Cohomologies of the Poisson superalgebra

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $\mathbb{R}^{2n}$ are investigated under suitable continuity restrictions on the cochains. The first and second cohomology spaces in the trivial representation and the zeroth and first cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a constant nondegenerate Poisson superbracket or arbitrary $n>0$. The third cohomology space in the trivial representation and the second cohomology space in the adjoint representation of this superalgebra are found for arbitrary $n>1$.