Chains of differential subvarieties in an algebraic variety.
A \(\delta\)-ring is a commutative ring \(A\) with \(1\) containing \(\mathbb{Q}\) together with a derivation \(\delta\) of \(A\). In the first part of his paper, the author shows that the lying over theorem, the going-up theorem, and the going-down theorem for \(\delta\)-integral domains still hold, if only \(\delta\)-invariant prime ideals are considered instead of arbitrary prime ideals. These results are applied to the theory of increasing chains of \(\delta\)-subvarieties of a \(\delta\)-variety \(X\) over some differentially closed field \(F\) of characteristic \(0\). The author calls such a chain a long chain, if it has at least \(d\) disjoint subchains of infinitely many subvarieties, where \(d\) is the \(\delta\)-transcendence degree of \(F\langle X\rangle\) over \(F\). He proves the so-called Kolchin catenary problem: Given any \(x\in X\), there is a long chain in \(X\) starting at \(x\).
scientific article; zbMATH DE number 1749650