On a Dade's conjecture

In [4], it was given an affirmative answer to Dade's conjecture: If G is a finite group and the 1-component R₁ of a G-graded ring R has finite block theory, then R has finite block theory. In this article, we will prove the same assertion in a more general context: G is an arbitrary group and R is a graded ring with the finite support. By [3], when G is an F E-group, the block theory of finitely supported gradings can be reduced to the block theory of finite group gradings. But in general, because there are non-F E-groups (cf. [3; Example 1.5]), the theory of finitely supported gradings cannot be included in the theory of finite group gradings. As by passing to the ring of fractions of a graded ring with the finite support with respect to a multiplicative system S ⊂ R₁ ∩ Z(R) we obtain a graded ring with the finite support, we may take over a part of the technique in [4].