  Let R be a commutative Noetherian d-dimensional complete equicharacterisitc
regular local ring and let I be an ideal of R such that every minimal prime
over I has height at most c. Let v=d - [(d-2)/c]-1 and v'=d - [(d-1)/c]. It has
been known that the i-th local cohomology module of any R-module M with support
in I vanishes for i>v', and if I is prime, for i>v; both results are sharp. The
purpose of this paper is to prove a necessary and sufficient condition, for a
not necessarily prime I, for the vanishing in the range i>v, i.e. in the same
range as for a prime ideal. The condition is in terms of some combinatorial
properties of the set of the minimal primes of I whose sum is zero-dimensional.
A version for non-regular local rings is also proven.
