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Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(Why is the Euler characteristic of powers of a line bundle a polynomial in the power?Why is the Euler characteristic of powers of a line bundle a polynomial in the power?), which works for locally free $\mathcal F$.

Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(Why is the Euler characteristic of powers of a line bundle a polynomial in the power?), which works for locally free $\mathcal F$.

Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(Why is the Euler characteristic of powers of a line bundle a polynomial in the power?), which works for locally free $\mathcal F$.

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David
David
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Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(Why is the Euler characteristic of powers of a line bundle a polynomial in the power?), which works for locally free $\mathcal F$.

Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(Why is the Euler characteristic of powers of a line bundle a polynomial in the power?), which works for locally free $\mathcal F$.

Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(Why is the Euler characteristic of powers of a line bundle a polynomial in the power?), which works for locally free $\mathcal F$.

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David
David
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Hilbert polynomial for any invertible sheaf

Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(Why is the Euler characteristic of powers of a line bundle a polynomial in the power?), which works for locally free $\mathcal F$.