On invariants and endomorphism rings of certain local cohomology modules

Let (R,m) denote an n-dimensional Gorenstein ring. For an ideal I⊂R with we define new numerical invariants τi,j(I) as the socle dimensions of . In case of a regular local ring containing a field these numbers coincide with the Lyubeznik numbers λi,j(R/I). We use τd,d(I), , to characterize the surjectivity of the natural homomorphism . As a technical tool we study several natural homomorphisms. Moreover we prove a few results on τi,j(I).