奇点范畴张量三角几何的研究

The study of singularity category draws a lot of attention in recent years. For a ring/scheme, its singularity category is defined to be the Verdier quotient of the bounded derived category of coherent sheaves by the subcategory of perfect complexes. It’s clear that the singularity category is trivial if and only if the underlying ring/scheme is smooth, however, how exactly the singularities are characterized requires further study. In an earlier paper, we show that the singular locus of a hypersurface ring can be reconstructed from its singularity category via the theory of tensor triangular geometry. In this project, we will focus on the reconstruction problem of general rings/schemes. First, consider the case of a complete intersection. We will study and develop the theory of its corresponding higher matrix factorizations category, then apply the theory of tensor triangular geometry to obtain the reconstruction of singular locus. Second, we will take the study further by considering the injective/projective stable derived category (these categories carry similar structures like matrix factorizations category) and the bounded derived category (it has well-established tensor product and support theory and we hope to bring it down to the singularity category) and obtain some reconstruction type results for more general ring/scheme. Finally, for an abstract singularity category, use (higher) comparison maps to compare its triangular spectrum to the singular locus of some concrete rings. This will give a somewhat direct explanation of the abstract triangular spectrum and find out what the theory of tensor triangular geometry can offer for singularity category in general.

奇点范畴是近年的一个研究热点,这类范畴反映了环与概形本身的奇异性,并在代数几何、表示论、数学物理上取得了重要的突破。然而,奇点范畴与奇点间的具体联系尚有待进一步的研究。申请人在前期研究中发现,借助张量三角几何理论,可以得到超曲面环奇点范畴的直接几何描述,即超曲面奇点范畴的三角谱与其奇点集合同胚,这是本项目的出发点。本项目计划继续深入上述研究。首先,考虑完全交（complete intersection）的情形,完善高阶矩阵分解（matrix factorization）范畴理论,建立其与奇点范畴的联系,得到该情形的重构；进一步,对内射/投射稳定导出范畴进行类似研究,推广应用导出范畴的支集理论,将重构结果推广到更一般的环与概形奇点范畴。最后,对抽象的奇点范畴,应用对比映射理论,将其三角谱与某些具体的环的奇点集合相关联,初步解释抽象的奇异性,为应用奇点范畴研究奇异性提供更多理论依据。

奇点范畴是近年的一个研究热点，本项目围绕奇点范畴进行了一些研究：1. 对于 complete intersection 的奇点范畴，给出了其三角几何的一个描述；2. 通过对阿贝尔范畴 support theory 的系统研究，给出了奇点范畴的一个推广，并证明了 Orlov 和 Chen 的一个结果对这种推广的奇点范畴也成立；3. 考虑到奇点范畴和 Gorenstein 投射模稳定范畴的关系，亦研究并推广了Gorenstein同调代数的一些结果，包括模型范畴和余绕对以及同调维数的关系；4. 研究了一般的带有某些特殊性质序列的范畴（如阿贝尔/exact范畴、三角范畴）的 Grothendieck 群的性质，并给出了此类范畴某些子范畴的一个分类结果。