Mathematical Sciences: Torsionfree Modules over Affine Coordinate Rings

这项研究将考虑仿射代数簇的坐标环上的有限生成的无挠模，特别是曲面和三折叠。主要研究人员将获得这些模块的直和抵消理论，并将其应用于特定的抵消问题。这将通过将仅具有有理双点的曲面上的所有不可分解的有限生成的扭自由模分类为奇点，并通过获得这些模的直和的不变量来实现。这项研究属于交换代数的一般领域，主要研究模的结构理论和消去问题。这项工作是经典模理论和几何技术的有趣结合。将要研究的问题是深刻而重要的。这个项目的结果将会引起数学界的兴趣。

This research will consider finitely generated torsionfree modules over coordinate rings of affine algebraic varieties, particularly surfaces and three-folds. The principal investigator will obtain a theory of direct-sum cancellation for these modules and will apply it to specific cancellation problems. This will be accomplished by classifying all indecomposable finitely generated torsionfree modules over surfaces with only rational double points as singularities and by obtaining invariants for direct sums of these modules. This research is in the general area of commutative algebra and is concerned with the structure theory of modules and cancellation problems. This work is an interesting combination of techniques from classical module theory and geometry. The questions that will be examined are deep and important. The outcome of this project will be of interest to the mathematical community.