光滑基底上的平坦性检测
Detecting flatness over smooth bases
摘要：
给定方案f：X --> Y的本质有限型态射和一个正整数d，设f^{d}：X^{d} --> Y表示X在Y上的d重纤维积X^{d}的自然映射，π_i：X^{d} --> X表示第i个标准投影.当Y在域上光滑且F是X上的凝聚层时，证明了F在Y上平坦当且仅当f^{d}将张量积层\otimes_{i=1}^d π_i^*（F）的相伴点映射到Y的泛点，其中d大于或等于dim Y.交换代数中的等价陈述是Auslander和Lichtenbaum关于正则局部环上有限生成模自由性的经典准则的一个类比--而不是一个推论。
英文摘要：
Given an essentially finite type morphism of schemes f: X --> Y and a positive integer d, let f^{d}: X^{d} --> Y denote the natural map from the d-fold fiber product, X^{d}, of X over Y and π_i: X^{d} --> X the i'th canonical projection. When Y smooth over a field and F is a coherent sheaf on X, it is proved that F is flat over Y if (and only if) f^{d} maps the associated points of the tensor product sheaf \otimes_{i=1}^d π_i^*(F) to generic points of Y, for some d greater than or equal to dim Y. The equivalent statement in commutative algebra is an analog---but not a consequence---of a classical criterion of Auslander and Lichtenbaum for the freeness of finitely generated modules over regular local rings.
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