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Following http://www.math.columbia.edu/~masdeu/files/notes/ModForms.pdf, define an effective Cartier divisor of an $S$-scheme $f: X \rightarrow S$ as a closed subscheme $Z \subseteq X$, such that $Z$ is flat and finite over $S$ via $f$.

$D$ defines an invertible sheaf $\mathcal{I}(D)$ on $X$; take $\mathcal{L}(D)$ to be its dual.

On page 20, we have "The degree of $D$ is defined as the rank of $f_*(i_*\mathcal{O}_D \otimes \mathcal{L}(D))$," where $i : Z \rightarrow X$ is the inclusion.

I understand that the latter sheaf is locally free, because $f$ is finite and flat. But I don't understand why it has a well-defined rank; why can't be different on affine subsets? It's not even required that $S$ should be connected...

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  • $\begingroup$ Perhaps the degree is supposed to be a function, rather than a number? After all, the LHS of the equation appearing in Theorem 8.3 is also a function... $\endgroup$
    – Zhen Lin
    Commented May 18, 2013 at 22:43
  • $\begingroup$ @ZhenLin You're right - I don't see why the left hand side would be constant either. I can deal with both only being defined locally, as long as we can do the construction in 8.3.2. $\endgroup$
    – Cocopuffs
    Commented May 18, 2013 at 23:04

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The degree of a finite rank locally free sheaf is locally constant, hence constant on connected components. For almost any problem involving the concept of rank, it will be no problem to restrict to $S$ connected, and then the rank will be constant.

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