# Normal schemes and Serre's criterion

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)?

## 1 Answer

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., $$\mathrm{Spec}(k[x,y]/xy)$$.

• Why is 1 not $S_2$? (And why is 2 $S_2$?) Aug 11 at 9:23
• In 1 take $Y$ to be the gluing point, so that $X - Y$ is the disjoint union of the punctured planes, and take the function which is equal to $0$ on one punctured plane and $1$ on the other; it obviously does not extend. Aug 11 at 9:50
• In 2 the dimension is 1, so there are no subschemes of codimension at least 2 and the condition is void. Aug 11 at 9:50