Exterior powers

Question

I was reading the following notes on tensor products: http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf

At some point (p. 39) there is the following example

In the last paragraph, he says that using exterior powers it can be proved that if $I\oplus I\simeq S^2$ as $S$-module, then $I\otimes_S I\simeq S$ as $S$-modules.

I do not know a lot about exterior powers (just the definition), but I would like to know what is the property being used here and what is the isomorphism he finds out.

Can you give me some hints?

As matter of fact I think it really proves that $I\otimes_S I$ is isomorphic to $S\otimes_S S\simeq S$, but I cannot construct a surjective map from $S\otimes_S S$ to $I\otimes_S I$.

Answer 1

He is using the isomorphism $${\bigwedge}^2(A\oplus B)\cong \left({\bigwedge}^2 A\right)\oplus (A\otimes B)\oplus\left( {\bigwedge}^2 B\right).$$ We take both $A=B=S$ and $A=B=I$ and use the fact that $\bigwedge^2S=\bigwedge^2I=0$. Then if $S\oplus S\cong I\oplus I$ then applying the exterior square gives $S\otimes S\cong I\otimes I$.