9. Algebraic Geometry

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a fixed algebraically closed field k

a n dimensions space A which's coordinates is denoted with k

a kind of functions from  A to k, which construct a polynomial ring R=F[x1, ..., xn] over k

to a polynomial, we have its zeros (solutions of a polynomial equation) which belong to A

to some polynomials, we have their zero set Z which's elements are common zeros of all these polynomials, which construct a subset T of R.

as a ring, we can get an ideal a of R that is generated by its subset T

the zero set of T = the zero set of a

a's generators is finite

the zero set of T = the common zeros of the finite set of a's generators (polynomials)

for any subset T of R, which's zero set is a algebraic set of space A

we denote the function that maps the subset T of R to the subset Y of A: Y=Z(T)

conversely, for any subset Y of A, we can get a subset T of R, its zero set cover Y, and the ideal that is generated by T is denoted with I(Y), its zero set cover Y too, which is a function which maps subsets of A to ideals of R.

then, we have two functions: Z(T) and I(Y), about the relationship about: T, a, Y, Z(T), I(Y)

proposition:

the complements of such algebraic sets endue A with a topology, name it the Zariski topology on A

for more appropriate with polynomial ring, we set affine structure on A

to base on a Zariski topology, an irreducible closed subset of A is named an affine algebraic variety, and the correspondence open subset of an affine algebraic variety is named a quasi-affine algebraic variety