Today is 20/03/2020 and I decided to learn some basics about affine varieties, following Gathmann’s note mainly. Other reference include Milne’s note.

The current goal is learn enough algebraic geometry to learn about (linear) algebraic groups.

Affine varieties

Convention: $K$ is algebraic closed field. (Algebraic geometry can be done over arbitrary fields or over rings but instead of varieties, the best language to use if that of scheme theory).

By affine $n$-space over field $K$, we mean simply the vector space $K^n$, which is usually denoted as $\mathbb{A}_K^n$ or just $\mathbb{A}^n$ (i.e. writing this means we ignore the addition and scalar multiplication that occur in $K^n$). Given subset $S\subset K[x_1,\ldots, x_n]$ of polynomials we call set of common roots in $\mathbb{A}^n$ of $S$ the (affine) zero locus (or algebraic set) of $S$, denoted as $V(S)={x\in \mathbb{A}^n: f(x)=0 \; \forall x\in S}$. An *affine variety $X\subset \mathbb{A}^n$ is the zero locus $V(S)$ of collection $S\subset K[x_1,\ldots, x_n]$.

Instead of viewing affine variety $X$ as zero locus of collection $S$ of polynomials, $X$ can be viewed as zero locus of ideal $\langle S\rangle$ of $K[x_1,\ldots, x_n]$ since $V(S)=V(\langle S\rangle)$. Furthermore, as any ideal in $K[x_1,\ldots, x_n]$ is finitely generated by Hilbert basis theorem, this means that any affine variety can be written as zero locus of finitely many polynomials.

Conversely, for each subset $X\subset \mathbb{A}^n$, the collection of all polynomials vanishing on $X$ is an ideal of $K[x_1,\ldots, x_n]$, denoted as $I(X)$. We have the inclusions $X \subset V(I(X))$ and $J\subset I(V(J))$ for $X\subset \mathbb{A}^n$ and ideal $J\subset K[x_1,\ldots, x_n]$. A question is that whether these are actually equal or not.

It is not always true that $J=I(V(J))$. A quick way to see $I(V(J))\ne J$ is notice that if we work with $K[x_1]$ (i.e. $n=1$) then with $f(x_1)=(x_1-a)^k$ then $V(f)={a}$, which does not contain the data about multiplicity $k$, so when we apply $I(.)$ to $V(f)$ we obtain $I(V(f))=\langle (x_1-a)\rangle$, which is not $J=\langle (x_1-a)^k\rangle$ if $k>1$. This example suggests us to define the radical $\sqrt{J}$ of ideal $J$ of ring $R$ to be

\[\sqrt{J}=\{f\in R: f^k \in J \; \text{for some } k\in \mathbb{N}\}\]

In fact, $I(V(J))=\sqrt{J}$, as shown by following theorem

(Hilbert’s Nullstellensatz) For any affine variety $X\subset \mathbb{A}^n$ we have $V(I(X))=X$. For any ideal $J$ of $K[x_1,\ldots, x_n]$, we have $I(V(J))= \sqrt{J}$. In particular, there is an inclusion-reversing bijection

\[\begin{align*} \{ \text{affine varieties in } \mathbb{A}^n \} & \longleftrightarrow \{ \text{radical ideal in } K[x_1,\ldots, x_n] \} \\ X & \longrightarrow I(X) \\ V(J) & \longleftarrow J. \end{align*}\]

A consequence of this theorem is the weak Nullstellensatz: For nonzero ideal $J$ in $K[x_1,\ldots, x_n]$ then $V(J)$ is nonempty, i.e. $J$ has a zero.

Let $X \subset \mathbb{A}^n$ be affine variety. A polynomial function on $X$ is a map $X\to K$ that is of the form $x \mapsto f(x)$ for some $f\in K[x_1,\ldots, x_n]$. The ring (actually $K$-algebra) $A(X)$ of polynomial functions on $X$ is just the quotient ring $A(X)=K[x_1,\ldots, x_n]/I(X)$, which is called coordinate ring of affine variety $X$.