By Antoine Chambert-Loir

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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. This bijection is the composition of the map ΦF ∶ C (M, N) → D(M, N) induced by F and of the map ΦG ∶ D(M, N) → C (M, N) induced by G. This implies that ΦF is injective and ΦG is surjective. By symmetry, ΦF is surjective too, so that it is bijection. In other words, the functor F is fully faithful. Let us now assume that F is fully faithful and essentially surjective.

When they exists, colimits of a diagram A are unique up to a unique isomorphism. A colimit of a diagram A is sometimes denoted by lim(A ). 5). — a) Let Q be the empty quiver (no vertex, no arrow). Let us consider the unique Q-diagram; it consists in nothing. By definition, a cone on this diagram is just an object A of C , and A is a limit if and only if there exists a unique morphism in C (B, A), for every object B of C . Consequently, a limit of this diagram in the category C is called an terminal object of C .

Moreover, qm+1 ∩ A = pm+1 . This concludes the proof. The following theorem lies at the ground of dimension theory in algebraic geometry. 6). — Let K be a field and let A be a finitely generated K-algebra. Assume that A is an integral domain and let F be its field of fractions. One has dim(A) = tr. degK (F). Proof. — We prove the theorem by induction on the transcendence degree of F. If tr. degK (F) = 0, then A is algebraic over K. Consequently, dim(A) = dim(K) = 0. Now assume that the theorem holds for finitely generated K-algebras which are integral domains and whose field of fractions has transcendence degree strictly less than tr.

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Algebraic Geometry of Schemes [Lecture notes] by Antoine Chambert-Loir

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