On the attached primes of top local cohomology modules

Abstract

Let a be an ideal of Noetherian ring R and M a finitely generated R-module such that cd ( a , M ) = c . In this paper, we investigate Att R ( H a c ( M ) ) . Among other things, it is shown that Max { p ∈ Supp R M | cd ( a , R / p ) = c } ⊆ Att R ( H a c ( M ) ) . We also show that Att R ( H a c ( M ) ) = { p ∈ Supp R M | cd ( a , R / p ) = c , p = Ann R ( H a c ( R / p ) ) } and { p ∈ Supp R M | cd ( a , R / p ) = c , dim R / p − 1 ≤ cd ( a , R / p ) ≤ dim R / p } ⊆ Att R ( H a c ( M ) ) . Finally, we prove that if ( R , m ) is a local ring and dim R / a = 1 then Att R ( H a c ( M ) ) = { p ∈ Supp R M | cd ( a , R / p ) = cd ( a , M ) } . Then by using this, it is shown that if ( R , m ) is a local ring then { p ∈ Supp R M | cd ( a , R / p ) = c , dim R / ( a + p ) = 1 } ⊆ Att R ( H a c ( M ) ) .