The Hilbert Polynomial and Degree
نویسنده
چکیده
Proof. The first assertion is Exercise III.5.2 of [1]. The degree assertion can be deduced by modifying the proof of the same exercise to remove the first term of the exact sequence, using Corollary 4.1 of [2] (and noting that invariance of cohomology under field extension means we can reduce immediately to the case of an infinite field). The last equality follows from Serre’s theorem on vanishing of higher cohomology. This then implies that χ(F (n)) ≥ 0 for n >> 0, hence that the leading coefficient is positive. Finally, the expression in terms of binomials is a general fact on polynomials taking integer values, see Proposition I.7.3 of [1].
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