Resumen de On a question of Hartshorne

Kamal Bahmanpour

• 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.