  In this work, the cohomology theory for partial actions of co-commutative
Hopf algebras over commutative algebras is formulated. This theory generalizes
the cohomology theory for Hopf algebras introduced by Sweedler and the
cohomology theory for partial group actions, introduced by Dokuchaev and
Khrypchenko. Some nontrivial examples, not coming from groups are constructed.
Given a partial action of a co-commutative Hopf algebra $H$ over a commutative
algebra $A$, we prove that there exists a new Hopf algebra $\widetilde{A}$,
over a commutative ring $E(A)$, upon which $H$ still acts partially and which
gives rise to the same cochain complex as the original algebra $A$. We also
study the partially cleft extensions of commutative algebras by partial actions
of cocommutative Hopf algebras and prove that these partially cleft extensions
can be viewed as a cleft extensions by Hopf algebroids.
