A correspondence between {H}ilbert polynomials and {C}hern polynomials over projective spaces (Q1765968)

Let \({\mathbb P}^{d}\) denote projective space over an algebraically closed field \(k\). Let \(K_{0}( {\mathbb P}^{d} )\) denote the Grothendieck group of locally free sheaves on \({\mathbb P}^{d}\). For \(x \in K_{0}( {\mathbb P}^{d} )\) with Chern classes \(c_{i}(x)\) the Chern polynomial is defined by \[ C_{x}(t) = 1 + c_{1}(x)t + c_{2}(x)t^{2} + \dots + c_{d}(x)t^{d} \] lying in \({\mathbb Z}[t]/(t^{d+1})\). On the other hand a graded, finitely generated module \(M\) over the graded ring \(k[x_{0}, \dots , x_{d}]\) gives a coherent sheaf on \({\mathbb P}^{d}\) and a class in the Grothendieck group \(G_{0}( {\mathbb P}^{d} ) \cong K_{0}( {\mathbb P}^{d} )\). The Hilbert polynomial \(P_{M}(t)\) is related to \(C_{M}(t)\) by the Hirzebruch-Riemann-Roch theorem [see \textit{D. Eisenbud}, ``Commutative algebra. With a view towards algebraic geometry'', Grad. Texts Math. 150 (1995; Zbl 0819.13001)]. The author shows that the homomorphism \[ \xi : K_{0}( {\mathbb P}^{d} ) \rightarrow ({\mathbb Z}[t]/(t^{d+1}))^{*} \times {\mathbb Z} \] given by \(\xi(M) = (C_{M}(t) , \text{rank}M))\) is injective. As an application she shows that the classes of \(M\) and \(N\) in \(K_{0}({\mathbb P}^{d})\) are equal if and only if \(C_{M}(t) = C_{N}(t)\) and \( \text{rank}M) = \text{rank}N)\) or if and only if \(P_{M}(t) = P_{N}(t)\). The paper concludes with a section detailing the precise relation between Chern and Hilbert polynomials, which appears in [\textit{D. Eisenbud}, loc. cit., Exercise 19.18].