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maybe it is a very trivial quetion but: suppose we have a smooth projective variety $X$ over $k$ and $\mathcal{A}$ an $\mathcal{O}_X$ algebra. We have the relative spectrum $Spec(Sym(\mathcal{A}))\rightarrow X$.

What is the canonical budle of $Spec(Sym(\mathcal{A}))$ as a $k$ scheme?

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  • $\begingroup$ Notice that $Spec(Sym(\mathcal{A}))$ is affine over $X$ and you have to put very tight restrictions to make it finite, i.e. proper, where the theory of the canonical module works. $\endgroup$
    – Leo Alonso
    Commented Oct 23, 2013 at 9:21
  • $\begingroup$ Do you really mean $Sym(\mathcal{A})$ or simply $\mathcal{A}$? In the former case, what is the point of taking an algebra and not an $\mathcal{O}_X$-module? $\endgroup$
    – abx
    Commented Oct 23, 2013 at 12:04
  • $\begingroup$ Oh I'm sorry! Here $\mathcal{A}$ is locally free so that we have finite morphism. Furthermore in this case we need the symmetric algebra. $\endgroup$
    – Zac
    Commented Oct 23, 2013 at 13:19
  • $\begingroup$ If $\mathcal{E}$ is a locally free sheaf on $X$, the canonical line bundle of $\mathrm{Spec}(\mathrm{Sym}(\mathcal{E}))$ is the pull back of $\omega _X\otimes \det(\mathcal{E})$. Is that what you are asking? $\endgroup$
    – abx
    Commented Oct 23, 2013 at 19:58

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