In Ravi Vakil (Foundations of Algebraic Geometry), $\S 13.7.4$, the rank of a quasicoherent sheaf $\mathscr{F}$ at a point $p$ of $X$ is defined as dim$_{k(p)}\mathscr{F}_p/m_p\mathscr{F}_p$.
A first question is: I need $\mathscr{F}_p$ to be finitely generated $\mathcal{O}_{X,p}$-module so that $\mathscr{F}_p/m_p\mathscr{F}_p$ is a vector space, right? Or just, is $\mathscr{F}_p$ to be finitely generated $\mathcal{O}_{X,p}$-module so that $\mathscr{F}_p/m_p\mathscr{F}_p$ it has a finite dimension? $\mathscr{F}_p/m_p\mathscr{F}_p$ will always be a vector space, independent if $\mathscr{F}_p$ to be finitely generated $\mathcal{O}_{X,p}$-module?
Also in section 13.7.4, if $X$ is irreducible, and $\mathscr{F}$ is a quasicoherent sheaf on $X$, then rank of $\mathscr{F}$ by convention means at generic point. However, in Hartshorne (Algebraic Geometry), page 148, it is defined the rank of $\mathscr{F}$ to be dim$_{k(p)}\mathscr{F}_p$, where $p$ is the generic point of $X$.
Question: What happened to $m_p\mathscr{F}_p$ in quotient $\mathscr{F}_p/m_p\mathscr{F}_p$? And what dimension is that for Hartshorne? Is $\mathscr{F}_p$ a vector space?
Best Answer
It's perfectly acceptable to have an infinite-dimensional vector space, so $\mathcal{F}_p$ need not be finitely generated over $\mathcal{O}_{X,p}$. For the rest of this answer I'll suppose that $X=\text{Spec } A$ is affine. Remember that $m_p$ is the maximal ideal in $A_p$, which coincides with $pA_p$. For A integral, the generic point $\xi$ corresponds to the zero ideal, so $A_{\xi}$ is the field of fractions of $A$ and $m_{\xi}=0$. So the two definitions actually coincide since $M_{\xi}=M_{\xi}/m_{\xi}M_{\xi}$ for any $A$-module $M$.
Finally, if $M$ is an $A$-module, then $M_p$ is an $A_p$-module for any prime $p$. $A_{\xi}$ is a field, so $M_{\xi}$ is an $A_{\xi}$ vector space.