A
theorem for you to prove: Let B be a local ring containing a
perfect field k...
50.1K
Verified Solution
Link Copied!
Question
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.
Answer & Explanation
Solved by verified expert
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
Get Answers to Unlimited Questions
Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!
Membership Benefits:
Unlimited Question Access with detailed Answers
Zin AI - 3 Million Words
10 Dall-E 3 Images
20 Plot Generations
Conversation with Dialogue Memory
No Ads, Ever!
Access to Our Best AI Platform: Zin AI - Your personal assistant for all your inquiries!
