Suppose we have two finitely generated modules $M$ and $N$ over a Noetherian ring $R$. If $M\otimes_RN$ is a nonzero projective $R$-module (or even better, a nonzero free $R$-module), can we conclude that both $M$ and $N$ are projective $R$-modules? If not, what condition should we put on $R$ to guarantee that?
2026-03-28 11:53:05.1774698785
Tensor product and projective modules
107 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in COMMUTATIVE-ALGEBRA
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- Extending a linear action to monomials of higher degree
- Tensor product commutes with infinite products
- Example of simple modules
- Describe explicitly a minimal free resolution
- Ideals of $k[[x,y]]$
- $k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements
- There is no ring map $\mathbb C[x] \to \mathbb C[x]$ swapping the prime ideals $(x-1)$ and $(x)$
- Inclusions in tensor products
- Principal Ideal Ring which is not Integral
Related Questions in TENSOR-PRODUCTS
- Tensor product commutes with infinite products
- Inclusions in tensor products
- How to prove that $f\otimes g: V\otimes W\to X\otimes Y$ is a monomorphism
- What does a direct sum of tensor products look like?
- Tensors transformations under $so(4)$
- Tensor modules of tensor algebras
- projective and Haagerup tensor norms
- Algebraic Tensor product of Hilbert spaces
- Why $\displaystyle\lim_{n\to+\infty}x_n\otimes y_n=x\otimes y\;?$
- Proposition 3.7 in Atiyah-Macdonald (Tensor product of fractions is fraction of tensor product)
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
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?