The contraction property on the relative weak normalization and Lipschitz saturation of algebras
Thiago da Silva
TL;DR
The paper addresses how the relative weak normalization $\widetilde{A}_{B,R}$ and the relative Lipschitz saturation $A^*_{B,R}$ contract along diagrams $R\to A\to B$, even when $A\to B$ need not be an inclusion. Building on Lipman and SR techniques, it extends these contraction results to Maranesi diagrams and analyzes them under universal injectivity, integrality, radiciality, and unramified conditions, using radicals of kernels and tensor-product behavior. Key contributions include establishing $f^{-1}(\widetilde{A'}_{B',R'})=\widetilde{A}_{B,R}$ in several settings, proving that weak normalization commutes with quotients by ideals, and deriving corresponding contraction results for Lipschitz saturation. These results provide base-change compatible control of weak normalization and Lipschitz saturation, with potential applications in normalization procedures and singularity theory in algebraic geometry.
Abstract
Inspired by the results obtained in \cite{SR}, in this work, we develop techniques to handle the contraction property for weak normalization and Lipschitz saturation of algebras for the following types of algebras: universally injective, integral, radicial, and unramified.