d-Transform Functor and Some Finiteness and Isomorphism Results

ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES

“…The functor Γ d (−) was originally defined in[2] and the modules H i d (M ) were called d-local cohomology modules associated to M were studied in[18,19]. After some preliminary results in Section 2, for an R-module M and an integer t we prove thatAss(H t d (M )) = I∈Φ Ass(Ext t R (R/I, M )) = I∈Φ Ass(H t I (M )), where Φ = {I : I is an ideal of R with dim R/I ≤ d} and H i d (M ) = 0 for all i < t. In Section 3, we shall provide some results concerning the vanishing and non-vanishing of d-local cohomology modules: we shall prove that, over a local ring R, if the non-zero finitely generated R-module M has (Krull) dimension n, then there exists an integer i with 0 ≤ i ≤ d such that H n−i d (M ) = 0.…”