Two remarks on integral morphisms of affine k-varieties (Q1071826)

The notion of integral morphism between affine algebraic k-varieties (k any field) has been studied by the author and \textit{M. Raimondo} [in Commutative algebra, Proc. Conf., Trento/Italy 1981, Lect. Notes Pure Appl. Math. 84, 183-196 (1983; Zbl 0508.14009)], where an affine variety defined over k is a couple (X,\({\mathcal O}_ X)\) with X homeomorphic to a Zariski closed subset of some \(k^ n\) and \({\mathcal O}_ X\) the sheaf of germs of regular functions on X. In this note we add two pinpoints on the topic. More precisely, let \({\mathcal V}_ k\) denote the category of affine k-varieties and relative morphisms. We first show that in \({\mathcal V}_ k\) any pair (\(\Phi\),\(\Psi)\) of morphisms such that \(\Phi\) is a closed immersion and \(\Psi\) is integral has a push-out. Secondly we give an example of X in \({\mathcal V}_ k\) for which the weak normalization (in the sense of Andreotti and Bombieri) of the affine scheme \((Spec \Gamma (X,{\mathcal O}_ X),\Gamma (X,{\mathcal O}_ X)^{\sim})\) is not in \({\mathcal V}_ k\).