What is the precise meaning of residue of a smooth variety?

Question

By variety we mean a finite-type, separated, geometrically integral scheme over a field $k$. Let $X$ be a smooth variety of dimension $n$ with $Y$ a smooth, irreducible closed subscheme of codimension $r$ with closed immersion $i: Y \hookrightarrow X$ defined by a sheaf of ideals $\mathcal{I}$. We have the short exact sequence relating the sheaves of differentials on $X$ and $Y$, $$ 0 \longrightarrow \mathcal{I}/\mathcal{I}^{2} \longrightarrow i^{*}\Omega_{X/k} \longrightarrow \Omega_{Y/k} \longrightarrow 0. $$ Then by taking the top exterior product of the terms in this sequence, and using the fact that $\mathcal{I}/\mathcal{I}^{2}$ is locally free, we obtain the famous adjunction formula, $$ \alpha: (\mathcal{I}//\mathcal{I}^{2})^{\wedge r} \stackrel{\simeq}{\rightarrow} \omega_{Y} \otimes i^{*} \omega_{X}. $$ I have seen several times a reference to the residue map giving a "section of $\omega_{Y}$" in this situation. In particular the text I have seen it in is the case of a quartic surface inside $\mathbb{P}^{3}$. But I was wondering what the residue map is in general. I am aware of a more abstract residue that I am not familiar with, but it seems like there is another definition that I am missing. It also seems to come up in terms of the trace map defined in Serre duality. Is anyone able to give me a concrete definition for the term? Is it the Poincare residue or is there something else as well?