可分Frobenius扩张下的Gorenstein余挠维数

Gorenstein Cotorsion Dimension under Separable Frobenius Extensions

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。

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