An adelic construction of Chern classes (Q2880045)

The author gives a formula for the second Chern class \(c_2(E)\) in terms of trivializations of a rank two vector bundle at scheme points of a surface \(X\) over a field.NEWLINENEWLINE Let \(X\) be a Noetherian irreducible scheme of finite type of dimension \(n\) over a field \(k\) and let \({\mathcal E}\) be the locally free sheaf of \({\mathcal O}_X\)-modules defined by a vector bundle \(E\). For each point \(x\in X\) let \(b_x\) be a basis of the free \({\mathcal O}_{X,x}\)-module \({\mathcal E}_x\). A collection \(b= (b_x)_{x\in X}\) is called an adelic trivialization of \(E\). One can associate with \(b\) a collection of matrices \(g= g_{x_0,x_1}\), where \(g_{x_0 x_1}\in \text{GL}({\mathcal O}_{X,x_0})\), by setting \(b_{x_1} g_{x_0x_1}= b_{x_0}\) if \(x_1\in\overline x_0\). In the case \(X\) is a smooth variety over a field \(k\) of characteristic \(p>0\) one gets a formula which the highest Chern number \(C_{n,X}(E)\pmod\). By \textit{S. Bloch} [Ann. Math. (2) 99, 349--379 (1974; Zbl 0298.14005)] the Chern class \(c_{2,X}(E)\) of a vector bundle \(E\) with trivial determinant was constructed using the cohomology group \(H^2(X,{\mathcal K}_2({\mathcal O}_X)\)) associated to Quillen K-groups. In this paper Bloch's approach is applied to adelic complexes.NEWLINENEWLINE The author also shows that Severi's formula for the second Chern class may be obtained as a special case of the formula proved in this paper.