Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules \(H^i_I(M)\) are I-cofinite, for all finitely generated R-modules M and all \(i\in {\mathbb {N}}_0\), if and only if the R-modules \(H^i_I(R)\) are I-cofinite with dimension not exceeding 1, for all integers \(i\ge 2\); in addition, under these equivalent conditions it is shown that, for each minimal prime ideal \({{\mathfrak {p}}}\) over I, either \({{\text {height}}}{{\mathfrak {p}}}\le 1\) or \(\dim R/{{\mathfrak {p}}}\le 1\), and the prime spectrum of the I-transform R-algebra \(D_I(R)\) equipped with its Zariski topology is Noetherian. Also, by constructing an example we show that under the same equivalent conditions in general the ring \(D_I(R)\) need not be Noetherian. Furthermore, in the case that R is a local ring, it is shown that the R-modules \(H^i_I(M)\) are I-cofinite, for all finitely generated R-modules M and all \(i\in {\mathbb {N}}_0\), if and only if for each minimal prime ideal \({\mathfrak {P}}\) of \({\widehat{R}}\), either \(\dim {\widehat{R}}/(I{\widehat{R}}+{\mathfrak {P}})\le 1\) or \(H^i_{I{\widehat{R}}}({\widehat{R}}/{\mathfrak {P}})=0\), for all integers \(i\ge 2\). Finally, it is shown that if R is a semi-local ring and the R-modules \(H^i_I(M)\) are I-cofinite, for all finitely generated R-modules M and all \(i\in {\mathbb {N}}_0\), then the category of all I-cofinite modules forms an Abelian subcategory of the category of all R-modules.

让我成为交换Noether环R的理想选择。对于所有有限生成的R模块M和所有\ {i \ in {\ mathbb {N}} _ 0 \）来说，表明R-模块\（H ^ i_I（M）\）是I-有限的，如果对于所有整数\（i \ ge 2 \），并且仅当R -modules \（H ^ i_I（R）\）为I-整数且维数不超过1时；另外，这些等同的条件下，示出了，对于每一个最小素理想\（{{\ mathfrak {P}}} \）超过予，无论是\（{{\ text {height}}} {{\ mathfrak {p}}} \ le 1 \）或\（\ dim R / {{\ mathfrak {p}}} \ le 1 \）以及素数的频谱中的我-transform ř代数\（D_I（R）\）配备有其Zariski拓扑是诺特。而且，通过构造一个示例，我们表明在相同的等效条件下，环\（D_I（R）\）通常不需要为Noetherian。此外，在R是局部环的情况下，对于所有有限生成的R模块M和所有\（i \ in ），表明R-模块\（H ^ i_I（M）\）是I- cofinite 。 {\ mathbb {N}} _ 0 \），当且仅当对于每个最小素理想\（{\ mathfrak {P}} \）的\（{\ widehat {R}} \），无论是\（\暗淡{\ widehat {R}} /（I { \ widehat {R}} + {\ mathfrak {P}}）\ le 1 \）或\（H ^ i_ {I {\ widehat {R}}}（{\ widehat {R}} / {\ mathfrak {P }} = 0 = 0），所有整数\（i \ ge 2 \）。最后，表明，对于所有有限生成的R模块M和所有\（i \ in {，，如果R是一个半局部环并且R-模块\（H ^ i_I（M）\）是I-有限的， \ mathbb {N}} _ 0 \），然后是所有我的类别-cofinite模块形成所有R -modules类别的Abelian子类别。