Euler difference for the ideal of a curve

Let $S$ be a surface with ample $A$, $C$ an integral curve, $J,\iota$ presenting the ideal of $C$, and $M$ a line bundle. Then $\chi(M)-\chi(J\otimes M)=\chi(C,M|_C)$.

Notation. $\chi$ is eulerCharacteristic: the alternating sum of $\mathbb C$-dimensions of $\operatorname{Ext}^n(\mathcal O,-)$, for $n\le2$ on a surface and $n\le1$ on a curve. PresentsPullbackIdeal I f J ι says that $\iota:J\to\mathcal O_Y$ is a monomorphism whose image on affine opens is the extension of $I$ along $f$. IsAmple means that for each point $x$ and open $V\ni x$, some positive power has a global section with nonvanishing open containing $x$, inside $V$, and affine. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$. An IntegralCurve on a surface is an integral closed subscheme of topological Krull dimension one, given by a closed immersion.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.