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On the new intersection theorem for totally reflexive modules
Collectanea Mathematica ( IF 1.1 ) Pub Date : 2019-09-26 , DOI: 10.1007/s13348-019-00264-3 Kamran Divaani-Aazar , Fatemeh Mohammadi Aghjeh Mashhad , Ehsan Tavanfar , Massoud Tousi
Collectanea Mathematica ( IF 1.1 ) Pub Date : 2019-09-26 , DOI: 10.1007/s13348-019-00264-3 Kamran Divaani-Aazar , Fatemeh Mohammadi Aghjeh Mashhad , Ehsan Tavanfar , Massoud Tousi
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.
中文翻译:
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。
更新日期:2019-09-26
中文翻译:
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。