Canonical sheaf of divisor

I have a question about canonical sheaf :

Let $$X$$ be a non-singular variety over a field $$K$$, and $$Y$$ a subvariety nonsingular of codimension $$r$$. So we have : $$\omega_{Y} \simeq \omega_X \otimes \bigwedge^r \mathcal{N}_{Y/X}$$ where $$\mathcal{N}_{Y/X}$$ is the normal sheaf od $$Y$$ in $$X$$. My question is about the case where $$Y=D$$ is a cartier divisor. I denote $$L = \mathcal{O}_X (D)$$.

So , Hartshone say that, in that case : $$\omega_D \simeq \omega_X \otimes L \otimes \mathcal{O}_D$$ My question may be stupid, but what is $$\mathcal{O}_D$$ here ? I have never seen this notation and it is not in the books that I have read.

Thanks !

• $O_D$ is the structure sheaf of $D$. (This notation is certainly in Hartshorne). Tensoring by $O_D$ is essentially the same as restricting to $D$. The point is just that the left-hand side is a sheaf on $D$, not on $X$, so the right-hand side should be too. Commented Oct 19, 2020 at 13:55
• Thanks a lot for answers :) Commented Oct 19, 2020 at 14:02

So we need to be careful about which space our sheaves are over. Let $$i:D\to X$$ be the inclusion. If we're working over $$X,$$ then the isomorphism should read $$i_*\omega_D\cong \omega_X\otimes L\otimes i_*\mathcal O_D.$$ So really $$\mathcal O_D$$ is abuse of notation for $$i_*\mathcal O_D,$$ the pushforward of the structure sheaf of $$D.$$ If we're working over $$D$$ then the isomorphism should read $$\omega_D\cong (\omega_X\otimes L\otimes \mathcal i_*O_D)|_D\cong (\omega_X\otimes L)|_D\otimes\mathcal O_D\cong (\omega_X\otimes L)|_D$$ where $$\mathcal F|_D=i^*\mathcal F$$ denotes the pullback sheaf. This is what's called the adjunction formula.