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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 associatedprimes of the local cohomology modules over $R$ are finite. Moreover, if $A$has dimension one and $\pi$ is an nonzero divisor, then the same propertieshold 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 thefiniteness properties of local cohomology over unramified regular local rings.In addition, we extend previous results on the injective dimension of localcohomology modules over certain regular rings of mixed characteristic.

Local cohomology modules of polynomial or power series rings over rings of small dimension