Serre's criterion says that for a scheme to be normal is equivalent to it being $R\_1$ (i.e. regular in codimension $1$) and $S\_2$ (i.e. regular functions on $X-Y$ extend to $Y$ if $Y$ has codimension at least $2$).


What would be examples of:


1. a scheme which is $R\_1$, but not $S\_2$ (i.e. not normal)?
2. a scheme which is $S\_2$, but not $R\_1$ (i.e. not normal)?



Normal schemes and Serre's criterion
1. Glue two planes at a point, i.e., take take $\mathrm{Spec}(A)$, where
$$
A = \{(f,g) \in k[x\_1,x\_2] \oplus k[y\_1,y\_2] \mid f(0,0) = g(0,0) \}.
$$
2. Take any singular curve, e.g., $\mathrm{Spec}(k[x,y]/xy)$.


