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This is to say, $\Lambda$ is a direct limit of the $\Lambda_N = R[x_1,\dots,x_N]^{S_N}$. As this limit is characterized by a universal mapping property, there is no choice in the definition of $\Lambda$.
thanks for catching this. in the $K$-theory of varieties $K^0(X) \to K^0(X-Y)$ is surjective (and the kernel is the image of $K^0(Y)$), whence my confusion.
