On local cohomology modules over ramified regular local rings

Abstract: In this paper, we show examples of local cohomology modules over ramified regular local ring, having finite set of associated primes. In doing so we consider our ramified regular local ring as Eisenstein extension of an unramified regular local ring sitting inside it and we use the Mayer-Vietoris spectral sequence to show the finiteness of the set of associated primes. In ramified regular local ring for extended ideal (from the unramified one) set of associative primes of a local cohomology module is always finite. Using this result, for non extended ideal, we show example of local cohomology module which has finite set of associative primes and where associative primes also contains prime number $p$.