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

2$\begingroup$ Okay, now I am happy. That formula follows, for instance, from Theorem 14.6 and Theorem 14.8, pp. 108109, 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

2$\begingroup$ 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