箭图Frobenius范畴、Gorenstein投射模和例外圈及其在表示论中的应用

本项目研究箭图Frobenius范畴，Gorenstein投射模，例外圈，及其应用。.对关系箭图(Q,I)和代数A的模范畴的任意子范畴X，本项目拟从组合数据引入双分单态射范畴Smon(Q,I,X)，建立它和其对偶之间的RSS等价；以此对称性，构造箭图Frobenius范畴。以箭图Frobenius范畴为工具，拟研究Gorenstein投射模中独立性公开问题；引入并研究更广的弱Gorenstein代数。确定箭图的导出范畴和丛范畴的球对象和例外圈，寻找其长度型与箭图的管状型和Coxeter群之间的关系。研究Frobenius范畴的稳定范畴的球对象、例外圈、自同构群、倾斜倾斜和丛倾斜对象。.本项目以箭图Frobenius范畴为纽带，对深入理解Frobenius范畴和三角范畴的多样性，以及它在表示论、Gorenstein同调代数、Coxeter群和代数几何中的应用，具有重要研究价值和科学意义。

This project is to study quiver Frobenius categories, Gorenstein-projective modules, exceptional cycles, with applications in the representation theory of algebras..For an arbitrary bound quiver (Q,I), an arbitrary algebra A over a field, and an arbitrary full subcategory X of the module category A-mod, we tends to introduce separated and graded monomorphism category Smon(Q,I,X), by using the quiver combinatorical datum, and to establish the Ringel-Schmidmeier-Simson (RSS) equivalence between Smon(Q,I,X) and its dual Sepi(Q,I,X). By using the symmetry of this RSS equivalence, we will construct the so-called quiver Frobenius categories. With the help of the quiver Frobenius categories, it is hopeful to solve an open problem of Independence in the definition of Gorenstein-projective modules. We will introduce weakly Gorenstein algebras, which contain Gorenstein algebras as a subclass, and we will give various descriptions of weakly Gorenstein algebras. We will determine all the spherical objects and exceptional cycles in the bounded derived category and in the cluster category, of a quiver, and explore their relations with the tubular type when the quiver is Euclidean, and with the Coxeter groups when the quiver is of Dynkin type. Finally, we will study spherical objects, exceptional cycles, the autoequivalence group, tilting objects and cluster tilting objects, of the stable categories of quiver Frobenius categories..Link quiver Frobenius category as a tie, this project should to be of research interest and scientific significance, in understanding the diversity of Frobenius categories and triangulated categories, and their applications in representation theory, Gorenstein homological algebra, the Coxeter group and algebraic geometry.