I wish to show that the injection $k[y^2, y^3] \rightarrow k[y]$ is not flat.
I know of geometric ways to see this, but I wish to see explicitly $k[y^2, y^3]$-modules (or localizations thereof) $0 \rightarrow M' \rightarrow M$ which does not remain injective upon tensoring with $k[y]$. Is such a search futile?
On
I'd like to add a short algebraic proof (although this is not exactly what you asked for) which shows again the usefulness of Exercise 7.2 in Matsumura, Commutative Ring Theory:
If $A\subset B$ are integral domains with the same field of fractions and $B$ is faithfully flat over $A$, then $A=B$.
In your case $A=k[y^2,y^3]$ and $B=k[y]$. If the extension is flat, then it is faithfully flat (since it is integral) and therefore $A=B$, a contradiction.
The "geometric proof" automatically produces an algebraic proof. The singularity on the cusp corresponds to the maximal ideal $(y^2,y^3) \subseteq k[y^2,y^3]$.
The inclusion $(y^2,y^3) \subseteq k[y^2,y^3]$ doesn't stay injective when tensored with $k[y^2,y^3] \to k[y]$. In fact, $y^2 \otimes y$ and $y^3 \otimes 1$ have the same image under $(y^2,y^3) \otimes_{k[y^2,y^3]} k[y] \to k[y]$, but are not equal.