Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Is it true that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) \cong H^0(j^{-1}\mathcal{N}_{X|\mathbb{P}^n})$?

EDIT Assume $Z$ is an irreducible component of $X$.

  • $\begingroup$ Your editing doesn't change the problem, Anton's example still applies. $\endgroup$
    – abx
    Jan 14, 2014 at 8:47
  • $\begingroup$ @abx: Sorry. Meant to write irreducible component. $\endgroup$
    – user45397
    Jan 14, 2014 at 8:53
  • 1
    $\begingroup$ OK, in this new version at least it makes sense. $\endgroup$
    – abx
    Jan 14, 2014 at 8:55

1 Answer 1


Take $X=\mathbb{P}^1\subset \mathbb{P}^2$ and let $Y$ be a single point $y\in X$. Then $j^*$ gives you a trivial bundle corresponding to $\mathrm{k}$, while $j^{-1}$ is isomorphic to the full stalk $\mathcal{N}_y\simeq \mathcal{O}_{X,y}\simeq \mathrm{k}[x]_{(x)}$ (here $\mathrm{k}$ denotes the base field and $\mathcal{N}$ is the normal bundle). This is exactly why one considers the pullback functor in algebraic geometry and not the inverse image functor.

  • $\begingroup$ @Fonarev: Thank you for the answer. I have edited the question. $\endgroup$
    – user45397
    Jan 14, 2014 at 8:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.