Cofiniteness of Local Cohomology Modules Over Regular Local Rings

It is well known that for a Noetherian ring R, an ideal I of R, and M a finitely generated R-module, the local cohomology modules H l i ( M ) are not always finitely generated. On the other hand, if R is local and m is its maximal ideal, then H m i ( M ) are Artinian modules, which is equivalent to the following two properties: (i) Supp R ( H m i ( M ) ) ⊆ { m } ; (ii) the vector space H o m R ( k , H m i ( M ) ) has finite dimension over k, where k = R/m. Taking these facts into account, Grothendieck [9] made the following conjecture. 1991 Mathematics Subject Classification 13H05, 13H10, 13D45.