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Locally Ringed Spaces and Manifolds

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Locally Ringed Spaces and Manifolds Locally ringed spaces and manifolds March 14, 2011 In this lecture we shall define the category of Manifolds Mat giving a k- differential atlas on a topological space X and the category of Manifolds MS h as a subcategory of the category of locally ringed spaces and then we shall prove that Mat and MS h are actually isomorphic. We shall reserve the symbol k to denote an integer k ≥ 0 or 1. k Definition 0.1. Let X be a topological space. A k-differentiable atlas AX on X is a family f(Ui; hi; Xi; ni)gi2I of quadruples, where I is a set and for each i 2 IXi is an ni open subset of X, ni ≥ 0 is an integer, Ui is an open subset of R and hi : Ui ! Xi is a homeomorphism such that X = [iXi, and for i and j in I, if we use use the notations: Xi j B Xi \ X j C X ji, the restrictions hi : Ui j ! Xi j and h j : U ji ! X ji −1 are compatible in the sense that the homeomorphism h j ◦ hi : Ui j ! U ji is k-differentiable. Whenever required for the sake of clarity, we shall denote an atlas whose data k is given using the symbols above by the symbol AX(I; U; h; n), but when there is k no risk of confusion we shall continue using AX. k k A pair (X; AX) of a set X and a k-differentiable atlas AX on X we call a topo- logical space with a k-differentiable atlas. A morphism, or k-differentiable map k k φ :(X; AX(I; U; h; n)) ! (Y; AY (J; V; g; m)) of such pairs is a continuous map φ : X ! Y such that for each (i; j) 2 I × J, −1 −1 −1 g j ◦ φ ◦ hi : hi (φ (Y j) \ Xi) ! V j is a k-differentiable map. k k k k Lemma 0.2. If φ :(X; AX) ! (Y; AY ) and :(Y; AY ) ! (Z; AZ) are morphisms of k k topological spaces with k-differentiable atlases on them, ◦ φ :(X; AX) ! (Z; AZ) k k k is also a morphism of the same. Also, for every (X; AX), 1X :(X; AX) ! (X; AX) is a morphism. 1 The Lemma above proves that the topological pairs, defined above, together with their morphisms form a category. Lets name it category of topological spaces with a k-differentiable atlas. k k Lemma 0.3. Let AX(I; U; h; n) and AX(J; U; h; n) are two of the k differentiable atlases on X, the relation ∼ given by k k k AX(I; U; h; n) ∼ AX(J; U; h; n) , AX(I t J; U; h; n) is also a k-differentiable atlas k is an equivalence relation. More over, the equivalence class of AX is partially ordered under the relation ⊂ and has a unique maximal element still denoted by k AX. We are ready to define what is called a k-differentiable manifold. Definition 0.4. k • Let (X; AX) be in the category of topological spaces with k-differentiable k atlases and let AX is a maximal atlas on X. A maximal atlas on X we call a k-differentiable structure on X and we define k-differentiable manifold to k be a pair (X; AX) of a topological space with a k-differentiable structure on it. k k • A morphism φ :(X; AX) ! (Y; AY ) of k-differentiable manifolds is a mor- phism in the the category of topological spaces with k-differentiable atlas. k 0 Definition 0.5. Let AX(I; U; h; n) be a k-differentiable manifold and let X ⊂ X open in X. A map f : X0 ! R is said to be a k-differentiable map if for every i 2 I −1 0 the map f ◦ hi : hi (Xi \ X ) ! R is a k differentiable. Definition 0.6. Let U ⊂ Rn open in Rn. The sheaf of k-differentiable functions k CU on U is the sheaf defined k CU (V) = f f : V ! R j f is a k-differentiable mapg for every V ⊂ U open in U. For W ⊂ V ⊂ U open in U the restriction maps are ρV f f f 2 Ck V : given by the usual restrictions functions; that is W ( ) = W for every U ( ) Definition 0.7. We define a category P whose objects are the pairs (X; FX) where X is a topological space and FX is a sheaf of R-algebras on X which is a subsheaf of sheaf of R-valued functions. ∗ A morphism ( f; f ):(X; FX) ! (Y; FY ) in P consists of a continuous map f ∗ and the morphism f : FY ! f∗FX of sheaves of R-algebras on Y induced by f and is defined by ∗ fV : FY (V) ! f∗FX(V); s 7! s ◦ f: ∗ Identity of an object (X; FX) is (1X; 1X) is given by continuous map 1X : X ! X and the composition ( f; f ∗) ◦ (g; g∗) = ( f ◦ g; ( f ◦ g)∗). 2 k Definition 0.8. A k- differentiable manifold is an object (M; CM) 2 P such that for every p 2 M there exists an open subset M of M, an integer n , an open subset p p np ∗ k k Up of R and an isomorphism (hp; h ):(Up; C ) ! (Mp; C ) in P. p Up M Mp ∗ k 0 k A morphism of k- differentiable manifolds ( f; f ):(M; CM) ! (M ; CM0 ) is a morphism in P. So we have another, I must say much better, way of defining a manifold. To dis- tinguish is from Mat we denote it by MS h. Now we shall show that two categories are isomorphic. Theorem 0.9. The categories Mat and MS h are isomorphic. k k Proof. Let (X; AX(I; U; h; n)) in Mat. Consider the sheaf of functions on AX on X defined as follows: Ak(X0) = f f : X0 ! R j f is a k-differentiable mapg for every X0 ⊂ X open in X and restrictions being usual restrictions of functions. k k AX is clearly a sheaf and we shall show that (X; AX) is an object of MS h. To show k that, the required data is available from the data of AX(I; U; h; n). All we need to show that for an arbitrary i 2 I ∗ k k (hi; h ):(Ui; C ) ! (Xi; A ) is an isomorphism in P; i Ui X Xi 0 that is, we need to prove that for arbitrary X ⊂ Xi open, ∗ k 0 k 0 hi 0 : A (X ) ! hi∗C (X ); f 7! f ◦ hi X X Xi Ui is a well defined isomorphism. X0 open in X and X open in X gives X0 open in i i X which further gives Ak (X0) = Ak (X0). By definition f 2 Ak (X0) means that X Xi X X k −1 0 k 0 ∗ f ◦ hi 2 CU (hi (X )) = hi∗CU (X ); that is, hi X0 well defined. The morphism of k ∗i i algebras hiX0 clearly an injection since hi is a homeomorphism and it is a surjection as g 2 h Ck (X0) is the image of g ◦ h−1 which clearly is in Ak (X0). So we have a i∗ Ui i X map k k E : Ob(Mat) ! Ob(MS h); (X; AX) 7! (X; AX): k k Now let φ :(X; AX(I; U; h; n)) ! (Y; AY (J; V; g; m)) is a morphism in Mat. We ∗ k k shall prove that (φ, φ ):(X; AX) ! (Y; AY ) is well defined morphism in P. φ is ∗ k k surely continuous so we just need to show that φ : AY ! φ∗AX is well defined. that is, we need to show that for arbitrary open subset Y0 ⊂ Y the morphism ∗ k 0 k 0 φY0 : AY (Y ) ! φ∗AX(Y ) which maps f 7! f ◦ φ is well defined. For the coming statements please keep in mind that, for A ⊂ B and a map l : B ! R when we, 3 abusing the notation, write l : A ! R we actually mean ljA : A ! B. Now observe the following: k 0 −1 0 f 2 AY (Y ) , 8 j 2 J f ◦ g j : g j (Y j \ Y ) ! R is differentiable −1 −1 −1 0 ) 8(i; j) 2 I × J f ◦ g j ◦ g j ◦ φ ◦ hi : hi (Xi \ φ (Y j \ Y )) ! R −1 is differentiable since g j ◦ φ ◦ hi is differentiable, as is φ given −1 −1 0 , 8(i; j) 2 I × J f ◦ φ ◦ hi : hi (Xi \ φ (Y j \ Y )) ! R is differentiable −1 −1 0 , 8i 2 I f ◦ φ ◦ hi : hi (Xi \ φ (Y )) ! R is differentiable k −1 0 k 0 , f ◦ φ 2 AY (φ (Y )) = φ∗AX(Y ): In the equivalences above we proved that if a continuous map φ : X ! Y of k k topological space induces a morphism φ :(X; AX) ! (Y; AY ) in Mat then it also ∗ k k induces a morphism (φ, φ ):(X; AX) ! (Y; AY ) in MS h. Thus we have a map k k k k ∗ E : HomMat ((X; AX); (Y; AY ) ! HomMS h ((X; AX); (Y; AY )); φ 7! (φ, φ ) ∗ ∗ ∗ ∗ E(φ) ◦ E( ) = (φ, φ ) ◦ ( ; ) = (φ ◦ ; (φ ◦ ) ) = E(φ ◦ ) and E(1X) = (1X; 1X). So we have a functor E : Mat ! MS h. It remains to prove that it is a it is an isomorphism.
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