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The rank of a coherent sheaf is defined in terms of the Hilbert polynomial (See Huybrechts-Lehn 1.2.2 or Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial).

Now let $\mathcal{F}\to X$ be a coherent sheaf over a projective manifold, so that there exists a resolution $E^{\bullet}\to\mathcal{F}$ of $\mathcal{F}$ by vector bundles.

Is there a way to define (or recover for instance) the rank of $\mathcal{F}$ using the resolution $E^{\bullet}$ (just like we can define the first Chern class for example)?

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Yes. Hilbert polynomials are additive over exact sequences: if $0\to F_n\to \cdots \to F_0\to 0$ is an exact sequence, then $\sum_{i=0}^n (-1)^iH_{F_i}(\lambda) =0$.

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  • $\begingroup$ Does that mean that $rk(\mathcal{F})=\sum_i(-1)^irk(E^i)=\chi(E^{\bullet})$? $\endgroup$
    – BinAcker
    May 15, 2022 at 13:01
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    $\begingroup$ @BinAcker Yes (this also follows really easily by the definition as rank at generic point, using generic freeness) $\endgroup$
    – Ben
    May 15, 2022 at 22:10

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