这些碎片和胶合数据来分析新的层。我们进行这种分析的各种经典和不那么经典的例子。

带模的动机，II：真模对的带转移模层

Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs

本研究构建了真模对范畴上的层与上同调理论，补充了此前[KMSY21]提出的非真模对范畴的相关理论，完善了带模动机的理论体系。

通过叠的等变广义上同调

Equivariant generalized cohomology via stacks

本文证明商叠上同调可由Borel构造计算，并推广至lisse扩张，进而证明高阶等变Grothendieck-Riemann-Roch定理，并给出等变层无穷范畴的Bernstein-Lunts型粘合描述。