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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)?
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1 Answer 1

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

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  • $\begingroup$ Why is 1 not $S_2$? (And why is 2 $S_2$?) $\endgroup$ Aug 11 at 9:23
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    $\begingroup$ 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. $\endgroup$
    – Sasha
    Aug 11 at 9:50
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    $\begingroup$ In 2 the dimension is 1, so there are no subschemes of codimension at least 2 and the condition is void. $\endgroup$
    – Sasha
    Aug 11 at 9:50

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