Prove that:
Let $R$ be a commuatative ring, let $T$ be total quotient ring of $R$. A finitely generated flat $R$-module $M$ is projective if and only if the scalar extension $T\otimes_R M$ is a projective $T$-module.
Thankyou so much.
2026-03-25 09:32:27.1774431147
Condition for a finitely generated flat module be projective
88 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in MODULES
- Idea to make tensor product of two module a module structure
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
- Example of simple modules
- $R$ a domain subset of a field $K$. $I\trianglelefteq R$, show $I$ is a projective $R$-module
- $S_3$ action on the splitting field of $\mathbb{Q}[x]/(x^3 - x - 1)$
- idempotent in quiver theory
- Isomorphism of irreducible R-modules
- projective module which is a submodule of a finitely generated free module
- Exercise 15.10 in Cox's Book (first part)
- direct sum of injective hull of two modules is equal to the injective hull of direct sum of those modules
Related Questions in PROJECTIVE-MODULE
- $R$ a domain subset of a field $K$. $I\trianglelefteq R$, show $I$ is a projective $R$-module
- projective module which is a submodule of a finitely generated free module
- Flat modules over a PID
- Projectivity of a module
- Proving a module is projective.
- Proving a specific module is projective.
- $M\otimes_A N\cong A$ implies $M$ is left $A$-projective?
- Proof of equivalence of two definitions of a projective module
- Problem based on Projective and Injective Module
- Splitting two exact sequences in a compatible way
Related Questions in FINITELY-GENERATED
- projective module which is a submodule of a finitely generated free module
- Ascending chain of proper submodules in a module all whose proper submodules are Noetherian
- Is a connected component a group?
- How to realize the character group as a Lie/algebraic/topological group?
- Finitely generated modules over noetherian rings
- Integral Elements form a Ring
- Module over integral domain, "Rank-nullity theorem", Exact Sequence
- Example of a module that is finitely generated, finitely cogenerated and linearly compact, but not Artinian!
- Computing homology groups, algebra confusion
- Ideal Generated by Nilpotent Elements is a Nilpotent Ideal
Related Questions in FLATNESS
- Difficulty understanding Hartshorne Theorem IV.4.11
- Flat modules over a PID
- Submodule of a flat module.
- When an integral extension of integral domains is flat?
- When flat submodule is direct summand?
- Flatness in a short exact sequence. If the
- Localization and flatness
- Computing $\operatorname{Tor}^R_1(M,N)$
- Does flatness ascend through a free ring map?
- Can it be that $R[[x]]$ is flat over $R$ but not over $R[x]$?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Forward direction is easy since taking total quotient is a localisation. For the reverse direction see Theorem 1 of the following paper https://projecteuclid.org/download/pdf_1/euclid.jmsj/1261060583