We give a new proof of some characteristic-free fundamental theorems in invariant theory first proved in C. De Concini and C. Procesi, A characteristic free approach to invariant theory, Adv. Math. 21 (1976), 330-354. We treat the action of the general linear group and the symplectic group. Our approach is geometric, and utilizes the fact that the categorical quotients are principal fiber bundles off codimension two or more.
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