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Theorem 1 follows from Theorem 2, applied to the symmetric algebra S(M). Call a commutative j£-algebra A invertible if, for some £-algebra B, A <®K B is a polynomial algebra K[XX, . . . , Xn]. Then A admits an augmentation, 0 —• A —> A —> K —• 0, and the AT-module J A = A/A depends, up to isomorphism, only on A. We say A is stably isomorphic to a J^-algebra B if A ®K C = B <8>K C for some invertible AT-algebra C

Locally polynomial algebras are symmetric