Let R be a commutative noetherian ring and I an ideal of R. Assume that for all integers i the local cohomology module \({\text {H}}_I^i(R)\) is I-cofinite. Suppose that \(R_\mathfrak {p}\) is a regular local ring for all prime ideals \(\mathfrak {p}\) that do not contain I. In this paper, we prove that if the I-cofinite modules form an abelian category, then for all finitely generated R-modules M and all integers i, the local cohomology module \({\text {H}}_I^i(M)\) is I-cofinite.

中文翻译：

---

局部上同调模和模子类的共有限性

设R为交换诺特环，I为R的理想。假设对于所有整数i，局部上同调模\({\text {H}}_I^i(R)\)是I -余有限。假设\(R_\mathfrak {p}\)是所有不包含I的素理想\(\mathfrak {p}\)的正则局部环。在本文中，我们证明如果I -余有限模形成交换范畴，则对于所有有限生成的R -模M和所有整数i，局部上同调模\({\text {H}}_I^i(M )\)是I -余有限。