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

By Antoine Chambert-Loir

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The dimension of E, denoted by dim(E), is the supremum of the lengths of chains in E. Let x ∈ E. The height (resp. the coheight) of x is the supremum of the length of chains ending (resp. starting) at x. They are denoted ht(x) and coht(x) respectively. 2). — Let X be a topological space. The Krull dimension of X, denoted dim(X), is the dimension of the set C(X) of all irreducible closed subsets of X, ordered by inclusion. Let Z be a closed irreducible subset of X. The codimension of Z in X, denoted codim(Z), is the coheight of Z in the partially ordered set C.

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. For every object M of D, let us choose an object G(M) of C and an isomorphism αM ∶ M → F ○ G(M). Let M, N be objects of D and let f ∈ D(M, N); since F is fully faithful, there exists a unique morphism f ′ ∈ C (G(M), G(N)) such that F( f ′ ) = −1 ; set G( f ) = f ′ . Since α ○ id ○α −1 = id αN ○ f ○ αM M M F○G(M) = F(idG(M) ), one has M G(idM ) = idG(M) .

FUNCTORS 53 Let us now construct an isomorphism of functors from IdD to G ○ F. Let M be an object of C . Since F is fully faithful, there exists a unique morphism βM ∈ C (M, G ○ F(M)) such that F(βM ) = αF(M) . Since αF(M) is an isomorphism, βM is an isomorphism as well. Moreover, if M, N are objects of C and f ∶ M → N is a morphism, then −1 F(G ○ F( f ) ○ βM ) = αF(N) ○ F( f ) ○ αF(M) ○ F(βM ) = αF(N) ○ F( f ) = F(βN ○ f ). Since F is fully faithful, one thus has βN ○ f = G ○ F( f ) ○ βM . In other words, the isomorphisms βM , for M ∈ ob(C ), define an isomorphism of functors from IdC to G ○ F.

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


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