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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

• n D, i.e, if (z1, . . . , zn) are coordinates in Cn, then f ∈ O(D)

iff near each point z0 ∈ D f can be represented by a convergent power series of the form f (z) = f (z1, . . . , zn) = Σ∞

α1,...,αnaα1,...,αn(z1 − z0 1)α1 . . . (zn − z0 n)αn

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Classes of functions and Sheaves

Definition An S structure, SM, 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 Kn, such that for any open set V ⊂ U f : V → K ∈ SM iff f ◦ h−1 ∈ S(h(V )) If f : U → K, where U =

i∈I Ui and Ui is open in M, then

f ∈ SM iff fUi ∈ SM for each i. A manifold with an S-structure is called an S-manifold, denoted by (M, SM), and elements of SM are called S-functions on M. An

• pen subset U ⊂ M and a homeomorphism h : U → U′ ⊂ Kn as in

(a) above is called an S coordinate system.

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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 ǫM are called C ∞ functions on open subsets of M. S = A : real-analytic manifold, and the functions in AM are called real analytic functions on an open subsets of M. S = O : complex-analytic manifold, and the functions in OM are called holomorphic (or complex analytic functions) on M. Now we will talk about S morphism between S manifolds.

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Introduction to sheaves

Definition An S morphism F : (M, SM) → (N, SN) is a continuous map, F : M → N, such that f ∈ SN = ⇒ f ◦ F ∈ SM 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, SM) together with two coordinate systems h1 : U1 → Kn and h2 : U2 → Kn such that U1 ∩ U2 = ∅, then h2 ◦ h−1

1

: h1(U1 ∩ U2) → h2(U1 ∩ U2) is an S isomorphism This is an S isomorphism on open subsets of (Kn, SKn).

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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 {Uα}α∈A of M, where M is a topological manifold. Consider the family of homeomorphisms: {hα : Uα → U′

α ⊂ Kn}

such that the family above are compatible. This defines an S structure on M by setting SM = {f : U → K} such that U is open in M and the functions in SM are pullbacks of functions in S by homeomorphisms {hα}α∈A. More precisely, define an S structure on M by setting SM = {f : U → K} such that U is open in M and f ◦ h−1

α

∈ S(hα(U ∩ Uα)) for all α ∈ A.

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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 ψf := f ◦ ψ is differentiable on ψ−1U ⊂ M. This can be written using the language of sheaves. We will see later how this is done.

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Examples of manifolds

Example Kn, (Rn, Cn). For every p ∈ Kn, U = Kn and h = identity. Then Rn becomes a real-analytic (hence differentiable) manifold and Cn is complex-analytic manifold.

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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 : Rn+1 − {0} → Pn(R) x → {subspace spanned x}

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Example Such mapping is onto. If we restrict on the circle it is also onto. Thus, we can equipp Pn with the quotient topology. Such π is continuous and it can be verified that Pn(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 Pn(R) is compact. Set π(x) = [x0, . . . , xn]. We say that (x0, . . . , xn) are homogeneous coordinates of [x0, . . . , xn]. This is well defined as one can easily

• verify. Using this homogeneous coordinates, we can define

differentiable structure (in fact, analytic) on Pn(R). Define analytic structure Pn(R) as follows: Uα = {S ∈ Pn(R) : S = [x0, . . . , xn] and xα = 0}

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Example It can be verified that each Uα is open. Finally, define hα([x0, . . . , xn]) = ( x0 xα , . . . , xα−1 xα , xα+1 xα , . . . , xn xα ) It is easy to verify that each hα and Uα are well defined. Each hα is a homeomorphism and that the transition maps hα ◦ h−1

β

are diffeomorphism (In fact, analytic).

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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 pV ,U : S(U) → S(V )

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Sheaves and Schemes

Definition satisfying the following conditions: pU,U = idS If W ⊂ V ⊂ U, then pW ,U = pW ,V ◦ pV ,U

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If we have two pre-sheaves S1 and S2, 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: S1(U)

pV ,U

• S2(U)

rV ,U

• S1(V )

S2(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.

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Definition A presheaf S is a sheaf if additionally it satisfies the following data: (How to glue) If si ∈ S(Ui) and if Ui ∩ Uj = ∅ we have that the following is satisfied: pUi∩Uj,Ui(si) = pUi∩Uj,Uj(sj) for all i, then there exists an s ∈ S(U) such that pUi,U(s) = si. (Local morphism) If s, t ∈ S(U) and pUi,U(s) = pUi,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 .

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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: G(N)

ψ# ψ⋆G(M)

G∞(N)

• ψ# ψ⋆C∞(M)
• 1337

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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 OX,x over

each x ∈ X forms a local ring. That is, the stalk OX,x = limx∈UOX(U) forms a local ring. The direct limit it taken with respect to the maps pU,V and inclusions.

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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.

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Introduction to schemes

Fix a ring R. Let X = Spec(R). Take the following set to form a basis for the space: Df = {P ∈ Spec(R) : f / ∈ P} Construct a sheaf O on X by defining O(Df ) = Rf . Rf is the localization with respect to the multiplicative set {1, f , f 2, f 3, . . .}. Given a point x ∈ Spec(R), the stalk at x is given by OX,x = Rx, the localization at the point x. (Spec(R), O) forms a locally ringed space, which is called affine

• scheme. Affine schemes are the atoms of schemes in the same way

as euclidean spaces are the atoms of abstract manifolds.

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Let us see what we mean by locally isomorphic to affine schemes. First, let us define the notion of morphisms in the category of locally ringed spaces. Definition Suppose we have two locally ringed spaces (X, OX) and (Y , OY ). A morphism between them is something which respects set structure, topological structure, and algebraic structure. More precisely, it is given by the following datum: A continuous map ψ : X → Y . For each open set V ⊂ Y , a homomorphism map ψU : OY (U) → OX(ψ−1(U)) such that the map defined above is a sheaf mapping, i.e it commutes with the restriction mapping. For each x ∈ X, the map ψx : O(Y ,ψ(x)) → O(X,x) maps maximal ideal to maximal ideal.

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Remark A natural question might be why do we require this notion of mapping of maximal ideal to maximal ideal. The reason for this is we would like compatibility between the structure sheaf of both

• schemes. This is a natural generalization for the case of manifolds

Definition A scheme (X, O) is a locally ringed space, which is locally an affine

• scheme. That is, there exists an open covering {Uα}α∈I of X such

that each of the open set Uα are isomorphic to an affine scheme (Spec(Rα), O).

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Definition Let R be a sheaf of commutative rings over a topological space X. Define Rp, for p ≥ 0, by the presheaf U → Rp(U) = R(U)

• . . . R(U).

Here we take the direct sum p-times. Rp, so defined, is clearly a sheaf of R-modules and is called the direct sum of R. If M is a sheaf of R-modules such that M ≃ Rp for some p ≥ 0, then M is said to be a free sheaf of modules. if M is a sheaf of R-modules such that each x ∈ X has a neighborhood U such that M|U is free, then M is said to be locally free. A natural question that might arise is that the definition of sheaves resemble the definition vector bundles over a topological space X. It is true that the notions locally free sheaves and vector bundles

• ver connected topological space X are precisely the same notions.

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Formally, we have the following statement: Proposition Fix a connected topological space X. The category of locally free sheaves and the category of vector bundles over X are equivalent. We require that the topological space X is connected is in order to have constant rank as we vary across sections.

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Algebraic way of viewing schemes

Before moving to algebraic cycles note that we can think of schemes using the following diagram: affine varieties

• finitely generated, N-F-R-O ACF
• affine schemes

commutative rings with identity

Each element f ∈ R defines a ”function” on the space Spec R, which is defined as follows: R → R/x → k(x) f (x) is the image of f in the above natural map For example the value of the function 15 at the point (7) ∈ SpecZ is 15 (mod 7) = 1. Note this doesn’t define a function in a natural sense.

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Algebraic way of viewing schemes

Remark The ideas above shows how varieties generalize to schemes in the natural sense. Moreover, it also provides motivation for why we require the local ring condition in the definition of the category of locally ringed spaces.

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Easter Eggs

Going back to morphism of locally ringed spaces we can rephrase it as follows. Definition A morphism, or map, between schemes X and Y is a pair (ψ, ψ#), where ψ : X → Y is a continuous map on the underlying topological spaces and ψ# : OY → ψ⋆OX is a map of sheaves on Y satisfying the condition that for any point p ∈ X and any neighborhood U of q = ψ(p) in Y. A section f ∈ OY (U) vanishes at q iff the section ψ#f ∈ ψ⋆OX(U) = OX(ψ−1(U)) vanishes at p. The last condition is a formulation in terms of local rings OX,p and OY ,q.

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Easter Eggs

Any map of sheaves ψ# : OY → ψ⋆OX induces on passing to the limit a map OY ,q = lim − →

q∈U⊂Y

OY (U) → lim − →

q∈U⊂Y

OX(ψ−1(U)), this last ring maps to the limit lim − →

p∈V ⊂X

OX(V )

• ver all open subsets V containing p, which is OX,p. This

vanishing condition says that the map OY ,q → OX,p sends the maximal ideal MY ,q into MX,p. In other words, it is a homomorphism of local rings.

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Vector Bundles and Topological K-theory

Definition Fix a field K = C or R. A S-vector bundle (π, E, K) of rank r over an S-manifold X is given by the following datum: a continuous surjective map π : E → X. Each fiber with respect to a point is a K vector space. Ex := π−1(x) is a vector space

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Definition Each fiber over an open set is a trivial bundle. If x ∈ X is a

• point. There exists a neighorhood Ux and a homeomorphism

hx : π−1(Ux) → Ux × Kr such that hx(Ex) ⊂ {x} × Kr. Finally, we have that the composition below is an isomorphism: hx p : Ex → {p} × Kr → Kr

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Given the data of an S vector bundle (π, E, K) of rank r. This defines gluing data. If x ∈ Uα ∩ Uβ. We have the following commutative diagram: Uα ∩ Uβ × Kr

π1

• h−1

α

π−1(Uα ∩ Uβ)

π

• hβ

Uα ∩ Uβ × Kr

π1

• Uα ∩ Uβ

From the commutative diagram above we get a smooth map gαβ as follows: gαβ : Uα ∩ Uβ → Glr(K) These maps satisfies the following compatibility conditions: gαβgβηgηα = Ir on Uα ∩ Uβ ∩ Uη. gαα = Ir on Uα. gαβgβα = Ir.

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The first condition is called Cech cocycle condition. These maps are called transition maps. Geometrically, what is happening here is that locally the vector bundle is locally trivial. Globally things are twisted and the twists is captured precisely through this transition maps gαβ. It is easy to show that the data of vector bundles is the same as gluing data. This shows why isomorphism classes is a set instead of a proper class.

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If we impose that our base space X is paracompact then we can classify vector bundles. Theorem For paracompact space X, the map [X, Gn] → Vectn(X), [f ] → f ⋆Un is a bijection. Moreover, Gn is called the classifying space.

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Topological K-theory

We are now ready to define topological K-theory. Recall that VectF(X) is the isomorphism classes of vector bundles over X. This is an abelian monoid with respect to the operation of direct sum.

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Definition Suppose X is either a complex or real manifold. Define K 0

T(X) to

be the group completion of the vector bundles VectF(X). We could either take real vector bundles or complex vector bundles depending on the situation. We will see later why this is a group. In order to see the picture better. We will go through the construction.It turns out this this K0 construction is a homotopy invariant property. This construction will motivate things when we define K-theory for varieties and schemes. Since we have algebraic structure on the

• stalks. We can define short exact sequence of vector bundles.

Suppose we have a short exact sequence of X vector bundles below: 0 → A1 → A2 → A3 → 0

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We can think of the group S(vect(X)) as being the subgroup generated by all elements of the form [A2] − [A1] − [A0]. Let I(vect(X)) be the free abelian group of isomorphism classes of vect(S). We have the following triviality: K0(vect(X)) ∼ = I(vect(S))/S(vect(X)) Interesting remark is the case for real vector bundle. Whitney’s embedding theorem tells you that we can embed any real manifold M inside of R2N+1, where N is the dimension of the real manifold

• M. Thus, given a real vector bundle (π, E, K) we can find another

bundle (π′, E ′, K) such that E ⊕ E ′ ∼ = T. The geometric idea is as follows:

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We can embedd a vector bundle E into the trivial bundle by working over the fibers, then after that we can patch up things in

• rder to get the desired E ′.

Let us recall an important theorem by Bott. First, recall that that we can define the infinite matrix version of matrix groups as follows: U = limnU(n) GL(C) = limnGL(n, C) Theorem Bott periodicity theorem πn(U) = πn(GL(C) = 0 if n is even Z if n is odd If we consider real matrices. Then, πn(O) = π(GL(R) = Z2, Z2, 0, Z, 0, 0, 0, Z(mod8) Things work out good for topological K-theory because of Bott

• periodicity. That is, we can think of Bott periodicity somewhat like

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Van-Kampen theorem, which provides a way of computing π1 given that we know π1 of some familiar spaces. One can use this to show that: K0(S2k) = K(S0) = Z × Z K0(S2k+1) = K0(S1) = Z

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General K-theory

Definition Before introducing Higher K-theory. We will introduce K0 for general abelian category. It is constructed in the same fashion as for vector bundles. Suppose C is an abelian category. Let I(C) denote free abelian group generated by the isomorphism classes of C. Define S(C) in the same fashion as topological K-theory, so we get the group: K0(C) = I(C)/S(C) Suppose C is an abelian category and let QP be the category

• btained from C by applying the Q− construction, then

Ki(C) := πi+1(BQP) Where BQP is Milnor’s geometric realization functor.

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Definition Suppose C is an abelian category. We shall define a new category

• QC. The objects of QC are the same objects of C. The morphisms

are defined as HomQC(P, Q) = {P ← X → Q}/ ∼ Such that the map P ← X is epimorphism and the morphism X → Q is a monomorphism. Two morphisms P ← X → Q and P ← X ′ → Q are related if they fit into the following commutative diagram: P

=

• X

p

• ψ
• i

Q

=

• P

X ′

p

• i

Q

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Definition (Milnor’s geometric functor) Suppose that we have a simplicial set S, then construct the following set [S] = ∐n≥0(Sn × ∆n)/ ∼ The equivalence relation is the one which glues together the

• simplices. We put the quotient topology on this final object [S].

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Algebraic cycles

Definition Suppose X is a scheme. We form a group of algebraic cycles on X by considering the free abelian group of subvarieties of X. Such group is denoted by Z(X). A cycle α ∈ Z(X) can be written as ΣiniYi where Yi is a subvariety of X and of course the sum above is finite.

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Note, that to each subvariety Y ⊂ X we can associate the cycle generated by it and is denoted by (Y ). Namely, if Y ⊂ X is a subvariety, then we look at Y1, . . . , Ys the irreducible components

• f the reduced scheme Yred. Then, because the Noetherian

condition each local ring OY ,Yi has finite composition series. So, if we write li for the length, then we can define the cycle associated to Y as (Y ) := ΣiliYi Therefore, we can see that algebraic cycles are approximations of

• Schemes. We will later see how we can approximate Chow group

using K-theory. The group defined above is very large. Also, we don’t have intersection theory. So, in order to fix those issues we will have to mod out by an appropriate equivalence relation. There is two ways to do this. One is more clear for people who think more

• geometrically. I will not introduce the algebraic one here.

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We will define equivalence of cycles in similiar fashion. The following definition is taken from Eisenbudd book.

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Definition Two cycles α1 and α2 are rationally equivalent if there is family of cycles interpolating between them. That is, if we have a cycle ψ in P1 × X whose restriction on the fibers {t0} × X and {t1} × X are α1 and α2. More precisely, Let R(X) ⊂ Z(X) be the subgroup generated by the following formal difference (ψ ∩ ({t0} × X)) − (ψ ∩ ({t1} × X)) where t0, t1 ∈ P1 and ψ is a subvariety of P1 × X not contained in any fiber {t} × X. Then two cycles α1 and α2 are rationally equivalence if their difference is in Rat(X), and two subschemes are rationally equivalent if their associated cycles are equivalent. Definition The Chow group A(X) of X is the following quotient: A(X) = Z(X)/Rat(X)

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First of all, let us try to digest what it means to be rationally

• equivalent. Rational equivalence can be thought as very rigid

homotopy between algebraic cycles. Using this intuition can get us little far in computing simple examples of Chow group. Any two path in Rn are homotopic to each. It is natural to ask the question what about two points in An. Given any two points p1 and p2 ∈ An, then we can connect them by a line, so any two points are rationally equivalent. A hyper surface is defined by the zeroes of a polynomial function. That is, we can think of it as a map φ : An → K. In fact, we can think of φ as a map φ : An → P1. So, we can see that the fiber

• ver {∞} is empty. Therefore, any hyper surface is actually

rationally equivalent to the empty set. We can think of this as throwing the hyper surface away to infinity.

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The first real example of Chow group is that of affine space. We have the following proposition which we shall prove. The idea is that we will show that any irreducible subvariety can be thrown away to infinity. That is, there is only one generator. We will make this more formally in the proof. The way we do this intuitively is that we project along arrows until we meet the irreducible

• subvariety. See figure below:

From the above information. We can see the that we get the

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following theorem. Theorem A(An) ∼ = Z[An] Finally, we have the following theorems, which links Algebraic cycles and K-theory. This provides us with a dictionary between algebraic cycles and K-theory: Theorem We have the following isomorphism which follows from Grothiendieck-Riemann-Roch:

• i CHi(X) ⊗ Q ∼

= K0(X) ⊗ Q

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Invariants

We will give the idea of how to detect arithmetic invariants on the level of Chow group. To do that we need to give introduction to spectral sequence. Definition A cohomological spectral sequence is a sequence {Er, dr}(r ≥ 0) of bigraded objects in an Abelian category: Er =

• p,q≥0

E p,q

r

together with differentials: dr : E p,q

r

→ E p+r,q+1−r

r

, dr = 0

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Definition such that H⋆(Er) = Er+1 Essentially with most spectral sequence as r gets big enough, then the spectral sequence converges. That is Er = Er+1 = Er+2 = . . . for r ≥ r0 and we call the convergent limit E∞. We can think of spectral sequence as three-dimensional grid, where for each r we have a plane of cohomological data. The planes are related to each other from the fact that we build them in an inductive fashion. a natural question if given a cohomology, does there exist a spectral sequence that converges to it ? The answer is given by the following proposition:

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Proposition Let K ⋆ be a filtered complex. Then there exists a spectral sequence {Er} that converges to the cohomology as r gets big

• enough. More precisely:

E p,q = F pK p+q F p+1K p+q E p,q

1

= Hp+q(GrpK ⋆) E p,q

∞ = Grp(Hp+q(K ⋆))

Proof idea: The whole idea is based on the fact that spectral sequence is algebraic discrete Riemann sum. Since we have already the initial term being defined, we can define the second term by using the cohomology of the first term which is precisely Hp+q(GrpK ⋆). We keep doing this process inductively, because the complex is bounded eventually everything collapses to E p,q

∞ = Grp(Hp+q(K ⋆)).

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Theorem Let X/K be a smooth projective variety of dimension d. Then for all r, there is a filtration, depending on k ⊂ K, Ar(XK, m, Q) = F 0 ⊃ F1 ⊃ . . . ⊃ F v ⊃ . . .

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Proof idea: Essentially the main idea is as follows. First of all, we spread the variety over a family of varieties such that the fibers capture pieces of the original variety. After that, we glue all the pieces together using spectral sequence in order to pass from local to global data.

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Thank You!

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