AI 中文总结
研究代数几何中平坦性，在半环背景下获离散性结果等，构造不可下降的忠实平坦环映射并展示下降指数与基数关系，证明特定态射极大平展轨迹的仿射性，还有相关引理变体、\(D -\)模刻画及Tor - 独立性结果。
AI 中文摘要
本论文研究代数几何中平坦性的各个方面。首先在半环背景下研究平坦性，证明关于单纯半环上同伦理论模型结构的导出群化函子的一些离散性结果，比较扎里斯基拓扑和fppf拓扑，研究正实数上的fppf代数。接着研究忠实平坦环映射的下降性质，构造不可下降的忠实平坦环映射，并通过一种通用方法构建示例展示下降指数与基数的精确关系。最后证明在\(X\)和\(Y\)是局部诺特概型、\(X\)正规且\(Y\)正则的情况下，态射\(X \to Y\)的极大平展轨迹的仿射性。还包括非诺特环的Artin - Rees引理变体、非分歧正则局部环上\(D -\)模的刻画以及通过不要求完备化的倾斜对应变体证明的平展概型整体截面的Tor - 独立性结果等。
英文摘要
This thesis studies various aspects of flatness in algebraic geometry. We first study flatness in the context of semi-rings. We prove some discreteness results for a derived groupification functor with respect to the homotopy theoretic model structure on simplicial semirings, compare the Zariski and fppf topology, and study fppf algebras over the positive reals. We then study descendibility properties of faithfully flat ring maps; we in particular construct a non-descendable faithfully flat ring map, and then construct examples demonstrating the precise the relationship between the exponent of descendibility and cardinality by developing a rather general method to convert module-theoretic non-vanishing "cup-products" to descendable faithfully flat ring maps with sufficiently high exponent. Finally, we prove the affineness of the maximal etale locus of morphism $X \to Y$ of schemes with $X$ and $Y$ locally Noetherian schemes, $X$ normal and $Y$ regular in fully generality. Some notable aspects of this chapter is a variant of Artin-Rees' lemma that holds for non-Noetherian rings, a characterization of $D$-modules over unramified regular local rings (similar to those obtained by O. Gabber and W. Zhang), and Tor-independence result for global sections of étale schemes which we prove by a variant of the tilting correspondence which doesn't require completion.
