Pure subrings of commutative rings (Q2828005)

Let \(A,B\) be commutative rings and \(A\subset B.\) \(A\) is a pure subrings of \(B \) if the canonical homomorphism \(M\otimes _{A}A\rightarrow M\otimes _{A}B\) is also injective for any \(A\)-module \(M.\) The authors investigate pure subalgebras of affine and local domains from algebraic and geometric viewpoints. The main results of the paper are:NEWLINENEWLINE1. Let \(R\subset S\) be algebraic local rings (i.e. localization of an affine algebra over an algebraically closed field \(k,\) \(\text{char}(k)=0,\) at a maximal ideal). Assume that this is a pure extension. If \(R\) is either reduced, or Gorenstein, then the induced map of completions \(\hat{R}\subset \hat{S}\) is also pure.NEWLINENEWLINE2. Let \(R\subset S\) be algebraic local rings such that \(\hat{R}\) is an integral domain. If the morphism \(\text{Spec}S\rightarrow \text{Spec}R\) is a surjection, then \(\text{Spec}\hat{S}\rightarrow \text{Spec}\hat{R} \) is also a surjection.NEWLINENEWLINE3. Let \(R\subset S\) be a pure extension of analytic local rings over \( \mathbb{C}\) such that \(R\) is an integral domain. Then the image of a Euclidean neighbourhood of the closed point of \(\text{Spec}S\) contains a Euclidean neighbourhood of the closed point of \(\text{Spec}R.\)NEWLINENEWLINE4. Let \(A\subset B\) be an inclusion of normal affine domains over a field \(k\) such that the morphism \(\text{Spec}B\rightarrow \text{Spec}A\) is generically finite and surjective. Then the exists a normal affine domain \( C/k\) such that \(A\subset C\subset B,\) the morphism \(\text{Spec} C\rightarrow \text{Spec}A\) is quasi-finite and surjective, and \(C\) is the largest affine subring of \(B\) containing \(A\) with these properties. Further, \(C\) is a pure extension of \(A.\)