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.