Intereting Posts

How do I find two integers – $x$ and $y$ – whose values satisfy the expression $x^2 + y^2 = z$, where $z$ is a perfect square?
Computing irrational numbers
problem books in functional analysis
The Physical Meaning of Tetration with fractional power tower
Explanation of proof of Gödel's Second Incompleteness Theorem
Calculate the infinite sum $\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)} $
Trace and Norm of a separable extension.
Calculating fundamental group of the Klein bottle
Ratio of the circumferance inside adjacent circle if two circles touch each others mid points? how about shperes?
Is there a domain in $\mathbb{R}^3$ with finite non-trivial $\pi_1$ but $H_1=0$?
Question about proof of $A \otimes_A A \cong A $
Extension of bounded convex function to boundary
How to interpret Hessian of a function
Prove that $0^0 = 1$ using binomial theorem
Need help finding a good book on Riemann Geometry

All rings are commutative. Suppose $B$ is a flat $A$-algebra, and that $M$ and $N$ are flat $B$-modules.

Is there a way to compare the two $A$-modules $M \otimes_A N$ and $M \otimes_B N$?

Thanks

- $S^{-1}A \cong A/(1-ax)$
- Direct sum of non-zero ideals over an integral domain
- faithfully flat ring extensions where primes extend to primes
- Does $R$ a domain imply $\operatorname{gr}(R)$ is a domain?
- Sufficient conditions for being a PID
- Prove that in a Noetherian ring, no invertible maximal ideal properly contains a nonzero prime ideal

- Is there a relation between $End(M)$ and $M$ under tensor products?
- Homogeneous ideal and degree of generators
- Is $V$ a simple $\text{End}_kV$-module?
- $k$-algebra homomorphism of the polynomial ring $k$
- Prove that $\Gamma_I(\frac{M}{\Gamma_I(M)})=0$
- (Ir)reducibility criteria for homogeneous polynomials
- Proof that $K\otimes_F L$ is not noetherian
- Explicit example of Koszul complex
- Modules $M$ such that the automorphism of $M \otimes M \otimes M$ induced by the permutation $(123)$ is the identity
- Why is the completion of the ring of germs of smooth functions isomorphic to $\mathbb{R}]$?

If $B$ is a localization of $A$ then the natural map $M\otimes_A N \to M\otimes_B N$ will be an isomorphism (without any assumption on $M$ and $N$).

How definitive is this example?

Well, one way to think about your question is to take $M = N = B$. (You can’t get much flatter $B$-modules than this!) Then you are asking that the natural map $B\otimes_A B \to B$ be an isomorphism, which is to say that the diagonal map

$$\mathrm{Spec} \, B \hookrightarrow \mathrm{Spec \, B}\otimes_{\mathrm{Spec} \, A} \mathrm{Spec} \, B$$

be an isomorphism. This is asking that $A \to B$ be an epimorphism.

It is not so easy to find flat epimorphisms that are not localizations

(see the answers here, especially this one),

and so in practice I think that you should consider “$B$ is a localization of $A$” to be the most reasonable answer to the question of when this map is

an isomorphism.

- The spectrum of normal operators in $C^*$-algebras
- Splitting of the tangent bundle of a vector bundle and connections
- Express roots in polynomials of equation $x^3+x^2-2x-1=0$
- Matrices $B$ that commute with every matrix commuting with $A$
- Second Countability of Euclidean Spaces
- Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$
- Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?
- Probability that two sets are disjoint? the same?
- Verification for the solution following differential equation!
- Is the relation on integers, defined by $(a,b)\in R\iff a=5q+b$, a function?
- Approximating $\pi$ with least digits
- Counting ways to partition a set into fixed number of subsets
- AM-GM-HM Triplets
- Proving that $\sec\frac{\pi}{11}\sec\frac{2\pi}{11}\sec\frac{3\pi}{11}\sec\frac{4\pi}{11}\sec\frac{5\pi}{11}=32$
- The relation between order isomorphism and homeomorphism