The Theory of Motives

The theory of motives originated from the observation, sometime in the 1960’s, that in algebraic geometry there were several different cohomology theories (see Homology and Cohomology and Cohomology in Algebraic Geometry), such as Betti cohomology, de Rham cohomology, l-adic cohomology, and crystalline cohomology. The search for a “universal cohomology theory”, such that all these other cohomology theories could be obtained from such a universal cohomology theory is what led to the theory of motives.

The four cohomology theories enumerated above are examples of what is called a Weil cohomology theory. A Weil cohomology theory, denoted H^{*}, is a functor (see Category Theory) from the category \mathcal{V}(k) of smooth projective varieties over some field k to the category \textbf{GrAlg}(K) of graded K-algebras, for some other field K which must be of characteristic zero, satisfying the following axioms:

(1) (Finite-dimensionality) The homogeneous components H^{i}(X) of H^{*}(X) are finite dimensional for all i, and H^{i}(X)=0 whenever i<0 or i>2n, where n is the dimension of the smooth projective variety X.

(2) (Poincare duality) There is an orientation isomorphism H^{2n}\cong K, and a nondegenerate bilinear pairing H^{i}(X)\times H^{2n-i}(X)\rightarrow H^{2n}\cong K.

(3) (Kunneth formula) There is an isomorphism

\displaystyle H^{*}(X\times Y)\cong H^{*}(X)\otimes H^{*}(Y).

(4) (Cycle map) There is a mapping \gamma_{X}^{i} from C^{i}(X), the abelian group of algebraic cycles of codimension i on X (see Algebraic Cycles and Intersection Theory), to H^{i}(X), which is functorial with respect to pullbacks and pushforwards, has the multiplicative property \gamma_{X\times Y}^{i+j}(Z\times W)=\gamma_{X}^{i}(Z)\otimes \gamma_{Y}^{j}(W), and such that \gamma_{\text{pt}}^{i} is the inclusion \mathbb{Z}\hookrightarrow K.

(5) (Weak Lefschetz axiom) If W is a smooth hyperplane section of X, and j:W\rightarrow X is the inclusion, the induced map j^{*}:H^{i}(X)\rightarrow H^{i}(W) is an isomorphism for i\leq n-2, and a monomorphism for i\leq n-1.

(6) (Hard Lefschetz axiom) The Lefschetz operator

\displaystyle \mathcal{L}:H^{i}(X)\rightarrow H^{i+2}(X)

given by

\displaystyle \mathcal{L}(x)=x\cdot\gamma_{X}^{1}(W)

for some smooth hyperplane section W of X, with the product \cdot provided by the graded K-algebra structure of H^{*}(X), induces an isomorphism

\displaystyle \mathcal{L}^{i}:H^{n-i}(X)\rightarrow H^{n+i}(X).

The idea behind the theory of motives is that all Weil cohomology theories should factor through a “category of motives”, i.e. any Weil cohomology theory

\displaystyle H^{*}: \mathcal{V}(k)\rightarrow \textbf{GrAlg}(K)

can be expressed as the following composition of functors:

\displaystyle H^{*}: \mathcal{V}(k)\xrightarrow{h} \mathcal{M}(k)\rightarrow\textbf{GrAlg}(K)

where \mathcal{M}(k) is the category of motives. We can get different Weil cohomology theories, such as Betti cohomology, de Rham cohomology, l-adic cohomology, and crystalline cohomology, via different functors (called realization functors) from the category of motives to a category of graded algebras over some field K. This explains the term “motive”, which actually comes from the French word “motif”, which itself is already used in music and visual arts, among other things, as some kind of common underlying “theme” with different possible manifestations.

Let us now try to construct this category of motives. This category is often referred to in the literature as a “linearization” of the category of smooth projective varieties. This means that we obtain it from some sense starting with the category of smooth projective varieties, but we also want to modify it so that it we can do linear algebra, or more properly homological algebra, in some sense. In other words, we want it to behave like the category of modules over some ring. With this in mind, we want the category to be an abelian category, so that we can make sense of notions such as kernels, cokernels, and exact sequences.

An abelian category is a category that satisfies the following properties:

(1) The morphisms form an abelian group.

(2) There is a zero object.

(3) There are finite products and coproducts.

(4) Every morphism f:X\rightarrow Y has a kernel and cokernel, and satisfies a decomposition

\displaystyle K\xrightarrow{k} X\xrightarrow{i} I\xrightarrow{j} Y\xrightarrow{c} K'

where K is the kernel of f, K' is the cokernel of f, and I is the kernel of c and the cokernel of k (not to be confused with our notation for fields).

In order to proceed with our construction of the category of motives, which we now know we want to be an abelian category, we discuss the notion of correspondences.

The group of correspondences of degree r from a smooth projective variety X to another smooth projective variety Y, written \text{Corr}^{r}(X,Y), is defined to be the group of algebraic cycles of X\times Y of codimension n+r, where n is the dimension of X, i.e.

\text{Corr}^{r}(X,Y)=C^{n+r}(X\times Y)

A morphism (of varieties, in the usual sense) f:Y\rightarrow X determines a correspondence from X to Y of degree 0 given by the transpose of the graph of f in X\times Y. Therefore we may think of correspondences as generalizations of the usual concept of morphisms of varieties.

As we have learned in Algebraic Cycles and Intersection Theory, whenever we are dealing with algebraic cycles, it is often useful to consider them only up to some equivalence relation. In the aforementioned post we introduced the notion of rational equivalence. This time we consider also homological equivalence and numerical equivalence between algebraic cycles.

We say that two algebraic cycles Z_{1} and Z_{2} are homologically equivalent if they have the same image under the cycle map, and we say that they are numerically equivalent if the intersection numbers Z_{1}\cdot Z and Z_{2}\cdot Z are equal for all Z of complementary dimension. There are other such equivalence relations on algebraic cycles, but in this post we will only mostly be using rational equivalence, homological equivalence, and numerical equivalence.

Since correspondences are algebraic cycles, we often consider them only up to these equivalence relations, and denote the quotient group we obtain by \text{Corr}_{\sim}^{r}(X,Y), where \sim is the equivalence relation imposed, for example, for numerical equivalence we write \text{Corr}_{\text{num}}^{r}(X,Y).

Taking the tensor product of the abelian group \text{Corr}_{\sim}^{r}(X,Y) with the rational numbers \mathbb{Q}, we obtain the vector space

\displaystyle \text{Corr}_{\sim}^{r}(X,Y)_{\mathbb{Q}}=\text{Corr}_{\sim}^{r}(X,Y)\otimes_{\mathbb{Z}}\mathbb{Q}

To obtain something closer to an abelian category (more precisely, we will obtain what is known as a pseudo-abelian category, but in the case where the equivalence relation is numerical equivalence, we will actually obtain an abelian category), we need to consider “projectors”, correspondences p of degree 0 from a variety X to itself such that p^{2}=p. So now we form a category, whose objects are h(X,p) for a variety X and projector p, and whose morphisms are given by

\displaystyle \text{Hom}(h(X,p),h(Y,q))=q\circ\text{Corr}_{\sim}^{0}(X,Y)_{\mathbb{Q}}\circ p.

We call this category the category of pure effective motives, and denote it by \mathcal{M}_{\sim}^{\text{eff}}(k). The process described above is also known as passing to the pseudo-abelian (or Karoubian) envelope.

We write h^{i}(X,p) for the objects of \mathcal{M}_{\sim}^{\text{eff}}(k) that map to H^{i}(X). In the case that X is the projective line \mathbb{P}^{1}, and p is the diagonal \Delta_{\mathbb{P}^{1}}, we find that

h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=h^{0}\mathbb{P}^{1}\oplus h^{2}\mathbb{P}^{1}

which can be rewritten also as

\displaystyle h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=\mathbb{I}\oplus\mathbb{L}

where \mathbb{I} is the image of a point in the category of pure effective motives, and \mathbb{L} is known as the Lefschetz motive. It is also denoted by \mathbb{Q}(-1). The above decomposition corresponds to the projective line \mathbb{P}^{1} being a union of the affine line \mathbb{A}^{1} and a “point at infinity”, which we may denote by \mathbb{A}^{0}:

\displaystyle \mathbb{P}^{1}=\mathbb{A}^{0}\cup\mathbb{A}^{1}

More generally, we have

\displaystyle h(\mathbb{P}^{n},\Delta_{\mathbb{P}^{n}})=\mathbb{I}\oplus\mathbb{L}\oplus...\oplus\mathbb{L}^{n}

corresponding to

\displaystyle \mathbb{P}^{n}=\mathbb{A}^{0}\cup\mathbb{A}^{1}\cup...\cup\mathbb{A}^{n}.

The category of effective pure motives is an example of a tensor category. This means it has a bifunctor \otimes: \mathcal{M}_{\sim}^{\text{eff}}\times\mathcal{M}_{\sim}^{\text{eff}}\rightarrow\mathcal{M}_{\sim}^{\text{eff}} which generalizes the usual notion of a tensor product, and in this particular case it is given by taking the product of two varieties. We can ask for more, however, and construct a category of motives which is not just a tensor category but a rigid tensor category, which provides us with a notion of duals.

By formally inverting the Lefschetz motive (the formal inverse of the Lefschetz motive is then known as the Tate motive, and is denoted by \mathbb{Q}(1)), we can obtain this rigid tensor category, whose objects are triples h(X,p,m), where X is a variety, e is a projector, and m is an integer. The morphisms of this category are given by

\displaystyle \text{Hom}(h(X,p,n),h(Y,q,m))=q\circ\text{Corr}_{\sim}^{n-m}(X,Y)_{\mathbb{Q}}\circ p.

This category is called the category of pure motives, and is denoted by \mathcal{M}_{\sim}(k). The category \mathcal{M}_{\text{rat}}(k) is called the category of Chow motives, while the category \mathcal{M}_{\text{num}}(k) is called the category of Grothendieck (or numerical) motives.

The category of Chow motives has the advantage that it is known to be “universal”, in the sense that every Weil cohomology theory factors through it, as discussed earlier; however, in general it is not even abelian, which is a desirable property we would like our category of motives to have. Meanwhile, the category of Grothendieck motives is known to be abelian, but it is not yet known if it is universal. If the so-called “standard conjectures on algebraic cycles“, which we will enumerate below, are proved, then the category of Grothendieck motives will be known to be universal.

We have seen that the category of pure motives forms a rigid tensor category. Closely related to this concept, and of interest to us, is the notion of a Tannakian category. More precisely, a Tannakian category is a k-linear rigid tensor category with an exact faithful functor (called a fiber functor) to the category of finite-dimensional vector spaces over some field extension K of k.

One of the things that makes Tannakian categories interesting is that there is an equivalence of categories between a Tannakian category \mathcal{C} and the category \text{Rep}_{G} of finite-dimensional linear representations of the group of automorphisms of its fiber functor, which is also known as the Tannakian Galois group, or, if the Tannakian category is a “category of motives” of some sort, the motivic Galois group. This aspect of Tannakian categories may be thought of as a higher-dimensional analogue of the classical theory of Galois groups, which can be stated as an equivalence of categories between the category of finite separable field extensions of a field k and the category of finite sets equipped with an action of the Galois group \text{Gal}(\bar{k}/k), where \bar{k} is the algebraic closure of k.

So we see that being a Tannakian category is yet another desirable property that we would like our category of motives to have. For this not only do we have to tweak the tensor product structure of our category, we also need certain conjectural properties to hold. These are the same conjectures we have hinted at earlier, called the standard conjectures on algebraic cycles, formulated by Alexander Grothendieck at around the same time he initially developed the theory of motives.

These conjectures have some very important consequences in algebraic geometry, and while they remain unproved to this day, the search for their proof (or disproof) is an important part of modern mathematical research on the theory of motives. They are the following:

(A) (Standard conjecture of Lefschetz type) For i\leq n, the operator \Lambda defined by

\displaystyle \Lambda=(\mathcal{L}^{n-i+2})^{-1}\circ\mathcal{L}\circ (\mathcal{L}^{n-i}):H^{i}\rightarrow H^{i-2}

\displaystyle \Lambda=(\mathcal{L}^{n-i})\circ\mathcal{L}\circ (\mathcal{L}^{n-i+2})^{-1}:H^{2n-i+2}\rightarrow H^{2n-i}

is induced by algebraic cycles.

(B) (Standard conjecture of Hodge type) For all i\leq n/2, the pairing

\displaystyle x,y\mapsto (-1)^{i}(\mathcal{L}x\cdot y)

is positive definite.

(C) (Standard conjecture of Kunneth type) The projectors H^{*}(X)\rightarrow H^{i}(X) are induced by algebraic cycles in X\times X with rational coefficients. This implies the following decomposition of the diagonal:

\displaystyle \Delta_{X}=\pi_{0}+...+\pi_{2n}

which in turn implies the decomposition

\displaystyle h(X,\Delta_{X},0)=h(X,\pi_{0},0)\oplus...\oplus h(X,\pi_{2n},0)

which, writing h(X,\Delta_{X},0) as hX and h(X,\pi_{i},0) as h^{i}(X), we can also compactly and suggestively write as

\displaystyle hX=h^{0}X\oplus...\oplus h^{2n}X.

In other words, every object hX=h(X,\Delta_{X},0) of our “category of motives” decomposes into graded “pieces” h^{i}(X)=h(X,\pi_{i},0) of pure “weighti. We have already seen earlier that this is indeed the case when X=\mathbb{P}^{n}. We will need this conjecture to hold if we want our category to be a Tannakian category.

(D) (Standard conjecture on numerical equivalence and homological equivalence) If an algebraic cycle is numerically equivalent to zero, then its cohomology class is zero. If the category of Grothendieck motives is to be “universal”, so that every Weil cohomology theory factors through it, this conjecture must be satisfied.

In Algebraic Cycles and Intersection Theory and Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry, we have made mention of the two famous conjectures in algebraic geometry known as the Hodge conjecture and the Tate conjecture. In fact, these two closely related conjectures can be phrased in the language of motives as the conjectures stating that the realization functors from the category of motives to the category of pure Hodge structures and continuous l-adic representations of \text{Gal}(\bar{k}/k), respectively, be fully faithful. These conjectures are closely related to the standard conjectures on algebraic cycles as well.

We have now constructed the category of pure motives, for smooth projective varieties. For more general varieties and schemes, there is an analogous idea of “mixed motives“, which at the moment remain conjectural, although there exist several related constructions which are the closest thing we currently have to such a theory of mixed motives.

If we want to construct a theory of mixed motives, instead of Weil cohomology theories we must instead consider what are known as “mixed Weil cohomology theories“, which are expected to have the following properties:

(1) (Homotopy invariance) The projection \pi:X\rightarrow\mathbb{A}^{1} induces an isomorphism

\displaystyle \pi^{*}:H^{*}(X)\xrightarrow{\cong}H^{*}(X\times\mathbb{A}^{1})

(2) (Mayer-Vietoris sequence) If U and V are open coverings of X, then there is a long exact sequence

\displaystyle ...\rightarrow H^{i}(U\cap V)\rightarrow H^{i}(X)\rightarrow H^{i}(U)\oplus H^{i}(V)\rightarrow H^{i}(U\cap V)\rightarrow...

(3) (Duality) There is a duality between cohomology H^{*} and cohomology with compact support H_{c}^{*}.

(4) (Kunneth formula) This is the same axiom as the one in the case of pure motives.

We would like a category of mixed motives, which serves as an analogue to the category of pure motives in that all mixed Weil cohomology theories factor through it, but as mentioned earlier, no such category exists at the moment. However, the mathematicians Annette Huber-Klawitter, Masaki Hanamura, Marc Levine, and Vladimir Voevodsky have constructed different versions of a triangulated category of mixed motives, denoted \mathcal{DM}(k).

A triangulated category \mathcal{T} is an additive category with an automorphism T: \mathcal{T}\rightarrow\mathcal{T} called the “shift functor” (we will also denote T(X) by X[1], and T^{n}(X) by X[n], for n\in\mathbb{Z}) and a family of “distinguished triangles

\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]

 which satisfies the following axioms:

(1) For any object X of \mathcal{T}, the triangle X\xrightarrow{\text{id}}X\rightarrow 0\rightarrow X[1] is a distinguished triangle.

(2) For any morphism u:X\rightarrow Y of \mathcal{T}, there is an object Z of \mathcal{T} such that X\xrightarrow{u}Y\rightarrow Z\rightarrow X[1] is a distinguished triangle.

(3) Any triangle isomorphic to a distinguished triangle is a distinguished triangle.

(4) If X\rightarrow Y\rightarrow Z\rightarrow X[1] is a distinguished triangle, then the two “rotations” Y\rightarrow Z\rightarrow Z[1]\rightarrow Y[1] and Z[-1]\rightarrow X\rightarrow Y\rightarrow Z are also distinguished triangles.

(5) Given two distinguished triangles X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1] and X'\xrightarrow{u'}Y'\xrightarrow{v'}Z'\xrightarrow{w'}X'[1] and morphisms f:X\rightarrow X' an g:Y\rightarrow Y' such that the square “commutes”, i.e. u'\circ f=g\circ u, there exists a morphisms h:Z\rightarrow Z such that all other squares commute.

(6) Given three distinguished triangles X\xrightarrow{u}Y\xrightarrow{j}Z'\xrightarrow{k}X[1]Y\xrightarrow{v}Z\xrightarrow{l}X'\xrightarrow{i}Y[1], and X\xrightarrow{v\circ u}Z\xrightarrow{m}Y'\xrightarrow{n}X[1], there exists a distinguished triangle Z'\xrightarrow{f}Y'\xrightarrow{g}X'\xrightarrow{h}Z'[1] such that “everything commutes”.

A t-structure on a triangulated category \mathcal{T} is made up of two full subcategories \mathcal{T}^{\geq 0} and \mathcal{T}^{\leq 0} satisfying the following properties (writing \mathcal{T}^{\leq n} and \mathcal{T}^{\leq n} to denote \mathcal{T}^{\leq 0}[-n] and \mathcal{T}^{\geq 0}[-n] respectively):

(1) \mathcal{T}^{\leq -1}\subset \mathcal{T}^{\leq 0} and \mathcal{T}^{\geq 1}\subset \mathcal{T}^{\geq 0}

(2) \displaystyle \text{Hom}(X,Y)=0 for any object X of \mathcal{T}^{\leq 0} and any object Y of \mathcal{T}^{\geq 1}

(3) for any object Y of \mathcal{T} we have a distinguished triangle

\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]

where X is an object of \mathcal{T}^{\leq 0} and Z is an object of \mathcal{T}^{\geq 1}.

The full subcategory \mathcal{T}^{0}=\mathcal{T}^{\leq 0}\cap\mathcal{T}^{\geq 0} is called the heart of the t-structure, and it is an abelian category.

It is conjectured that the category of mixed motives \mathcal{MM}(k) is the heart of the t-structure of the triangulated category of mixed motives \mathcal{DM}(k).

Voevodsky’s construction proceeds in a manner somewhat analogous to the construction of the category of pure motives as above, starting with schemes (say, over a field k, although a more general scheme may be used) as objects and correspondences as morphisms, but then makes use of concepts from abstract homotopy theory, such as taking the bounded homotopy category of bounded complexes, and localization with respect to a certain subcategory, before passing to the pseudo-abelian envelope and then formally inverting the Tate object \mathbb{Z}(1). The triangulated category obtained is called the category of geometric motives, and is denoted by \mathcal{DM}_{\text{gm}}(k). The schemes and correspondences involved in the construction of \mathcal{DM}_{\text{gm}}(k) are required to satisfy certain properties which eliminates the need to consider the equivalence relations which form a large part of the study of the category of pure motives.

Closely related to the triangulated category of mixed motives is motivic cohomology, which is defined in terms of the former as

\displaystyle H^{i}(X,\mathbb{Z}(m))=\text{Hom}_{\mathcal{DM}(k)}(X,\mathbb{Z}(m)[i])

where \mathbb{Z}(m) is the tensor product of m copies of the Tate object \mathbb{Z}(1), and the notation \mathbb{Z}(m)[i] tells us that the shift functor of the triangulated category is applied to the object \mathbb{Z}(m) i times.

Motivic cohomology is related to the Chow group, which we have introduced in Algebraic Cycles and Intersection Theory, and also to algebraic K-theory, which is another way by which the ideas of homotopy theory are applied to more general areas of abstract algebra and linear algebra. These ideas were used by Voevodsky to prove several related theorems, from the Milnor conjecture to its generalization, the Bloch-Kato conjecture (also known as the norm residue isomorphism theorem).

Historically, one of the motivations for Grothendieck’s attempt to obtain a universal cohomology theory was to prove the Weil conjectures, which is a higher-dimensional analogue of the Riemann hypothesis for curves over finite fields first proved by Andre Weil himself (see The Riemann Hypothesis for Curves over Finite Fields). In fact, if the standard conjectures on algebraic cycles are proved, then a proof of the Weil conjectures would follow via an approach that closely mirrors Weil’s original proof (since cohomology provides a Lefschetz fixed-point formula –  we have mentioned in The Riemann Hypothesis for Curves over Finite Fields that the study of fixed points is an important part of Weil’s proof). The last of the Weil conjectures were eventually proved by Grothendieck’s student Pierre Deligne, but via a different approach that bypassed the standard conjectures. A proof of the standard conjectures, which would lead to a perhaps more elegant proof of the Weil conjectures, is still being pursued to this day.

The theory of motives is not only related to analogues of the Riemann hypothesis, which concerns the location of zeroes of L-functions, but to L-functions in general. For instance, it is also related to the Langlands program, which concerns another aspect of L-functions, namely their analytic continuation and functional equation, and to the Birch and Swinnerton-Dyer conjecture, which concerns their values at special points.

We recall in The Riemann Hypothesis for Curves over Finite Fields that the Frobenius morphism played an important part in counting the points of a curve over a finite field, which in turn we needed to define the zeta function (of which the L-function can be thought of as a generalization) of the curve. The Frobenius morphism is an element of the Galois group, and we recall that a category of motives which is a Tannakian category is equivalent to the category of representations of its motivic Galois group. Therefore we can see how we can define “motivic L-functions” using the theory of motives.

As the L-functions occupy a central place in many areas of modern mathematics, the theory of motives promises much to be gained from its study, if only we could make progress in deciphering the many mysteries that surround it, of which we have only scratched the surface in this post. The applications of motives are not limited to L-functions either – the study of periods, which relate Betti cohomology and de Rham cohomology, and lead to transcendental numbers which can be defined using only algebraic concepts, is also strongly connected to the theory of motives. Recent work by the mathematicians Alain Connes and Matilde Marcolli has also suggested applications to physics, particularly in relation to Feynman diagrams in quantum field theory. There is also another generalization of the theory of motives, developed by Maxim Kontsevich, in the context of noncommutative geometry.

References:

Weil Cohomology Theory on Wikipedia

Motive on Wikipedia

Standard Conjectures on Algebraic Cycles on Wikipedia

Motive on nLab

Pure Motive on nLab

Mixed Motive on nLab

The Tate Conjecture over Finite Fields on Hard Arithmetic

What is…a Motive? by Barry Mazur

Motives – Grothendieck’s Dream by James S. Milne

Noncommutative Geometry, Quantum Fields, and Motives by Alain Connes and Matilde Marcolli

Algebraic Cycles and the Weil Conjectures by Steven L. Kleiman

The Standard Conjectures by Steven L. Kleiman

Feynman Motives by Matilde Marcolli

Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Periodes) by Yves Andre

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2 thoughts on “The Theory of Motives

  1. Pingback: Adjoint Functors and Monads | Theories and Theorems

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