单态射范畴及其在代数和几何中的应用

最近研究表明子模范畴可描述G-投射模和奇点。对有向图Q和范畴C，本项目拟系统研究一般的单态射范畴Mon(Q,C)及其在代数和几何中的应用。当C取代数A的模范畴且Q取A_2型图时，它就是子模范畴；一般地它是Q的路代数与A的张量代数的模范畴的一个性质奇特的全子范畴。.本项目拟研究单态射范畴的函子有限性、Frobenius性、相对AR理论、和其对称拆合；利用它和其它条件确定相应的张量代数的G-投射模；通过它的稳定三角范畴描述相应代数的奇点及其类型、范畴化消解或形式消解奇点；由单态射范畴构造分数维数的Calabi-Yau范畴；建立模稳定范畴、G-投射模稳定范畴和G-稳定范畴以及3种稳定等价之间的关系。.这些研究对进一步揭示G-同调代数、表示论、奇点理论、稳定三角和CY范畴之间的联系，具有重要科学意义。由于单态射范畴的明确可构性，这些联系为在相关方向取得新成果，将提供独特视野和有力工具。

The recent studies hint that submodule categories possibly describe Gorenstein-projective modules and singularities. For a given quiver Q and category C, this project will systematically study the monomorphism category Mon(Q,C) with applications in algebra and geometry. If C is the module category of an algebra A and Q is the quiver of type A_2, then Mon(Q,A) is just the submodule category of A. In general, Mon(Q,A) is a full subcategory of deep properties in the module category of the tensor product of the path algebra of Q and A...This project will study the functorially finiteness、the Frobenius property, the relative Auslander-Reiten theory, and the symmetric recollements, of monomorphism categories. We will explicitly determine the category of the Gorenstein-projective modules of the corresponding tensor algebra, via the monomorphism category together with other conditions; describe the singularity cateogry of the corresponding algebra via the stable category of monomorphism category, and categorically resolve or formally resolve the singularities. We will also construct Calabi-Yau categories of fractional dimension from monomorphism category; and establish the possible relations among the three kinds of stable equivalences,arsing from the module stable category, the stable category of G-projective modules, and the G-stable category. ..These researches will hopefully be of importance to understand and explore the relations among Gorenstein homological algebra, representation theory, singularity theory, stable triangulated categories, and Calabi-Yau categories. Since the monomorphism categories have explicit constructions, these relations will provide a peculiar insight and a powerful tool for obtaining new results in the fields mentioned above.