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Example of non injective module over Noetherian local ring with trivial vanishing against residue field?

If you take $R=k[[x,y]]$ and $M=k[[x]][x^{-1}]$, this should give you an example. $M$ is not injective because there is a non-split injective map $M \to R[x^{-1},y^{-1}]/yR[x^{-1}]$. But if you ...
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