Regular map
Definition
Let $K$ be an algebraically closed field.
In affine space
Algebraic set > field
Definition. Let $V$ be an algebraic subset. A map $f:V\to K$ is called regular if there exists $F\in K[x_1,\dots,x_n]$ such that $F\mid_V=f$.
In other words, $f$ is regular if it is the restriction of a polynomial.
The regular maps form an algebra, denoted $K[V]$. There is a surjective restriction map $$ \phi:K[x_1,\dots,x_n]\longrightarrow K[V]. $$ Its kernel is denoted $I(V)$, and consists of those polynomials in the domain which restrict to 0 on $V$. We sometimes call $K[V]$ a coordinate ring.
It is natural to ask whether for any ideal $I$ if the image $K[x_1,\dots,x_n]/I$ is the coordinate ring of some algebraic set. This is not the case. In fact, it is not even true in general that $I$ is the ideal generated by some algebraic set. The conditions under which such things behave better is formulated in the Nullstellensatz.
Quasi-affine algebraic set > field
Definition. Let $U$ be a quasi-affine algebraic set. A map $f:U\to K$ is called regular if there exists $F,G\in K[x_1,\dots,x_n]$ such that $f=\frac{F}{G}\mid_U$, where $G(x)\neq 0$ for all $x\in U$.
In other words, $f$ is regular if is the restriction of a rational polynomial.
These regular maps form an algebra, denoted $K[U]$. The assignment of a quasi-affine algebraic set to its algebra of regular maps determines a contravariant functor, in the sense that a map $$ \phi:X\to Y $$ between quasi-affine algebraic subsets induces a map
\begin{gather}
\phi^\ast:K[Y]\to K[X] \
f\mapsto f\circ \phi
\end{gather}
between algebras of regular maps.
Algebraic set > algebraic set
Definition. A map $\phi:X\to Y$ between algebraic subsets is called regular if $$ \phi(x_1,\dots,x_n)=(\phi_1(x_1,\dots,x_n),\dots,\phi_m(x_1,\dots,x_n)) $$ where $\phi_i\in K[x]$.
In other words, it is regular if each component function is a polynomial in $K$.
Projective
Let $X$ be a projective algebraic set. Let ${U_i}$ be the covering of $\mathbb{P}^n$ where each $U_i$ is homeomorphic to $\mathbb{A}^n$.
Algebraic set > field
Definition. A function $f:X\to k$ is called regular if for all $i$ the restriction $$f\mid_{X\cap U_i}:X\cap U_i\to K$$ is a regular map (from an affine algebraic set to a field).
Variations
Operations
Derivatives
Given a morphism $F\in K[U]$ for some variety $U$, we can formally define the differential of $F$ at $p=(a_1,\dots,a_n)$, denoted $\text{d}F|_p$ or $\text{d}_pF$, as $$\text{d}pF=\sum{i=1}^n\frac{\partial F}{\partial x_i}\big|_p\cdot (x_i-a_i).$$ Observe this is the degree 1 term of the Taylor expansion of $F$, and in this sense agrees with our other notions of derivatives.
This may be exteded to a linear function into the cotangent space: \begin{gather}\text{d}_p:K[U]\to (T_pU)^\ast\ f\mapsto \text{d}_pf\end{gather} which restricts to an isomorphism $$\text{d}_p:\mathfrak{m}_p/\mathfrak{m}^2_p\overset{\sim}{\longrightarrow}(T_pU)^\ast$$ (see that entry for more).
We have the analagous properties:
- $\text{d}_p(F+G)=\text{d}_pF+\text{d}_pG$
- $\text{d}_p(FG)=F(p)\text{d}_pG+G(p)\text{d}_pF$
Properties
Affine
- Given a regular map $f:X\to Y$, then $f^\ast:K[Y]\to K $ is injective iff $f$ is dominant (\overline{f(X)}=Y).
- For an affine algebraic set $X$, $$K ={f\mid f=\frac{F}{G}\mid_X}.$$
Projective
- Let $X$ be a quasi-projective algebraic set. Let $F,G\in K[x_0,\dots,x_n]$. Then if $F,G$ are homogeneous of degree $d$ then $f=\frac{F}{G}\mid_X$ is regular (if $G(X)\neq 0$ for all $x\in X$).