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Let $K$ be a field. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^n_K$ whose scheme theoretic support is a reduced, closed subscheme $X \subseteq \mathbb{P}^n_K$ of dimension $k$. Let $X_1, \ldots, X_r$ be the irreducible components of $X$ of dimension $k$. I think that then we always have $$\deg \mathcal{F}=\sum_{i=1}^r \textrm{rank}(\mathcal{F}|_{X_i}) \cdot \deg X_i.$$ Does somebody know a reference where this can be found?

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  • $\begingroup$ When you say "subvariety", what do you mean? You specify that $X$ has several irreducible components. Are you using "subvariety" to mean closed subscheme? In that case, your formula does not make sense. $\endgroup$ – Jason Starr Dec 2 '15 at 15:23
  • $\begingroup$ The support of $\mathcal{F}$ is a closed subscheme of $\mathbb{P}^n$ and I assume that this is in fact a variety (thus reduced). Why does the formula not make sense? $\endgroup$ – Hans Dec 2 '15 at 15:31
  • $\begingroup$ The usual definition of "variety" means both "reduced" and "irreducible". $\endgroup$ – Jason Starr Dec 2 '15 at 15:33
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    $\begingroup$ Okay, now I am happy. That formula follows, for instance, from Theorem 14.6 and Theorem 14.8, pp. 108--109, of Matsumura's "Commutative Ring Theory". You need to take the affine cones, and then you choose the local ring to be the local ring of a polynomial subring over which each $\mathcal{F}|_{X_i}$ is (generically) flat. $\endgroup$ – Jason Starr Dec 2 '15 at 15:41
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    $\begingroup$ Another reference is Proposition VI.2.7, p. 295, of Koll'ar, "Rational Curves on Algebraic Varieties". The formula there is more immediately recognizable as your formula than is the formula in Matsumura, $\endgroup$ – Jason Starr Dec 2 '15 at 15:51

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