Babylonian tower theorems on the punctured spectrum
Horrocks has posed the following problem: Let \(...\to R_{n+1}\to R_ n\to...\to R_ 0\) be a ''tower'' of regular local rings, i.e. \(R_ n=R_{n+1}/(t_{n+1})\) for some nonzero divisor \(t_{n+1}\in R_{n+1}\). For every n, let \({\mathcal E}_ n\) be a vector bundle on the punctured spectrum of \(R_ n\) such that \({\mathcal E}_ n\) is the restriction of \({\mathcal E}_{n+1}\). Is then \({\mathcal E}_ n\) trivial for each n? In this paper an affirmative answer is given. Moreover, in most of the geometrically interesting cases there is a bound for how far \({\mathcal E}_ 0\) can be extended at most.- This generalizes results of Evans-Griffiths resp. Barth-Van de Ven, Sato and Tyurin in the projective case. The main idea of the proof consists in applying formal deformation theory. In the second part of the paper a similar result is derived for locally complete intersections on the punctured spectrum of a regular local ring.
scientific article; zbMATH DE number 3859282