Gorenstein平坦模与Frobenius扩张

设R■A是环的Frobenius扩张,其中A是右凝聚环,M是任意左A-模.首先证明了_AM是Gorenstein平坦模当且仅当M作为左R-模也是Gorenstein平坦模.其次,证明了Nakayama和Tsuzuku关于平坦维数沿着Frobenius扩张的传递性定理的"Gorenstein版本":若_AM具有有限Gorenstein平坦维数,则GfdA(M)=GfdR(M).此外,证明了若R■S是可分Frobenius扩张,则任意A-模(不一定具有有限Gorenstein平坦维数),其Gorenstein平坦维数沿着该环扩张是不变的.

Gorenstein Flat Modules and Frobenius Extensions

Let R⊂A be a Frobenius extension of rings, where A is right coherent. Let M be any left A-module. We first show that _AM is Gorenstein flat if and only if the underlying R-module _RM is Gorenstein flat. Then we prove a "Gorenstein version" of Nakayama and Tsuzuku's theorem on transfer of flat dimensions along Frobenius extensions:if _AM has finite Gorenstein flat dimension, then Gfd_A(M)=Gfd_R(M). Moreover, it is proved that if R ⊂ S is a separable Frobenius extension, then for any A-module (not necessarily of finite Gorenstein flat dimension), its Gorenstein flat dimension is invariant along such ring extension.