flat morphism

Constant-dimensional fibers.

For ${\mathcal{F}}\in {}_{{\mathcal{O}}_X}{\mathsf{Mod}} $ then $\mathcal{F}$ is a flat sheaf relative to $Y$ at a point $x \in X$ if ${\mathcal{F}}_{x}\in {}_{{\mathcal{O}}_{Y, f(x)}}{\mathsf{Mod}} ^\flat$ is flat as a module.
- A sheaf $\mathcal{F}$ is a flat sheaf if it is flat at every point of $X$.
A morphism $X \xrightarrow{f} Y$ is a flat morphism at a point $x \in X$ if ${\mathcal{O}}_{X, x}\in {}_{{\mathcal{O}}_{Y, f(x)}}{\mathsf{Mod}} ^\flat$ is flat as a module.
- A morphism $f$ is a flat morphism if $f$ is flat at every point $x \in X$.
A scheme $X$ is flat if its structure sheaf ${\mathcal{O}}_X$ is a flat sheaf.

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proper morphism

- Tags: - #AG/basics - Refs: - #todo/add-references - Links: - flat morphism - projective morphism - separated morphism
fpqc

fidèlement plat et quasi-compact, i.e. a flat morphism which is quasicompact.
fppf

fidèlement plat de présentation finie, i.e. a flat morphism of finite presentation. An fppf covering is a covering $T_i\to X$ by flat morphisms which are locally of finite presentation.