Apologize if this is a newbie question.

Let $i: Z\hookrightarrow X$ be a closed immersion with ideal sheaf $\mathscr{I}$. The conormal sheaf of $Z$ in $X$ is defined as $\mathscr{I}/\mathscr{I}^2$, regarded as a sheaf on $Z$.

Can the conormal sheaf be interpreted as $i^*\mathscr{I}$?


1 Answer 1


Yes, $$\mathscr I/\mathscr I^2=\mathscr I\otimes_{\mathcal O_X}\mathcal O_X/\mathscr I=i^*\mathscr I.$$ This is mentioned here.

  • $\begingroup$ Thank you. Guess a follow up question is why is this not mentioned/remarked in most introductory text(s)? $\endgroup$
    – aaa acb
    Jun 24, 2023 at 15:26
  • $\begingroup$ @aaaacb you are not alone. When I was beginning with AG, I also had the same question and follow-up question for a while! $\endgroup$ Jun 26, 2023 at 1:59

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