Riemann-Roch theorem for locally principal orders
Let C be a smooth integral projective curve over a field k algebraically closed in the field K of rational functions on C. Let M and L be C-vector bundles of \(rank\quad n\) on \(V=K^ n\). Then the Riemann-Roch theorem says that \(\ell (L)=\deg_ ML+\ell (L^{\omega})-g_ M+1,\) where \(\ell (L)=\dim_ MH^ 0(C,L)\), \(L^{\omega}={\mathcal H}om(L,\omega_ C)\), \(\omega_ C\) is the canonical sheaf on C, \(g_ M=\dim_ kH^ 1(C,M)- \dim_ kH^ 0(C,M)+1\), and \(\deg_ ML\) is the appropriately defined degree of L with respect to M. Usually, \(M\cong {\mathcal O}^ n_ C\), \(\deg_ ML=\deg \bigwedge^ n(L)\), and then \(g_ M=ng_ C-n+1\), where \(g_ C\) is the genus of C. If V has an additional structure, the choice of M may depend on it. For example, Witt's version of the Riemann-Roch theorem [\textit{E. Witt}, Math. Ann. 110, 12-28 (1934; Zbl 0009.19301)] is concerned with the case of central simple K-algebras V and maximal C-orders M in V (that is, C- bundles M such that \(M_ x\) is a subring of V containing the local ring \({\mathfrak O}_{x,C}\) for each \(x\in C\), and not properly contained in any M' having the same properties). In this case, \(\deg_ ML=\deg N(L)\), where L is an M-ideal on V, and N is the norm function from V to K. Moreover, \(L^{\omega}\) can be computed in terms of the inverse \(L^{- 1}\) to L. Here we prove an extension of the Riemann-Roch-Witt theorem to arbitrary finite dimensional semisimple separable K-algebras and to a class of orders M which still allow to compute \(\deg_ ML\) and \(L^{\omega}\) in terms of the multiplicative structure of M as for maximal orders. These orders are characterized in different ways and a specific structural description of them is given. One of the characterizations says that M is such an order if and only if for each M-ideal L on V, which is not an ideal for an order properly containing M, the completion \(\hat L_ x\) is a principal \(\hat M_ x\)-ideal for each \(x\in C\).
finite dimensional semisimple separable K-algebras
scientific article; zbMATH DE number 3959611
scientific article