On the generalized local cohomology and its duality

L e t (R, K ) b e a commutative, Noetherian lo ca l rin g w ith the non-zero multiplicative identity and all the modules considered be unitary throughout. The m ain purpose o f th is p a p e r is to s tu d y t h e generalized local cohomology H 6 (* , * ) introduced by J. Herzog : H 6 (M , N )=lim for R-m odules M and N , 5 (1 . 1 . 1 ). T h is i s in fa c t a generalized one of the usual local cohomology H 6 (* ): fo r any R -m odule N, 1-P"(R, N )= H 6 (N ). A s is w e ll k n o w n , the vanishing (o r non-vanishing) of th e local cohomology module lifin (N ) o f a fin itely g en era ted (a b b revia ted to f. -g . fro m n o w o n ) R -m o d u le N reflects som e important character of N , say dimension and depth o f N . It is quite reasonable to ask when th e generalized local cohomology module H 6 ( M , N ) of R -m odu les M and N vanishes (o r never.)