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I see now that the correct way was to go with the strict Henselization, not k((t)). As for $R^0$ what you suggest is exactly how I deduceded the isomorphism. Thanks Will.
Sorry for possible confusion. If $M$ is my module then $\overline k[M]$ is the group ring (probably I shouldn't have used word "algebra"). Then $T = Spec(\overline k[M]^G)$ is the torus I want to compute.
