I do not understand why the following does not give a counterexample to Problem I.5.4(a) in Hartshorne's Algebraic Geometry book. The question of how to solve this problem has already been posted but no complete solution has been provided, and a solution I managed to find online has a gap. Any help, comments, references or solutions to the problem are greatly appreciated.
Relevant definitions:
If $Y \subseteq \mathbb{A}^2$ is a curve given by $f(x,y) = 0$ and $P = (0,0)$ is a point, write $f(x,y) = f_0 + f_1 + \cdots + f_d$, where $f_i$ is a homogeneous polynomial of degree $i$. The multiplicity $\mu_P(Y)$ of $Y$ at $P$ is the smallest $r$ such that $f_r \neq 0$.
If $Y, Z \subseteq \mathbb{A}^2$ are two distinct curves given by equations $f = 0$ and $g= 0$ and $P \in Y \cap Z$, then the intersection multiplicity $(Y \cdot Z)_P$ of $Y$ and $Z$ at $P$ is the length of the $\mathcal{O}_P$-module $\mathcal{O}_P/(f,g)$.
The problem states that $(Y \cdot Z)_P \geq \mu_P(Y) \cdot \mu_P(Z)$.
Consider the curves $Y$ and $Z$ given by $f = x^2 - y^3$ and $g = y^2 - x^3$ and the point $P = (0,0)$. We have $\mu_P(Y) = \mu_P(Z) = 2$, but I claim that $(Y \cdot Z)_P \leq 3$.
Let $R$ be the local ring $k[x,y]_m$, where $m$ is the maximal ideal $m = (x,y)$ and consider the module $M = R/(f,g)$. By definition $(Y \cdot Z)_P = \ell_R(M)$. To show that $(Y \cdot Z)_P \leq 3$, it suffices to show that $m^3M = 0$, for which it suffices to show that $m^3 \subseteq (f,g)$. I claim even that $(x^2, y^2) \subseteq (f,g)$ and since $m^3 = (x^3, x^2 y, xy^2, y^3) \subseteq (x^2, y^2)$ the result will follow.
For $x^2$ observe that \begin{align*} yg + f & = y(y^2 - x^3) + x^2 - y^3 \\ & = x^2(1 - xy), \end{align*} and since $1 - xy$ is a unit in $R$, it follows that $x^2 \in (f,g)$.
For $y^2$ observe that \begin{align*} xf + g & = x(x^2 -y^3) + y^2 - x^3 \\ & = y^2(1 - xy), \end{align*} and again since $1 - xy$ is a unit in $R$, we have $y^2 \in (f,g)$.
If it is the case that $(f,g)=(x^2,y^2)$ in the local ring $R$, then this means the intersection multiplicity is $4$, since the images of $1$, $x$, $y$ and $xy$ in the quotient $R/(x,y)$ form a $k$-basis.
But as $x^2-y^3$ has tangents just $x=0$ with multiplicity $2$ at $(0,0)$ and as $y^2-x^3$ has tangents just $y=0$ with multiplicity $2$ at $(0,0)$, these curve intersect transversally (without a common tangent) at $(0,0)$, and the intersection multiplicity is the product of their multiplicities there, viz., $4$.