可分 Frobenius 扩张下的 Gorenstein 余挠维数

针对环变化下的 Gorenstein 同调性质,提出模的 Gorenstein 余挠性质及相应维数在环的可分 Frobenius 扩张下的保持性质。 首先证明对可分 Frobenius 扩张 R→S,S-模 M 是 Gorenstein 余挠模当且仅当 M 是 Gorenstein 余挠的R-模,从而可得模的 Gorenstein 余挠维数沿着该环扩张保持不变; 作为应用,证明了若该环扩张是可裂的,则环的整体 Gorenstein 余挠维数也保持不变;此外,讨论了群环上的 Gorenstein 余挠维数,进一步验证 Gorenstein 余挠维数在环的可分 Frobenius 扩张下的不变性。

Gorenstein Cotorsion Dimension under Separable Frobenius Extensions

For the Gorenstein homological properties under changes of rings the Gorenstein cotorsion property of modulesand the preserving property of the corresponding dimension under a separable Frobenius extension of the ring wereproposed. It was first proved that for a separable Frobenius extension R→S the S-module M was a Gorenstein cotorsionmodule if and only if M was an R-module of a Gorenstein cotorsion module and thus the Gorenstein cotorsion dimension ofmodules remain invariant along such ring extension. As an application it was shown that if this ring extension wassplittable the overall Gorenstein cotorsion dimension of the ring remained invariant as well. In addition the Gorensteincotorsion dimensions over group rings were discussed and the invariance of the Gorenstein cotorsion dimensions under theseparable Frobenius extensions was further verified