Let $K$ be a field.
Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^n_K$ whose scheme theoretic support is a subvariety $X \subseteq \mathbb{P}^n_K$ 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?