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A theorem for you to prove: Let B be a local ring containing a perfect field k...

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Electrical Engineering

Atheorem for you to prove: Let B be a local ring containing aperfect field k that is isomorphic to its residue field B/m, andsuch that B is a localization of a finitely generated k-algebra.Then the module of relative differential forms M_B/k is a freeB-module of rank equal to the dimension of B if and only if B is aregular local ring.

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3.7 Ratings (417 Votes)
PROOF Let B is noetherian Suppose that M Bk is a free Bmodule of rank dimB Then by Proposition A Let B be a local ring with field of representatives k Then the canonical morphism of kmodules mm2 M Bk B k is an isomorphism rank kmm2 dimB so B is a regular local ring    See Answer
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