The Miracle of Flatness in Algebraic Geometry
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 \'{e}tale schemes which we prove by a variant of the tilting correspondence which doesn't require completion.