Invariants Associated to K-theoretic Methods and Complexity of Algebraic Cycle Groups Karim Mansour University of Alberta Department of Mathematics Alberta, Canada abdelgal@ualberta.ca December 04, 2100
Detecting geometry by its interaction with its enviroment Definition K = R ǫ ( D ) will denote the real-valued infinitely differentiable functions on D, which we shall simply call C ∞ functions on D; i.e, f ∈ ǫ ( D ) iff f is real valued function such that partial derivatives of all order exist and are continuous at all points of D. A ( D ) will denote the real-analytic functions on D; in particular we have that A ( D ) ⊂ ǫ ( D ), Recall, f ∈ A ( D ) iff the Taylor expansion of f converges to f in a neighborhood of any point of D. K = C O ( D ) will denote the complex-valued holomorphic functions on D, i.e, if ( z 1 , . . . , z n ) are coordinates in C n , then f ∈ O ( D ) iff near each point z 0 ∈ D f can be represented by a convergent power series of the form f ( z ) = f ( z 1 , . . . , z n ) = 1 ) α 1 . . . ( z n − z 0 α 1 ,...,α n a α 1 ,...,α n ( z 1 − z 0 Σ ∞ n ) α n 1337 2 / 33
Classes of functions and Sheaves Definition An S structure, S M , on a topological manifold M is a family of K valued functions defined on the open sets of M such that: For every p ∈ M , there exists an open neighborhood U of p and a homeomorphism h : U → U ′ , where U ′ is an open set in K n , such that for any open set V ⊂ U f : V → K ∈ S M iff f ◦ h − 1 ∈ S ( h ( V )) If f : U → K , where U = � i ∈ I U i and U i is open in M, then f ∈ S M iff f U i ∈ S M for each i. A manifold with an S -structure is called an S -manifold, denoted by ( M , S M ), and elements of S M are called S -functions on M. An open subset U ⊂ M and a homeomorphism h : U → U ′ ⊂ K n as in (a) above is called an S coordinate system. 1337 3 / 33
Introduction to sheaves The above will be formalized more when we talk about sheaves. We will be interested in three classes S -structure on M. Consider the following three classes of functions: S = ǫ : differentiable ( or C ∞ ) manifold, and the functions in ǫ M are called C ∞ functions on open subsets of M. S = A : real-analytic manifold, and the functions in A M are called real analytic functions on an open subsets of M. S = O : complex-analytic manifold, and the functions in O M are called holomorphic (or complex analytic functions) on M. Now we will talk about S morphism between S manifolds. 1337 4 / 33
Introduction to sheaves Definition An S morphism F : ( M , S M ) → ( N , S N ) is a continuous map, F : M → N , such that f ∈ S N = ⇒ f ◦ F ∈ S M An S -isomorphism is an S -morphism that is a homeomorphism such that the inverse is an S morphism If we have S -manifold ( M , S M ) together with two coordinate systems h 1 : U 1 → K n and h 2 : U 2 → K n such that U 1 ∩ U 2 � = ∅ , then h 2 ◦ h − 1 : h 1 ( U 1 ∩ U 2 ) → h 2 ( U 1 ∩ U 2 ) is an S isomorphism 1 This is an S isomorphism on open subsets of ( K n , S K n ). 1337 5 / 33
Classes of functions The above says that this approach to geometry is similar to the classical way of doing geometry. In particular, it represents the same idea. Conversely, we can go the other direction. If we have an open covering { U α } α ∈A of M, where M is a topological manifold. Consider the family of homeomorphisms: { h α : U α → U ′ α ⊂ K n } such that the family above are compatible. This defines an S structure on M by setting S M = { f : U → K } such that U is open in M and the functions in S M are pullbacks of functions in S by homeomorphisms { h α } α ∈ A . More precisely, define an S structure on M by setting S M = { f : U → K } such that U is open in M and f ◦ h − 1 ∈ S ( h α ( U ∩ U α )) for all α ∈ A . α 1337 6 / 33
Detecting differentiability We would like to probe differentiability using family of functions. This will allow a very natural generalization to schemes. A continuous map ψ : M → N between differentiable manifolds is differentiable iff, for every differentiable function f on an open subset U ⊂ N , the pullback ψ f := f ◦ ψ is differentiable on ψ − 1 U ⊂ M . This can be written using the language of sheaves. We will see later how this is done. 1337 7 / 33
Examples of manifolds Example K n , ( R n , C n ). For every p ∈ K n , U = K n and h = identity . Then R n becomes a real-analytic (hence differentiable) manifold and C n is complex-analytic manifold. 1337 8 / 33
Example If V is a finite dimensional vector space over K . Consider: P ( V ) := { the set of one dimensional subspaces of V } is called the projective space of V. Consider π : R n +1 − { 0 } → P n ( R ) x �→ { subspace spanned x } 1337 9 / 33
Example Such mapping is onto. If we restrict on the circle it is also onto. Thus, we can equipp P n with the quotient topology. Such π is continuous and it can be verified that P n ( R ) is Hausdorff space with a countable basis. If we restrict, π on the circle, then it is also continuous and surjective. This tells us that P n ( R ) is compact. Set π ( x ) = [ x 0 , . . . , x n ]. We say that ( x 0 , . . . , x n ) are homogeneous coordinates of [ x 0 , . . . , x n ]. This is well defined as one can easily verify. Using this homogeneous coordinates, we can define differentiable structure (in fact, analytic) on P n ( R ). Define analytic structure P n ( R ) as follows: U α = { S ∈ P n ( R ) : S = [ x 0 , . . . , x n ] and x α � = 0 } 1337 10 / 33
Example It can be verified that each U α is open. Finally, define h α ([ x 0 , . . . , x n ]) = ( x 0 , . . . , x α − 1 , x α +1 , . . . , x n ) x α x α x α x α It is easy to verify that each h α and U α are well defined. Each h α is a homeomorphism and that the transition maps h α ◦ h − 1 are β diffeomorphism (In fact, analytic). 1337 11 / 33
Sheaves and Schemes Essentially sheaves capture local geometry. Using this local data it allows us to extend manifold theory to the algebraic settings. Definition Fix a ring K . A pre-sheaf S is given by the following datum: For each open set U ⊂ X there is a K-module S ( U ) If V ⊂ U , there exists K − module morphism p V , U : S ( U ) → S ( V ) 1337 12 / 33
Sheaves and Schemes Definition satisfying the following conditions: p U , U = id S If W ⊂ V ⊂ U , then p W , U = p W , V ◦ p V , U 1337 13 / 33
� � If we have two pre-sheaves S 1 and S 2 , then morphisms of sheaves is just the most natural way of preserving both algebraic and topogical structures. That is, for each open sets V ⊂ U ⊂ X , the following diagram commutes: � S 2 ( U ) S 1 ( U ) p V , U r V , U � S 2 ( V ) S 1 ( V ) In order for pre-sheaves to be useful we will need the notion of sheaves. Sheaves tells us how to pass from local to global data and vice versa. 1337 13 / 33
Definition A presheaf S is a sheaf if additionally it satisfies the following data: (How to glue) If s i ∈ S ( U i ) and if U i ∩ U j � = ∅ we have that the following is satisfied: p U i ∩ U j , U i ( s i ) = p U i ∩ U j , U j ( s j ) for all i, then there exists an s ∈ S ( U ) such that p U i , U ( s ) = s i . (Local morphism) If s , t ∈ S ( U ) and p U i , U ( s ) = p U i , U ( t ) for all i, then s = t . Example Let f : X → Y be a continuous map of topological spaces. For any sheaf F on X, we define the direct image sheaf f ⋆ F on Y by ( f ⋆ F )( V ) = F ( f − 1 ( V )) for ay open set V ⊂ Y . 1337 14 / 33
� � Capturing differentiability using sheaves Any continuous map ψ : M → N induces a map of sheaves on N: ψ # : G ( N ) → ψ ⋆ G ( M ) sending a continuous function f ∈ G ( N )( U ) on an open subset U ⊂ N to the pullback f ◦ ψ ∈ G ( M )( ψ − 1 ( U )) = ( ψ ⋆ G ( M ))( U ). Using these ideas, a differentiable map ψ : M → N may be defined as a continuous map ψ such that the induced map ψ # carries the subsheaf G ∞ ( N ) ⊂ G ( N ) into the subsheaf ψ ⋆ C ∞ ⊂ ψ ⋆ G ( M ). That is, we have the following commutative diagram: ψ # � ψ ⋆ G ( M ) G ( N ) ψ # � ψ ⋆ C ∞ ( M ) G ∞ ( N ) 1337 14 / 33
Schemes In analytic geometry we model things locally as Euclidean spaces. Such model only detect analytic structure of the object that we would like to study. In order to also detect algebraic structure associated to topological space we will define schemes. Definition If X is equipped with sheaf of rings O , then X is called a structure sheaf. X is called a locally ringed space if the stalks O X , x over each x ∈ X forms a local ring. That is, the stalk O X , x = lim x ∈ U O X ( U ) forms a local ring. The direct limit it taken with respect to the maps p U , V and inclusions. 1337 15 / 33
Let us see couple of simple examples of locally ringed spaces before we proceed: Suppose we consider X = C . Define sheaf of rings O on X as follows: For each open set U ⊂ X , O ( U ) is the ring of complex continuous functions ψ : U → C . Suppose X is a general complex manifold, and O ( U ) is the ring of holomorphic functions ψ : U → C . It is easy to see that the stalks at each x ∈ X are local rings, where the maximal ideals at x is given by the functions which vanish at x . Note that any manifold (smooth,analytic,and complex) is a locally ringed space, where the maximal ideal at x is given by the function which vanish at x. From this perspective, we will see that schemes are generalizations of manifold. 1337 16 / 33
Recommend
More recommend
