---
tag: 5M6KS8F47R
kind: concept
name: Flat morphism
msc: [14A15, 14B25, 13C11]
sources:
  - url: https://en.wikipedia.org/wiki/Flat_morphism
same_as:
  wikidata: Q3324325
  wikipedia: Flat morphism
---

## Summary

A flat morphism of schemes is a morphism whose local rings are flat modules over the local rings of the target.
Flatness is the algebraic expression of a family whose fibres vary continuously.
In a flat family of projective schemes over a connected base, numerical invariants such as the Hilbert polynomial stay constant from fibre to fibre.

## Explanations

### L1 human

A flat morphism is a family of spaces with no sudden jumps.
Open immersions and the projection $\mathbb{A}^2 \to \mathbb{A}^1$ are flat.
The blow-up of the plane at a point is not flat: all its fibres are points except one, which is a whole line.
Flatness forbids such a jump in the fibre dimension, and also subtler jumps, such as a fibre suddenly gaining an embedded point.