On a question of Hartshorne

Abstract

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.