Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination.
Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.2: Complex Numbers And Quaternions
Problem 47E
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![Let R be a commutative Noetherian ring and M, N be finitely generated R-modules.
a) Define the tensor product M R N and prove that it is also a finitely generated R-module.
Use the properties of finitely generated modules in your proof.
b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N.
Provide an explicit example and perform the computation.
c) Define the projective dimension of a module and prove that if M has finite projective
dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M.
Close
d) Explore the flatness of modules by proving that a module M is flat if and only if
Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this
equivalence.
e) Using the concept of homological dimension, determine the global dimension of a polynomial
ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your
determination.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9115f0db-70f3-45cd-bcf3-6e1e8244cd08%2Ff07e91d1-26e5-47fa-bc8c-740bfd3467fd%2Fqcnmc1m_processed.png&w=3840&q=75)
Transcribed Image Text:Let R be a commutative Noetherian ring and M, N be finitely generated R-modules.
a) Define the tensor product M R N and prove that it is also a finitely generated R-module.
Use the properties of finitely generated modules in your proof.
b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N.
Provide an explicit example and perform the computation.
c) Define the projective dimension of a module and prove that if M has finite projective
dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M.
Close
d) Explore the flatness of modules by proving that a module M is flat if and only if
Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this
equivalence.
e) Using the concept of homological dimension, determine the global dimension of a polynomial
ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your
determination.
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