Lemma 55.11.1. In Situation 55.9.3 suppose we have an effective Cartier divisors $D, D' \subset X$ such that $D' = D + C_ i$ for some $i \in \{ 1, \ldots , n\} $ and $D' \subset X_ k$. Then

**Proof.**
The second equality follows from the definition of the bilinear form $(\ \cdot \ )$ in (55.9.6.1) and Lemma 55.9.6. To see the first equality we distinguish two cases. Namely, if $C_ i \not\subset D$, then $D'$ is the scheme theoretic union of $D$ and $C_ i$ (by Divisors, Lemma 31.13.10) and we get a short exact sequence

by Morphisms, Lemma 29.4.6. Since we also have an exact sequence

(Divisors, Remark 31.14.11) we conclude that the claim holds by additivity of euler characteristics (Varieties, Lemma 33.33.2). On the other hand, if $C_ i \subset D$ then we get an exact sequence

by Divisors, Lemma 31.14.3 and we immediately see the lemma holds. $\square$

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