Atiyah-Macdonald 2.3

Question

In solving question 2.3 from Atiyah & Macdonald's commutative algebra textbook, I run into the following difficulty:

Let $A$ be a local ring with $k:= A/mA$ its residue field and let $M$ and $N$ be $A-$modules.

Note that for an arbitrary $A-$module $P$, we can view $P \otimes_A k$ as a $k-$ module by letting $\lambda (p \otimes_A x) = (p \otimes_A \lambda x),$ for $p \in P$ and $\lambda, x \in k.$

I want to show that $(M \otimes_A k) \otimes_k (N \otimes_A k) \approx (M \otimes_A N) \otimes_A k,$ where $\approx$ denotes an isomorphism.

Answer 1

This is associativity of the tensor product: as $A$-modules, $(M \otimes_A k) \otimes_k (N \otimes_A k) \cong M \otimes_A (k \otimes_k N) \otimes_A k \cong M \otimes_A N \otimes_A k \cong (M \otimes_A N) \otimes_A k$

Answer 2

Hint: When $A$-modules $M,N$ have the property that $\mathcal mM=\mathcal mN=0$ then $M\otimes_A N \cong M\otimes_k N$ as $A$-modules.