Hopf-Galois扩张的Gorenstein性质和表示不变量的研究

本项目把Hopf代数理论与Gorenstein 同调代数和代数表示论相结合,以Hopf-Galois扩张为突破口,利用同调和表示手段来研究它的Gorenstein性质和表示不变量.设A/B为Hopf代数H上的Hopf-Galois扩张,本项目主要研究的问题有:(1) 利用伴随对,Morita 等价,导出范畴和对偶等工具得到扩张的一些Gorenstein性质,并通过构造Hopf-Galois扩张关于扩张函子的谱序列来给出A、B和H的整体维数之间的关系; (2) 通过比较代数A和子代数B的有限生成模投射维数的关系,来给出有限维数为扩张下的不变量的一些充分条件; (3) 通过研究Hopf-Galois扩张的整体Gorenstein投射维数, 整体Gorenstein内射维数,整体维数,弱维数,表示维数和有限维数等之间的关系, 得到一些同调不变量和表示不变量及更多的Gorenstein同调性质.

In this project, combining the theory of Hopf algebras with Gorenstein homological algebra and representation theory of algebras, via Hopf-Galois extensions, using homological and representative methods, we will study a series of Gorenstein properties and invariant properties of representations under Hopf-Galois extensions. Let A/B be a Hopf-Galois extension over a Hopf algebra H. Then the project is organized as follows: .(1) Using the tools of adjoint pairs, Morita equivalent, derived category and duality, we obtain some Gorenstein properties of Hopf-Galois extensions A/B. And the spectral sequence for Ext will be constructed which yields an estimate for the global dimension of A in terms of the corresponding data for H and B..(2) By comparing the projective dimensions of finitely generated modules over the algebra A and its subalgebra B, we give some sufficient conditions making the finitistic dimension become an invariant under Hopf-Galois extensions..(3) The relationship of global Gorenstein projective dimension, global Gorenstein injective dimension, global dimension, weak dimension, representation dimension and finitistic dimension under the Hopf-Galois extension A/B will be studied. Furthermore, some homological invariants, invariant properties of representations and more Gorenstein homological properties will be obtained.