1
$\begingroup$

I am reading lemma 27.27.3 in stacks project.


In the proof it seems it seems to claim:

If $X$ is a scheme, where $f_1,\ldots, f_n \in \Gamma(X, O_X)$ generates the ring. Then $X= \bigcup X_{f_i}$ where $$ X_f:= \{x \in X \, : \, f_x \not= 0 \in O_{X,x} \}$$

How so? Suppose false, then pick $x$ not in the union. Then $$ 1_x =\sum (g_if_i)_x =0 \in O_{X,x} $$ But there is nothing wrong this either...

$\endgroup$
1
$\begingroup$

Your argument does not work since the stalks cannot be zero: They are local rings and the zero ring is not local. Therefore you reach a contradiction, which proves the claim.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.