Thiếu úy
Give a counter example to the following statement: If $f:X\to Y$ is flat, of finite type with $Y$ locally Noetherian then $f$ is smooth if and only if the fibers $X_y\to \mathrm{Spec}(k(y))$ is smooth for every $y$ ‘’closed’’ in $Y$.
Remark: the above statement is true if $X,Y$ are schemes over a field $k$.
Bài viết đã được chỉnh sửa nội dung bởi Nxb: 05-07-2023 - 13:17
Thiếu úy
Give a counter example to the following statement: If $f:X\to Y$ is flat, of finite type with $Y$ locally Noetherian then $f$ is smooth if and only if the fibers $X_y\to \mathrm{Spec}(k(y))$ is smooth for every $y$ ‘’closed’’ in $Y$.
Remark: the above statement is true if $X,Y$ are schemes over a field $k$.
Em thấy có một ví dụ ở đây:https://mathoverflow...hemes-imply-pro
Nếu thêm điều kiện $f$ là proper morphism thì kết quả trên là đúng (Bài 6.2.8 trang 227 trong sách của Qing Liu). Ở link trên họ cũng nói properness ảnh hưởng đến kết quả.
"The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!" - Grothendieck
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