  Let $A$ be a ring and $R$ be a polynomial or a power series ring over $A$.
When $A$ has dimension zero, we show that the Bass numbers and the associated
primes of the local cohomology modules over $R$ are finite. Moreover, if $A$
has dimension one and $\pi$ is an nonzero divisor, then the same properties
hold for prime ideals that contain $\pi.$ These results do not require that $A$
contains a field. As a consequence, we give a different proof for the
finiteness properties of local cohomology over unramified regular local rings.
In addition, we extend previous results on the injective dimension of local
cohomology modules over certain regular rings of mixed characteristic.
