Flat Modules and Intersection

[Euge]

Let $M$ be a flat module over a commutative ring $A$. Suppose $X_1$ and $X_2$ are submodules of an $A$-module $X$. Prove that $(X_1 \cap X_2) \otimes_A M = (X_1 \otimes_A M) \cap (X_2 \otimes_A M)$ as submodules of $X\otimes_A M$.

 

[Euge]

Spoiler

There is a short exact sequence $0 \to X_1 \cap X_2 \to X \to X/X_1 \oplus X/X_2 \to 0$. Tensoring with $M$ gives a short exact sequence $$0 \to (X_1 \cap X_2) \otimes_A M \to X \otimes_A M \to \frac{X\otimes_A M}{X_1 \otimes_A M} \oplus \frac{X \otimes_A M}{X_2 \otimes_A M}\to 0$$ The kernel of the third map is $(X_1 \otimes_A M) \cap (X_2 \otimes_A M)$ so indeed $$(X_1 \cap X_2) \otimes_A M = (X_1\otimes_A M) \cap (X_2 \otimes_A M)$$

 

[mathwonk]

Spoiler

The difference of the inclusion maps from the direct sum of two submodules into the ambient module, has kernel equal to the diagonal map from their intersection. Then tensoring with $M$ preserves direct sums, kernels, and cokernels, hence gives this result.