Abstract: We consider and construct a Grothendieck spectral sequence on the Ext groups to reveal the cotorsion dimensions of weak HopfGalois extensions. Let H be a finite dimensional weak Hopf algebra and A/B  a weak H-Galois extension, we give the relationship between the left cotor
sion dimensions of A,B and the right global dimensions of H, and discuss the properties of the left cotorsion dimensions of A,B when B is commutative or H* is semisimple.

Key words: weak Hopf Galois extension, weak Hopf comoudle algebra, cotorsion dimension; , weak crossed product