Logarithmic Structures of Fontaine-Illusie. II ---Logarithmic Flat Topology

This is a continuation of the paper [K1] on the foundation of log geometry in the sense of Fontaine-Illusie. Here we discuss mainly log flat topologies, especially log flat descent theory. This paper was started around 1991, and was circulated as an incomplete preprint for a long time. Since then, some contents of this paper have been reproduced by several authors with proofs ([Ha], [KS], [Ni], [Na2], [Ol], ...). A. Moriwaki [M] also studied flat descents in the category of log schemes. In the parts which were incomplete in the circulated preprint, we sometimes referred to these papers instead of completing the original proofs. In particular, the author does not claim the results with ∗ (i.e., two theorems 7.1, 7.2 and one proposition 6.5) are his results. Since the paper is already referred to in many published works, in the other parts, we preferred to preserve the original, circulated form. In both parts, we tried to preserve the original numberings of definitions and propositions. The author wishes to express his special thanks to Chikara Nakayama who helped him a lot in the completion of this paper. He is also thankful to Luc Illusie, Takeshi Saito, Takeshi Kajiwara, and Takeshi Tsuji for helpful discussions. The author is partially supported by NSF Award 1601861.