16LECTURE 3.MORPHISMS OF AFFINE ALGEBRAIC VARIETIESNote that it suffices to check the previous condition only for generators of theidealI(Y), for example for the polynomials defining the system of equationsY.In terms of the polynomials(P1(T), . . . , Pm(T))satisfying (3.5), the morphismf:X→Yis given as follows:fK(a) = (P1(a), . . . , Pm(a))∈Y(K),∀a∈X(K).It follows from the definitions that a morphismφgiven by polynomials((P1(T), . . . , Pm(T))satisfying (3.5) is an isomorphism if and only if there existpolynomials(Q1(T), . . . , Qn(T))such thatG(Q1(T), . . . , Qn(T))∈I(Y),∀G∈I(X),Pi(Q1(T), . . . , Qn(T))≡TimodI(Y), i= 1, . . . , m,Qj(P1(T), . . . , Pm(T))≡TjmodI(X), j= 1, . . . , n.The main problem of (affine) algebraic geometry is to classify affine algebraicvarieties up to isomorphism. Of course, this is a hopelessly difficult problem.Example 3.3.1. LetYbe given by the equationT21-T32= 0,andX=A1kwithO(X) =k[T]. A morphismf:X→Yis given by the pair of polynomials(T3, T2). For everyk-algebraK,fK(a) = (a3, a2)∈Y(K), a∈X(K) =K.The affine algebraic varietiesXandYare not isomorphic since their coor-dinate rings are not isomorphic.The quotient field of the algebra
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