Let \((R,\mathfrak {m},k)\) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen–Macaulay R-module \(B_M\) such that the socle of \(B_M\otimes _RM\) is zero. When R is a quasi-specialization of a \({\text {G}}\)-regular local ring or when M has complete intersection dimension zero, we show the existence of such a big Cohen–Macaulay R-module. It is conjectured that if R admits a non-zero Cohen–Macaulay module of finite Gorenstein dimension, then it is Cohen–Macaulay. We prove this conjecture if either R is a quasi-specialization of a \({\text {G}}\)-regular local ring or a quasi-Buchsbaum local ring.

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关于完全自反模块的新交定理

令\（（R，\ mathfrak {m}，k）\）为本地环。我们建立了新相交定理的完全自反类比，为每个完全自反R模块M提供一个大Cohen–Macaulay R模块\（B_M \）使得\（B_M \ otimes _RM \）的唯一是零。当R是\（{\ text {G}} \） -规则局部环的准专业化时，或者当M具有完整的交集维数为零时，我们表明存在这样大的Cohen–Macaulay R-模块。据推测，如果R接受有限Gorenstein维数的非零Cohen–Macaulay模块，则为Cohen–Macaulay。如果R是\（{\ text {G}} \） -规则局部环的准专业化或拟Buchsbaum局部环的准专业化，我们证明了这一猜想。