AI 中文总结
该文研究虚Gorenstein环上的Cartan-Eilenberg复形范畴，证明其C-E正合结构的性质，给出C-E-容许平衡对的刻画，建立Tate上同调相关的C-E类似正合序列与平衡结果。
AI 中文摘要
设R为环，Ch(R)为左R-模的复形范畴。我们考虑Ch(R)上的C-E（Cartan-Eilenberg）正合结构，证明其是一个有效的正合范畴。我们证明：若R是左虚Gorenstein环，则(C-E(GProj),C-E(GInj))是C-E-容许平衡对；在适当附加假设下，逆命题也成立。我们还研究Tate上同调的C-E版本，建立了Avramov-Martsinkovsky正合序列的C-E类似物，以及Ext和Tor的平衡结果。
英文摘要
Let R be a ring and let Ch(R) be the category of complexes of left R-modules. We consider the C-E (Cartan-Eilenberg) exact structure on Ch(R) and show that it is an efficient exact category. We prove that if R is a left virtually Gorenstein ring, the pair (C-E(GProj),C-E(GInj)) is a C-E-admissible balanced pair. Under suitable additional hypotheses, the converse holds. We also consider the C-E version of Tate cohomology and establish C-E analogues of the Avramov-Martsinkovsky exact sequences and balance results for both Ext and Tor.
