Lemma 19.7.19.680A0890C1EE Given $S,S',S''\in \hat{\mathcal{S}}(C)$, we have
\[ d_{\mathrm{sg}}(S\cap S'',S'\cap S'')\leq d_{\mathrm{sg}}(S,S'). \]
In particular, if $S^ i,S'^ i\in \hat{\mathcal{S}}(C)$ ($i\in \mathbb {N}$) and $S^ i\to S$, $S'^ i\to S'$, then
\[ S^ i\cap S'^ i\to S\cap S'. \]
Proof.
Observe that for any $k\in \mathbb {N}$,
\[ |S_ k\cap S_ k''|-|S_ k\cap S_ k'\cap S_ k''|\leq |S_ k|-|S_ k\cap S_ k'|. \]
The same holds if we interchange $S$ with $S'$. It follows that
\[ |S_ k\cap S_ k''|+|S_ k'\cap S_ k''|-2|S_ k\cap S_ k'\cap S_ k''|\leq |S_ k|+|S'_ k|-2|S_ k\cap S_ k'|. \]
The first assertion follows.
Next we compute
\[ \begin{split} d_{\mathrm{sg}}(S^ i\cap S'^ i,S\cap S')\leq & d_{\mathrm{sg}}(S^ i\cap S'^ i,S^ i\cap S')+d_{\mathrm{sg}}(S^ i\cap S',S\cap S')\\ \leq & d_{\mathrm{sg}}(S'^ i,S')+d_{\mathrm{sg}}(S^ i,S) \end{split} \]
and the second assertion follows.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.