# Algebraic geometry and Ring theory

Yoshifumi Tsuchimoto

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

DEFINITION 08.2   A local ring is called regular if its Krull dimension is equal to >

PROPOSITION 08.3   Let be a field of characterictic . Then every quadratic curve (a curve defined by a homogeneous polynomial of degree ) in over is isomorphic to a curve of the form

In particular, every quadratic curve in over is isomorphic to a curve .

PROPOSITION 08.4   Let be a field of characterictic . Then every cubic curve (a curve defined by a homogeneous polynomial of degree ) in over is isomorphic to a curve of the form

It should be meaningful to point out:

PROPOSITION 08.5   Let be a imaginary number in . defines a lattice (a discrete subgroup of rank in ) . A complex manifold may be embedded to the complex projective plane by the Weierstrass -function and its derivative . Namely, a rational map defined by

gives a holomorphic map . moreover, satisfy a cubic relation so that gives an isomorphism of and a cubic curve in .