Logarithmic Flatness
Logarithmic Flatness
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对数平坦度
DOI：
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发表时间：
2016
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通讯作者：
W. D. Gillam
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文献类型：
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作者：
W. D. Gillam
细对数方案的映射X→Y诱导出从X下的方案到Y上的细对数方案的严格态射的Olsson代数堆栈的映射。X上的一个叶称为在Y上对数平坦当且仅当它在这个代数堆栈上是平的。本文研究了么半群和分次环映射的对数平坦性及其相关的平坦性概念。结果表明，对数平坦度等价于更一般的“形式对数平坦度”概念，该概念对于任意对数环拓扑映射是有意义的。给出了许多“自然”出现的X→Y的具体对数平坦度标准，例如环面变种、节点曲线等。对于非常简单的X→Y，证明了对数平坦度等价于已有的“完美”概念，从而为更复杂的X→Y提供了推广，有助于通过退化技术研究滑轮的模数。
A map of fine log schemes X → Y induces a map from the scheme underlying X to Olsson’s algebraic stack of strict morphisms of fine log schemes over Y . A sheaf on X is called log flat over Y iff it is flat over this algebraic stack. This paper is a study of log flatness and the related notions of flatness for maps of monoids and graded rings. It is shown that log flatness is equivalent to a more general notion of “formal log flatness” that makes sense for an arbitrary map of log ringed topoi. Concrete log flatness criteria are given for many X → Y that occur “in nature,” such as toric varieties, nodal curves, and the like. For very simple X → Y it turns out that log flatness is equivalent to previously extant notions of “perfection,” thus it provides a generalization for more complicated X → Y useful for studying moduli of sheaves via degeneration techniques.