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...

  • $\begingroup$ 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 May 18 '13 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 May 18 '13 at 23:04

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.


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.