关于Gorenstein投射复形的进一步研究

Further study on Gorenstein projective complexes

Enochs E和Garcia Rozas J R在"Gorenstein Injective and Projective Complexes"一文中证明了在n-Gorenstein环R上,若左R-模复形C为Gorenstein投射复形当且仅当它的每一项左R-模Cm为Gorenstein投射模。弱化了此结论的必要性条件,得到在任意环R上,若左R-模复形C为Gorenstein投射复形,则它的每一项左R-模Cm为Gorenstein投射模。并且最后给出Gorenstein投射复形C与任意投射复形上合冲L的关系,即Exti(C,L)=0。

In paper of Gorenstien injective and projective complexes,Enochs E and Garcia Rozas J R show that over n-Gorenstein ring R,a complex C of left R-module is Gorenstein projective if and only if Cm is Gorenstein projective in left R-module for allm∈Z.In this paper,we weaken the necessary conditions for this conclusion and prove that over an arbitrary ring R,a complex C of left R-module is Gorenstein projective,then Cm is Gorenstein projective in left R-module for all m∈Z.At last,we show the relationship between Gorenstein projective complex C and the cosyzygy of projective complex L,i.e.Exti（C,L）=0.