A remark on the Riemann-Roch formula on affine schemes associated with Noetherian local rings
Let \(A\) be a homomorphic image of a regular local ring and \(r= \dim A\). Let \(A_*(Z)_{\mathbb{Q}}= \bigoplus_{i=0}^rA_i(Z)_{\mathbb{Q}}\) be the Chow group of the affine scheme \(Z= \text{Spec}(A)\) with rational coefficients and \(K_0(Z)\) the Grothendieck group of finitely generated \(A\)-modules. There is a natural isomorphism \(\tau: K_0(Z)_{\mathbb{Q}} \to A_*(Z)_{\mathbb{Q}}\) called the Riemann-Roch map. Let \(\tau([A])= q_r+ q_{r-1}+ \cdots+ q_0, q_i \in A_i(Z)_{\mathbb{Q}}.\) The author is interested in the question when \(\tau([A])= q_r\), that is, \(q_{r-1}= \ldots= q_0= 0\). The motivation comes from the following facts. \textit{P. Roberts} [Trans. Am. Math. Soc. 300, 583-591 (1987; Zbl 0636.13005)] has shown that the vanishing theorem holds if \(\tau([A])= q_r\). If \(A\) is a local ring of characteristic \(p> 0\) with perfect residue field, \textit{L. Szpiro} [in: Commutative Algebra, Sympos. Durham 1981, Lond. Math. Soc. Lect. Note Ser. 72, 83-90 (1982; Zbl 0554.13005)] has conjectured that for any complex \(\mathbf F_{\bullet}\) of finitely generated free \(A\)-modules such that \(H_i({\mathbf F_{\bullet}})\) has finite length for every \(i \geq 0\), \[ \sum_i\ell_A(H_i({\mathbf F_{\bullet}} \otimes_A{^eA}))= p^{re}\sum_i\ell_A(H_i({\mathbf F_{\bullet}}), \] where \(^eA\) denotes the \(A\)-module \(A\) induced by the Frobenius map \(x \to x^{p^e}\), \(x \in A\). This formula holds if \(\tau([A])= q_r\). It is known that \(\tau([A])= q_r\) if \(A\) is a complete intersection. The author gives an example of a Gorenstein ring \(A\) such that \(\tau([A]) \neq q_r\). The main result is the following theorem: Let \(X\) be a smooth projective variety over a field \(k\). Let \(Z\) be the affine cone of \(X\) and \(c\) the Cartier divisor corresponding to the line bundle \({\mathcal O}_X(1)\). Then there is an isomorphism \(\xi: A_*(X)_{\mathbb{Q}}/(c) \to A_*(Z)_{\mathbb{Q}}\) such that \(\xi\pi(\text{td} (\Omega_X^{\vee}))= \tau([A])\), where \(\text{td} (\Omega_X^{\vee})\) denotes the Todd class of the locally free sheaf \(\Omega_X^{\vee}\) and \(A\) is the local ring of \(Z\). As a consequence, \(\tau([A])= q_r\) if and only if \(\text{td}(\Omega_X^{\vee}) \equiv 1\text{ mod}(c)\).
scientific article; zbMATH DE number 889577
scientific article