---- Let $k \subset \Omega$ be [[field|fields]], where $\Omega$ is [[algebraically closed]]. > [!definition] Definition. ([[algebraic set]]) > 1. Let $S$ be a subset of $k[T_{1},\dots,T_{n}]$. Put $V(S):= \{ \boldsymbol x \in \Omega^{n}: f(\boldsymbol x)=0 \ \forall f \in S \}$. A set of the form $V(S)$ is called an **(affine)** **$k$-algebraic subset of $\Omega^{n}$**, or an **algebraic subset of $\Omega^{n}$ defined over $k$**. > > >2. Let $X$ be a subset of $\Omega^{n}$. Put $I(X):=\{ f \in k[T_{1},\dots,T_{n}]: f(X)=0 \}.$ A set of the form $I(X)$ is called the **$k$-ideal of $X$**. > Clearly, if $\langle S \rangle$ is the [[ideal]] of $k[T_{1},\dots,T_{n}]$ [[ideal generated by a subset|generated by]] $S$, then $V(\langle S \rangle)=V(S)$. > By **algebraic subset $V(S)$ of $\Omega^{n}$** (without the $k$- prefix) one means a $\Omega$-algebraic subset of $\Omega^{n}$. > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \definecolor{thistle}{RGB}{216,191,216} > \usepackage{xcolor} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGUQBfU9TXfIRQBGclVqMWbANbIAKgH1RAAgA6qqBBxxlisBW68QGbHgJEATGOr1mrRCACSACgAaASkN9Tgy6WHitlIOrl7G-GZCyKIBNpL2IOoA8gC2MADmdAB6hDzeAuYoZLESdmwAas7snlziMFDp8ESgAGYAThApSACs1DgQSAAs1AxYYAmaODj1INQAFjB0UGyQ47OlwYmqcEwARnAwOOsMdLswDAAKEb4OY9iwYe2dSGQg-UiiIAtLKwSsI2MJlppss4mUHOodvtDsdTucrj5CiA7lgHnkQE8uogrG8BohPt9lg5Vv8QKM1g5JiD1kEEpCmGgDkcRnDLtckSi0UZMUgAMx9PHDL6LIngP7HQFsKkzMGbemMmEss5sxFCZFge6sdE8xCvd6IbrajpY-m4pAWWpcIA > \begin{tikzcd} > S \arrow[r, "\subset" description, no head, dotted] \arrow[d] & {k[T_1, \dots, T_n]} & \color{thistle}I(X) \arrow[l, "\supset" description, no head, dotted] \\ > \color{thistle}V(S) \arrow[r, "\subset" description, no head, dotted] & \Omega^n & X \arrow[l, "\supset" description, no head, dotted] \arrow[u] > \end{tikzcd} > \end{document} > ``` > > [!basicproperties] > > 1. (Inclusion reversal) For subsets $X \subset Y$ of $\Omega^{n}$, we have $I(X) \supset I(Y)$. Similarly, for subsets $S \subset T$ of $k[T_{1},\dots,T_{n}]$, we have $V(S) \supset V(T)$. > 2. It follows that $V(I(\cdot))$ and $I(V(\cdot))$ *preserve* inclusions. > 3. For $S \subset k[T_{1},..,T_{n}]$, we have $S \subset I(V(S))$. > > [!proof]- > > $I(V(S))$ is the set of [[polynomial 4|polynomials]] vanishing on the common zeros of elements of $S$. Certainly any $f \in S$ is one of them. > > > If $f \in S$ and $\boldsymbol x \in V(S)$, then $f(\boldsymbol x)=0$ by the definition of $V(S)$, and so $f$ vanishes on all of $V(S)$, i.e., $f \in I(V(S))$. > 3. For $X \subset \Omega^{n}$, we have $X \subset V(I(X))$. > > [!proof]- > > $V(I(X))$ is the common vanishing of the set of polynomials vanishing on $X$. Certainly any $\boldsymbol x \in X$ belongs here. > > > > Explicitly, if $\boldsymbol x \in X$ and $f \in I(X)$ then $f(\boldsymbol x)=0$ by the definition of $I(X)$, and so $\boldsymbol x$ is a [[root of a polynomial|root]] of all $I(X)$, i.e., $\boldsymbol x \in V(I(X))$. > 4. In fact, $V(I(X))$ is the smallest $k$-algebraic subset of $\Omega^{n}$ that contains $X$. > > [!proof]- > > > > **(a)** Clearly $V(I(X))$ is $k$-algebraic, and we've just seen it contains $X$. > > **(b)** Suppose $X \subset Y$ for some $k$-algebraic set $Y \subset \Omega^{n}$. Claim: $V(I(X)) \subset Y$. Indeed, assuming $Y=V(\mathfrak{a})$ for some [[ideal]] $\mathfrak{a}$ of $k[T_{1},\dots,T_{n}]$, one has $V(I(X)) \subset V(I(Y))=V\left( \underbrace{ I\big(V(\mathfrak{a})\big) }_{ \supset \mathfrak{a} } \right) \subset V(\mathfrak{a})=Y.$ > > > > 5. In particular, $V(I(X))=X$ if $X (\subset X)$ is $k$-algebraic. > 6. For every set $X \subset \Omega^{n}$, the [[ideal]] $I(X)$ of $k[T_{1},\dots,T_{n}]$ is [[radical of an ideal|radical]]. > > [!proof]- > > Always $I(X)\subset \sqrt{ I(X) }$. To see the reverse inclusion, suppose $f \in \sqrt{ I(X) }$, say, $f^{\ell} \in I(X)$ for some $\ell \geq 0$. Then $f^{\ell}(\boldsymbol x)=0$ for all $\boldsymbol x \in X$, and so $f( \boldsymbol x)=0$ since $\Omega$ is an [[integral domain]]. Hence $f \in I(X)$. > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```