On the structure presheaf of a differential ring
The author considers an extension to differential algebra of an equivalent definition of the usual structure sheaf \(\tilde A\) of a commutative ring A. The new definition is obtained by consideration of a family C of ideals of A. One then defines a presheaf \(\tilde A_ C\) by means of rings of fractions \(S^{-1}A\) relative to C. When the family C consists of all ideals in A, the usual definition of \(\tilde A\) is recovered. Properties of the family C are reflected in the presheaf \(\tilde A_ C\), and it is shown that under certain conditions on C, the presheaf \(\tilde A_ C\) is a sheaf. Implications of this result are considered when A is a differential ring and C is the family of differential ideals in A. In particular, one is able to identify the differential ring \(A_ D\) of global sections of \(\tilde A\) as a ring of fractions when A is ''very nice'', and even show that \(A_ D\cong A\) under certain conditions on A. In the final section the usual pair of adjoint functors given by Spec and global sections between the categories of commutative rings and local ringed spaces is extended to the differential case. One problem which arises in the differential case is that a morphism \(Spec_ DB\to Spec_ DA\) of local differential ringed spaces, where A and B are differential rings, may not correspond to a differential ring homomorphism \(A\to B\), unlike the usual situation in algebraic geometry. However, when one restricts attention to the case when A and B are very nice, the correspondence between morphisms \(Spec_ DB\to Spec_ DA\) and differential ring homomorphisms \(A\to B\) is recovered.
scientific article; zbMATH DE number 3865455