Canonical sheaf of divisor

Question

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 !

Answer 1

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.