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Let x1, …, xd be a system of parameters for R, let F• be a free R-resolution of the residue field of R with F0 = R, and let K• denote the Koszul complex of R with respect to x1, …, xd. Lift the identity map R = K0 → F0 = R to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from R = Kd → Fd is not 0.
Any questions?
Any questions?