Timeline for $\deg \mathcal{F}=\sum_{i=1}^r \textrm{rank}(\mathcal{F}|_{X_i}) \cdot \deg X_i$

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Dec 2, 2015 at 15:51	comment	added	Jason Starr		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,
Dec 2, 2015 at 15:41	comment	added	Jason Starr		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.
Dec 2, 2015 at 15:36	comment	added	Hans		Fair enough, I have edited the question.
Dec 2, 2015 at 15:34	history	edited	Hans	CC BY-SA 3.0	added 15 characters in body
Dec 2, 2015 at 15:33	comment	added	Jason Starr		The usual definition of "variety" means both "reduced" and "irreducible".
Dec 2, 2015 at 15:31	comment	added	Hans		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?
Dec 2, 2015 at 15:23	comment	added	Jason Starr		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.
Dec 2, 2015 at 15:15	history	asked	Hans	CC BY-SA 3.0