  We define currents on a Grassmann algebra $Gr(N)$ with $N$ generators as
distributions on its exterior algebra (using the symmetric wedge product). We
interpret the currents in terms of ${\Z}_2$-graded Hochschild cohomology and
closed currents in terms of cyclic cocycles (they are particular multilinear
forms on $Gr(N)$). An explicit construction of the vector space of closed
currents of degree $p$ on $Gr(N)$ is given by using Berezin integration.
