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Mar 11, 2017 at 3:03 comment added pinaki What is the condition on $I$? E.g. your question has a negative answer if $I$ is allowed to be the ideal generated by all $x_{i,j}$.
Mar 10, 2017 at 23:50 history edited Ashvin Swaminathan CC BY-SA 3.0
Added in assumption on L
Mar 8, 2017 at 2:01 comment added Ashvin Swaminathan @nfdc23 Thanks so much, I have now edited the question accordingly!
Mar 8, 2017 at 2:00 history edited Ashvin Swaminathan CC BY-SA 3.0
Made improvements based on comment
Mar 8, 2017 at 1:44 comment added nfdc23 The description of the completion requires assuming $k(p)=k$ (e.g., OK if $k$ is algebraically closed), and then appealing to the Cohen Structure Theorem is both too heavy (you have the coefficient field in hand!) and not precise enough (you want the isomorphism as $k$-algebras, whereas Cohen Structure Theorem is only as abstract local rings).
Mar 8, 2017 at 0:11 history asked Ashvin Swaminathan CC BY-SA 3.0