Algebraic geometry and Ring theory

Yoshifumi Tsuchimoto

DEFINITION 08.1   For any ring $A$ , we define its Krull dimension to be the maximum of ascending chains of primes in $A$ . Namely,

$$\operatorname{Krulldim}(A) =\max\{n;A... ...p}_n \supsetneq \mathfrak{p}_{n-1}\supsetneq \dots \supsetneq \mathfrak{p}_0\}$$

DEFINITION 08.2   A local ring $A=(A,\mathfrak{m})$ is called regular if its Krull dimension is equal to $\dim_{A/\mathfrak{m}}(\mathfrak{m} /\mathfrak{m}^2)$ >

PROPOSITION 08.3   Let $\mathbbm{k}$ be a field of characterictic $\neq 2$ . Then every quadratic curve (a curve defined by a homogeneous polynomial of degree $2$ ) in $\P ^2$ over $\mathbbm{k}$ is isomorphic to a curve of the form

$$a X^2 + b Y^2+c Z^2=0. \qquad(a,b,c\in \mathbbm{k}).$$

In particular, every quadratic curve in $\P ^2$ over $\mathbb{R}$ is isomorphic to a curve $X^2+Y^2=Z^2$ .

PROPOSITION 08.4   Let $\mathbbm{k}$ be a field of characterictic $\neq 2,3$ . Then every cubic curve (a curve defined by a homogeneous polynomial of degree $3$ ) in $\P ^2$ over $\mathbbm{k}$ is isomorphic to a curve of the form

$$ZY^2=X^3 + p XZ^2 + q Z^3 \qquad(p,q\in \mathbbm{k}).$$

It should be meaningful to point out:

PROPOSITION 08.5   Let $\tau$ be a imaginary number in $\mathbb{C}$ . $\tau$ defines a lattice (a discrete subgroup of rank $2$ in $\mathbb{C}$ ) $L=\mathbb{Z}+\tau \mathbb{Z}$ . A complex manifold $\mathbb{C}/L$ may be embedded to the complex projective plane $\P ^2(\mathbb{C})$ by the Weierstrass $\wp$ -function $\wp(z;L)$ and its derivative $\wp'(z;L)$ . Namely, a rational map defined by

$$\mathbb{C}\ni z \mapsto [\wp(z;L):\wp'(z;L):1] \in \P ^2(\mathbb{C})$$

gives a holomorphic map $f: \mathbb{C}/L \to \P ^2(\mathbb{C})$ . moreover, $\wp, \wp'$ satisfy a cubic relation so that $f$ gives an isomorphism of $\mathbb{C}/\mathbb{Z}+\tau \mathbb{Z}$ and a cubic curve in $\P ^2(\mathbb{C})$ .