  Let K be a global function field of positive characteristic p and let M be a
(commutative) finite and flat K-group scheme. We show that the kernel of the
canonical localization map H^{1}(K,M)\to\prod_{all v}H^{1}(K_{v},M) in flat
(fppf) cohomology can be computed solely in terms of Galois cohomology. We then
give applications to the case where M is the kernel of multiplication by p^{m}
on an abelian variety defined over K.
