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Let we have the following exact sequence of vector bundles on an complex smooth projective algebraic surface $X$ in $\mathbb P^3$ given by :

$0 \to E \to \mathcal O_X^{\oplus r} \to F \to 0$

where, it's given that $E$ is vector bundle and $F$ is a coherent sheaf on $X$ supported on a codimension $1$ subscheme of $X$.

Then since rank is additive we must have rank$(E) =r-\text{rank}(F)$. But I have come across situation where for $r=2$ it's mentioned that rank$(E) =r$.

Then is it so that $F$ being supported on a codimension $1$ subscheme means that it has always $0$ rank?

Here by rank I mean the complex dimension of the vector space which is obtained as the stalk at the generic point.

Any help from anyone is welcome.

HARRY
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    Have you tried to see the ranks at the generic point of $X$? What is the rank of $F$ at the generic point of $X$? – Mohan Mar 30 '20 at 13:42
  • @Mohan, If we follow the alternative definition of Lehn and Huybrechts book definition $1.2.2$, then can we say that in their language $\alpha_{dimX}(F) =0$ because $F$ has dimension $1$ less than $X$ and hence rank$(F)=\frac{\alpha_{dimX}(F)}{\alpha_{dimX}(\mathcal O_X)}=0$?(though the definition is for sheaves having full dimension) – HARRY Mar 30 '20 at 13:54
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    Rank is computed over which ring makes a difference. For example, $\mathbb{Z}/p\mathbb{Z}$, $p$ a prime has rank one over itself but rank zero over $\mathbb{Z}$. – Mohan Mar 30 '20 at 14:33
  • @Mohan, I am working over $\mathbb C$. – HARRY Mar 30 '20 at 15:49
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    That is not important. What does rank of a sheaf mean to you? – Mohan Mar 30 '20 at 15:56
  • @Mohan, in my case the sheaf $F$ is basically pushforward of a line bundle on a curve in $X$ and by rank I mean rank at the generic point(In vakils book that's taken as a definition for curve for simplicity for proving additivity) – HARRY Mar 30 '20 at 16:10
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    Generic point of $X$? I think you could look up definition of the rank of a sheaf. – Mohan Mar 30 '20 at 16:44
  • @Mohan,https://math.stackexchange.com/q/1127935/234042 here in the question equivalence of two definitions (Huybrechts and Vakil)for an integral scheme is mentioned. If for any integral scheme $X$ rank is defined rank at the generic point then I don't know what else it could be other than generic point of $X$? Please correct me if I am making a fundamental mistake. – HARRY Mar 30 '20 at 17:13
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    If you take, say a line bundle on a curve $C\subset X$, with $\dim X>1$, what is the rank of the direct image of the line bundle on $X$? – Mohan Mar 30 '20 at 18:33
  • @Mohan, If the generic point of $X$ is not in $C$ then the stalk of the pushforward is $0$ and hence the rank is $0$. Now the only thing left to prove is the generic point of $X$ does not belong to $C$.(From the additivity of rank we must have its rank to be $0$).Please correct me if I am wrong. – HARRY Mar 30 '20 at 19:11
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    So, is the generic point of $X$ in $C$? – Mohan Mar 30 '20 at 19:25
  • @Mohan, I think it's not in $C$ as the singleton set containing it intersects every nonempty open subset of $X$ and hence in particular intersects $X-C$. – HARRY Mar 31 '20 at 04:58

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