Logo
Home Study Material Contact Search
AI Chat FREE
  1. engineering
  2. electrical engineering
  3. a theorem for you to prove: let b be a local ring

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

50.1K

Verified Solution

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: Flex AI - Your personal assistant for all your inquiries!
Become a Member

Other questions asked by students

Statistics
1.2K views

Present the regression output below noting the coefficients, assessing the adequacy of the model and the...

Biology
1.2K views

Malaria is a tropical disease caused by a single-celled parasite. People become infected with malaria...

Calculus
1.2K views

1 O Use the equation a Determine whe O The parabola O The parabola

Calculus
1.2K views

Find parametric equations for the tangent line at the point cos 2 sin 2 2...

Accounting
1.2K views

Income Statement Wayne Corporation had the following revenue and expense account balances for a recent...

Accounting
1.2K views

A. Alex purchased a CNC machine on 1st October 2019 at a cost of $110,000...

Accounting
1.2K views

3 Check my 3.34 po D Wences Silven...

Accounting
1.2K views

correct asnwers please Required: 1. Prepare journal entries...

Accounting
1.2K views

6.Suppose that a Plywood manufacturer processes wood pulp into four grades of Plywood totaling 500,000...

Accounting
1.2K views

A local government has the following transactions during the current fiscal period a. The budget...

Finance
1.2K views

Nevada Enterprises is considering buying a vacant lot that sells for $1.2 million. If the...

Finance
1.2K views

One safe investment pays 10% per year, and a more risky investment pays 18% per...