Structure sheaves on almost commutative algebras
An algebra \(R\) over a field \(k\) of characteristic \(0\) is said to be almost commutative if there exists a positive filtration \(0\subset F_0R\subset F_1R\subset\cdots\) of \(R\) such that \(F_0R=k\) and the associated graded ring \(G(R)\) is a commutative affine \(k\)-algebra generated by homogeneous elements of degree \(1\). In previous work, the author constructed a sheaf \({\mathcal O}^F_Y\) on the scheme \(Y=\text{Proj}(G(R))\), in such a way that for every \(p\in Y\), the stalk \({\mathcal O}^F_{Y,p}\) coincides with the localization \(S^{-1}_pR\) of \(R\) at the saturated subset \(S_p\) of \(R\) corresponding to the Ore set consisting of homogeneous elements in \(G(R)-p\). In this paper, the author first constructs the sheaf \({\mathcal O}^F_Y\) in an alternative way, modulo the extra assumption that \(G(R)\) be a domain and provides a detailed description of the rings of sections of this sheaf over basic open sets. It is then shown that the sheaf \({\mathcal O}^F_Y\) can really be very helpful to represent the base ring \(R\) geometrically: the author gives general conditions (satisfied by large classes of concrete rings) under which \(R\) reduces to the ring of global sections of the sheaf \({\mathcal O}^F_Y\). The remainder of the paper is essentially devoted to a thorough study of the lifting properties of the stalks \({\mathcal O}^F_{Y,p}\), the local study of \(R\) and the application of local-global results, using the fact that the induced map \(R\to\bigoplus S^{-1}_pR\) is faithfully flat. As a final result, the author generalizes to the present situation the continuity of the local-rank function.
scientific article; zbMATH DE number 1448609
scientific article