Let $S$ be a Noetherian scheme and $Y,Z$ two locally closed subschemes.
What is the scheme theoretic intersection of $Y$ and $Z$. I am asking because in Mumford's "Lectures on curves on an algebraic surface" page 59. He defines it as, I quote, topologically it is the intersection $Supp(Y) \cap Supp(Z)$ and you take the scheme structure to be the sum of the ideals defining $Y$ and $Z$.
Am I correctly understanding that this means the following : cover $S$ by opens $U_i$ such that $Z\cap U_i$ and $Y\cap U_i$ are closed defined by ideals $I_i$ and $J_i$ and define $W_i$ to be the closed subscheme of $U_i$ defined by $I_i+J_i$ and then define $W$, the scheme intersection of $Y$ and $Z$ to be the glueing of the $W_i$'s ?
By definition there exist a pair of open subschemes $Y_0$ and $Z_0$ such that: $Y$ and $Z$ are closed subschemes of $Y_0$ and $Z_0$, respectively; without loss of generality (wlog for short) one can assume that $Y_0$ and $Z_0$ are the biggest ones with this property ([B] Remark 7.3.12).
The scheme theoretic intersection $S_0=Y_0\times_SZ_0$ is exactly $Y_0\cap Z_0$ (cfr. [B] Lemma 7.2.5) considered as open subscheme of $S$!
Wlog, one can assume that $S_0,Y_0,Z_0,Y$ and $Z$ are affine; one needs to compute $Y\times_{Y_0}S_0$ and $Z\times_{Z_0}S_0$, that is: \begin{equation*} S_0=Spec(A_0),\,Y_0=Spec(B_0),\,Z_0=Spec(C_0),\,Y=Spec(B),\,Z_0=Spec(C) \end{equation*} where $B=B_{0\displaystyle/I_0}$ and $C=C_{0\displaystyle/J_0}$ for some ideals $I_0$ and $J_0$ of $B_0$ and $C_0$, respectively (see [B] Lemma 7.3.8).
From all this: \begin{gather*} Y\times_{Y_0}S_0=Spec\left(B_{0\displaystyle/I_0}\otimes_BA_0\right)=Spec\left(A_{0\displaystyle/I}\right),\\ Z\times_{Z_0}S_0=Spec\left(C_{0\displaystyle/J_0}\otimes_CA_0\right)=Spec\left(A_{0\displaystyle/J}\right),\\ \left(Y\times_{Y_0}S_0\right)\times_{S_0}\left(Z\times_{Z_0}S_0\right)=Spec\left(A_{0\displaystyle/I}\otimes A_{0\displaystyle/J}\right)=\\ =Spec\left(A_{0\displaystyle/I+J}\right)=Y\cap Z, \end{gather*} where $I$ and $J$ are obvious; this last one scheme is exacly the scheme theoretic intersection of $Y$ and $Z$ over $S_0$; but by universal property of fibre product of schemes, it is $Y\cap Z$ as affine $S$-scheme.
Gluing togheter these affine $S$-schemes, one has that the support of scheme theoretic intersection over $S$ of $Y$ and $Z$ is $Y\cap Z$.
[B] S. Bosch (2013) Algebraic Geometry and Commutative Algebra, Springer