Cohomology of Arbitrary Spin Currents in $AdS_3$

We study conserved currents of any integer or half integer spin built from massless scalar and spinor fields in $AdS_3$. 2-forms dual to the conserved currents in $AdS_3$ are shown to be exact in the class of infinite expansions in higher derivatives of the matter fields with the coefficients containing inverse powers of the cosmological constant. This property has no analog in the flat space and may be related to the holography of the AdS spaces. `Improvements' to the physical currents are described as the trivial local current cohomology class. A complex of spin $s$ currents $(T^s, {\cal D})$ is defined and the cohomology group $H^1(T^s, {\cal D}) = {\bf C}^{2s+1}$ is found. This paper is an extended version of hep-th/9906149.