nLab regular map

Idea

A regular map is a morphism of algebraic varieties.

Definition

If the varieties $M\subset\mathbb{A}^m$, $N\subset\mathbb{A}^n$ are affine, then a regular map is a restriction-corestriction of a polynomial map from $\mathbb{A}^m\to \mathbb{A}^n$. It automatically induces the map on the level of structure sheaves.

For general varieties, a map is regular if there is a cover of the domain by the affines $U_i, i\in I$ and the codomain by the affines $V_i, i\in I$ such that $f(U_i)\subset V_i$ and the restriction-corestriction to a map $f| : U_i\to V_i$ is regular.

References

• Andreas Gathmann, Section 4 of: Algebraic Geometry, University of Kaiserslautern, 2021 pdf

• J. S. Milne, Sect. 3.g of: Algebraic geometry, 2017 pdf

category: algebraic geometry