Abstract
Let R be a Gorenstein local ring and I be an ideal of R. Denote by H(I,R) the assertion: ''The socle of the local cohomology module H_I^n(R) is finitely generated for each n''.
We translate H(I,R) into a property of the minimal injective coresolution of R, and then use this translation to prove H(I,R) for all regular local rings R and all ideals I with level prime avoidance.