By Antoine Chambert-Loir
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Additional info for Algebraic Geometry of Schemes [Lecture notes]
One says that a functor F∶ C → D is an equivalence of categories if there exists a functor G∶ D → C such that F and G are quasi-inverse functors. 7). — For a functor F∶ C → D to be an equivalence of categories, it is necessary and sufficient that it be fully faithful and essentially surjective. Proof. — Let G∶ D → C be a functor such that F and G are quasi-inverse. For every object P of D, F ○ G(P) is isomorphic to P, hence F is essentially surjective. Moreover, for every objects M, N of C , the functor G ○ F, being isomorphic to idC , induces a bijection from C (M, N) to itself.
A similar definition applies for contravariant functors. A functor F is essentially surjective if for every object P of D, there exists an object M of C such that F(M) is isomorphic to P in the category D. 3) (Forgetful functors). — Many algebraic structures are defined by enriching other structures. Often, forgetting this enrichment gives rise to a functor, called a forgetful functor. For example, a group is already a set, and a morphism of groups is a map. There is thus a functor that associates to every group its underlying set, thus forgetting the group structure.
For every noetherian ring A, the ring A[X] is noetherian. In particular, for every field K and every integer n ⩾ 0, the ring K[X1 , . . , Xn ] is noetherian. Proof. — Let I be an ideal of A[X]. For every integer m, let Jm be the set of leading coefficients of elements of I whose degrees are equal to m (the leading coefficient of the zero polynomial being 0); one checks that is an ideal of A. 10. IRREDUCIBLE COMPONENTS 27 For every integer m, the ideal Jm is finitely generated. We may thus fix a finite set Qm of polynomials belonging to Im whose leading coefficients generate Jm .