Flatness testing over singular bases

Janusz Adamus¹ ; Hadi Seyedinejad²

¹ Department of Mathematics The University of Western Ontario London, Ontario, Canada N6A 5B7 and Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland
² Department of Mathematics The University of Western Ontario London, Ontario, Canada N6A 5B7

Annales Polonici Mathematici, Tome 107 (2013) no. 1, pp. 87-96.

Résumé

We show that non-flatness of a morphism $\varphi:X\to Y$ of complex-analytic spaces with a locally irreducible target of dimension $n$ manifests in the existence of vertical components in the $n$-fold fibred power of the pull-back of $\varphi$ to the desingularization of $Y$. An algebraic analogue follows: Let $R$ be a locally (analytically) irreducible finite type $\mathbb C$-algebra and an integral domain of Krull dimension $n$, and let $S$ be a regular $n$-dimensional algebra of finite type over $R$ (but not necessarily a finite $R$-module), such that $\mathop{\rm Spec} S\to\mathop{\rm Spec} R$ is dominant. Then a finite type $R$-algebra $A$ is $R$-flat if and only if $(A^{\otimes^n_R})\otimes_RS$ is a torsion-free $R$-module.

Détail

DOI : 10.4064/ap107-1-6

Keywords: non flatness morphism varphi complex analytic spaces locally irreducible target dimension manifests existence vertical components n fold fibred power pull back varphi desingularization algebraic analogue follows locally analytically irreducible finite type mathbb c algebra integral domain krull dimension regular n dimensional algebra finite type necessarily finite r module mathop spec mathop spec dominant finite type r algebra r flat only otimes otimes torsion free r module