Flatness testing over singular bases

作者：Adamus Janusz^*; Seyedinejad Hadi

来源：Annales Polonici Mathematici, 2013, 107(1): 87-96.

DOI：10.4064/ap107-1-6

• flat
• torsion-free
• fibred power
• tensor power
• vertical component
• vertical element
• analytic tensor product

We show that non-flatness of a morphism phi : X -%26gt; 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 phi to the desingularization of Y. An algebraic analogue follows: Let R be a locally (analytically) irreducible finite type 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 Spec S -%26gt; 4 Spec R is dominant. Then a finite type R-algebra A is R-flat if and only if (AR(circle times Rn))circle times(R) S is a torsion-free R-module.

• 出版日期2013