Prolongation of regular-singular connections on punctured affine line over a Henselian ring (Q6561420)

The paper under review generalizes Deligne's equivalence between the categories of regular-singular connections on the formal punctured disk and on the punctured affine line (see [\textit{P. Deligne}, Publ., Math. Sci. Res. Inst. 16, 79--297 (1989; Zbl 0742.14022)]) to the case where the base is a strictly Henselian discrete valuation ring of equal characteristic zero. The main result of the paper is as follows. Let \(R\) be a Noetherian Henselian local \(C\)-algebra, where \(C\) is an algebraically closed field of characteristic zero, and \(x\) a variable. Let \(\mathbf{MC}_{rs}(\ast/R)\) denote the category of regular-singular connections on \(\ast\) over \(R\) and let \(\mathbf{MC}^{0}_{rs}(\ast/R)\) denote the full subcategory of objects whose underlying modules are \(R\)-flat. Then the restriction functor \(\mathbf{r:\,MC}^{0}_{rs}(R[x^{\pm}]/R)\rightarrow \mathbf{MC}^{0}_{rs}(R((x))/R)\) is an equivalence provided that \(R\) is a \(G\)-ring. If \(R\) is, moreover, a discrete valuation ring, then \(\mathbf{ r:\,MC}_{rs}(R[x^{\pm}]/R)\rightarrow\mathbf{MC}_{rs}(R((x))/R)\) is an equivalence.\N\NThe authors' approach is based on Deligne's equivalence as presented in [\textit{P. H. Hai} et al., ``Algebraic theory of regular-singular connections with parameters'', Rend. Seminario Mat. Università Padova (to appear), Theorem 10.1] and Popescu approximation (see [\textit{D. Popescu}, Nagoya Math. J. 104, 85--115 (1986; Zbl 0592.14014)]). The authors also provide a weaker result when the base is higher dimensional.