Proposition. Suppose that $I$ and $J$ are ideals sheaves on $\mathbb{P}^{n}$ with regularities $m_{1}$ and $m_{2}$, respectively. Suppose also que the zeros of $I$ and $J$ intersect in a set of dimension at most one. Then $I \cdot J$ is $(m_{1} + m_{2})$-regular.
1) What does the bold part in the above proposition mean?
Since $I\cdot J$ is the image of $I \otimes J$ in $\mathcal{O}_{\mathbb{P}^{n}}$ we have the following exact sequence
$$0 \longrightarrow K \longrightarrow I \otimes J \longrightarrow I \cdot J \longrightarrow 0$$
Now, at this stage of the demonstration, it is stated that the support of $K$ is contained in a set of dimension at most one. Why?
$(2)$ This result is found in the following work: On the Castelnuovo-Mumford Regularity Of Products of Ideal sheaves- Jessica Sidman.
Any help is most welcome.
Thanks in advance for your support.
The support of $K$ is contained in the intersections of the zeros of $I$ and $J$. Indeed if $p$ is not in the zeros of $J$ the sequence on the stalks becomes $$ 0 \longrightarrow K_p \longrightarrow I_p \otimes \mathcal{O}_{\mathbb{P}^n,p} \longrightarrow I_p \longrightarrow 0 $$ and the multiplication map is an isomorphism. The same holds if $p$ is not a zero for $I$.
ADDED: Since we are taking a tensor product of $\mathcal{O}_{\mathbb{P}^n,p}$-modules, we can write $$ I_p \otimes \mathcal{O}_{\mathbb{P}^n,p} \ni \sum_j f_j \otimes g_j =\left(\sum_jf_jg_j \right)\otimes 1.$$ Hence every element of $I_p \otimes \mathcal{O}_{\mathbb{P}^n,p}$ is given by $g\otimes 1$ for some $g\in I_p$. and the multiplication map sends $g\otimes 1$ to $g$ which is clearly an isomorphism.
Let me add an example where the multiplication map $I_p \otimes J_p \rightarrow I_p\cdot J_p$ is not an isomorphism. Let $p$ be the origin of $\mathbb{C}^3$ (with coordinates $(x,y,z)$) and let $$I_p= (x,y,z), \, J_p = (xz-y^2, y-z^2, x-yz).$$ Note that the element $$ x\otimes (z^2-y) + y\otimes (x-yz) + z\otimes (y^2-xz) \in I_p\otimes J_p $$ is in the kernel of the multiplication map.