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# Algebraic Cycles Anand Sawant Abstract Algebraic cycles arose in the study of intersection theory for algebraic varieties

This note based on a a lecture in the Mathematics Students Seminar at TIFR on September 7 2012 is meant to give an intorduction to algebraic cycles and various adequate equivalence relations on them We then state the Standard Conjecture D and state

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## Algebraic Cycles Anand Sawant Abstract Algebraic cycles arose in the study of intersection theory for algebraic varieties

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Algebraic Cycles Anand Sawant Abstract Algebraic cycles arose in the study of intersection theory for algebraic varieties. This note, based on a a lecture in the Mathematics Students’ Seminar at TIFR on September 7, 2012 is meant to give an intorduction to algebraic cycles and various adequate equivalence relations on them. We then state the Standard Conjecture D and state some of its consequences including the conjectural theory of motives. 1 Intersection theory and algebraic cycles Intersection theory deals with the problem of studying intersection of two closed subschemes

of a scheme. Perhaps the ﬁrst result in intersection theory was Bezout’s theorem , which says that given two plane curves of degree and in having no component in common, their intersection consists of at most mn points (exactly mn points if counted with multiplicity). A systematic study of intersection theory involves the study of the group of algebraic cycles on a scheme . In this section, we give a brief account of basic deﬁnitions and properties of algebraic cycles; for details, the reader is referred to Fulton’s book . For simplicity, all the schemes considered in this

note will be of ﬁnite type over a ﬁeld Deﬁnition 1.1. An algebraic cycle on a scheme is an element of the free abelian group on the set of all the closed integral subschemes of . We shall denote the group of algebraic cycles by ). Since closed integral subschemes of are determined uniquely by their generic points, we have We usually grade this group by dimension, by letting ) denote the algberaic cycles of dimension . If denotes the set of points of of dimension , then ) : . A point is said to have codimension , if the local ring has Krull dimension . We may grade ) by

codimension as well: writing for the set of points of of codimension , we set ) : . If is equidimensional of dimension , we have If is a closed integral subscheme, we denote its class in ) by [ ], and call it a prime cycle
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Deﬁnition 1.2. An algebraic cycle associated to a closed subscheme Y is deﬁned to be ] : )[ where the sum is taken over all the irreducible components of and ) denotes the length of as module over itself. Note that the length is ﬁnite, being an artinian ring. For a well-working theory of algebraic cycles, we need to have a notion of

intersection product of the cycles associated to two subvarieties and of a scheme of dimension , which will closely reﬂect the geometry of their intersection . Just taking the set-theoretic intersection of subvarieties in question will not work, because subvarieties may occur in bad position - for instance, two parallel lines in a plane or the case of intersecting a subvariety with itself. In case the cycles intersect properly , that is, if dim dim dim , we can deﬁne In order to take care of the bad cases, we must have an adequate equivalence relation on which is broad enough to

ensure that given any two cycles on we can ﬁnd equivalent cycles intersecting properly such that their intersection is equivalent to the original intersection. The most important such equivalence relation which is essential for the purposes of intersection theory on smooth varieties is rational equivalence Deﬁnition 1.3 (Rational equivalence) An -dimensional cycle on a scheme is said to be ratio- nally equivalent to zero and we write rat 0 if there exist ( 1)-dimensional subvarieties ,... of , such that the projection maps are dominant and we have (0)] )] where [ )] )] for a

point . Here denotes the projection . The group of -dimensional cycles that are rationally equivalent to zero is denoted by Rat ). We use the notation Rat ), if the cycles are graded by codimension. Thus, in other words, an -dimensional cycle on a scheme is rationally equivalent to zero if there is an ( 1)-dimensional cycle on such that (0)] )] Deﬁnition 1.4. The Chow group of dimension -cycles on a scheme is deﬁned to be CH ) : Rat We will henceforth assume that all the schemes considered here are equidimensional and will write CH ) : CH ), where is the dimension of Theorem 1.5.

If X is a smooth scheme of dimension n over a ﬁeld F, then the intersection product makes CH CH into a commutative ring.
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This theorem was ﬁrst proved in the Chevalley seminar for smooth quasiprojective varieties using the moving lemma and was later generalized by Fulton by replacing the moving lemma by using the techniques of reduction to the diagonal and of deformation to the normal cone. We summarize the imporatant properties of Chow groups in the next proposition: Proposition 1.6. (1) If f X is proper, there is a pushforward map : CH CH for all p. (2) If g X

is ﬂat of relative dimension d, then there is a pullback map : CH CH for all p. (3) Let i X be a closed subscheme and let U Z. Let j X be the inclusion of U into X. Then the sequence CH CH CH is exact for all p. (4) If X is an a ne bundle of rank d, then the ﬂat pullback : CH CH is an isomorphism for all p. 2 Examples of adequate equivalence relations We now see examples of some other adequate equivalence relations on algebraic cycles on smooth projective varieties over an algebraically closed ﬁeld . We shall denote the category of smooth projectivve varieties by SmProj

Deﬁnition 2.1 (Algebraic equivalence) We say that ) is algebraically equivalent to zero and write alg 0 if there exists a smooth curve , points and a cycle such that )] )] The subgroup of ) generated by cycles algebraically equivalent to zero is denoted by Alg ). We use the notation Alg ), if the cycles are graded by codimension. Clearly, it can be seen that Rat Alg ). The inclusion is proper in general. We next introduce homological equivalence of algebraic cycles. This depends on the notion of Weil cohomology theory , which we deﬁne next. Deﬁnition 2.2. Weil cohomology

theory is a contravariant functor : SmProj Graded -algebras for a ﬁeld such that if SmProj is of dimension , then ) satisﬁes the following properties:
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1. ) is a ﬁnite dimensional -vector space for each 2. 0 for 0 and for 3. 4. (Poincar e duality) There exists a nondegenerate pairing for each 5. (K unneth isomorphism) There is a canonical isomorphism induced by the projection maps. 6. (Cycle map) There exists a group homomorphism that is functorial with proper pushforwards and ﬂat pullbacks, respects products and direct sums and such that it

coincides with the canonical map in the case Spec 7. (Weak Lefschetz property) For any smooth hyperplane section (that is, for some hyperplane ), is an isomorphism for 2 and a monomorphism for 1. 8. (Hard Lefschetz property) For any smooth hyperplane section , let ([ ]) ). Let ) be the Lefschetz operator given by 7 . Then is an isomorphism for ,..., The classical examples of Weil cohomology theories include the singular (Betti) cohomology, de Rham cohomology, -adic cohomology, crystalline cohomology and Hodge cohomology. We are now set ot deﬁne the adequate equivalence relation of

homological equivalence. Deﬁnition 2.3 (Homological equivalence) Fix a Weil cohomology theory on SmProj . A cycle ) is said to be homologically equivalent to zero and we write hom 0 if 0. The subgroup of ) generated by cycles homologiically equivalent to zero is denoted by Hom ). We use the notation Hom ), if the cycles are graded by codimension. It is a fact that Alg Hom ) for each . The inclusion can be proper, as was shown ﬁrst by Gri ths and later by Clemens and others. (Gri ths showed that Hom Alg ) is nonzero and actually has an element of inﬁnite order, if is a

generic hypersurface of degree 5 in .) We now deﬁne the coarsest of the adequate equivalence relations. Recall that the degree map 〈· is deﬁned by if ); otherwise.
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Deﬁnition 2.4 (Numerical equivalence) Let dim . We say that ) is numerically equivalent to zero and we write num 0, if 0, for all ). The subgroup of generated by cycles numerically equivalent to zero is denoted by Num ). If is equidimensional, we write Num Num ). Note that for each , we have Hom Num ); this follows from the fact that the cycle class map induces a ring homomorphism CH ).

To see this, let be the diagonal; then for any ) and any ), we have = )) as desired. Grothendieck’s Standard Conjecture D says that up to torsion, homological and numerical equivalence coincide. Standard Conjecture D: For each , we have Hom Num Note that the Standard Conjecture D implies that homological equivalence is independent of the choice of a Weil cohomology theory! Remark 2.5. The standard conjecture is known to hold for divisors, that is, the case 1. If then the Standard Conjecture D is implied by the Hodge Conjecture. Thus, by virtue of this, the Standard Conjecture D is true for

abelian varieties in characteristic 0. Remark 2.6. The standard conjectures on algebraic cycles were ﬁrst stated by Grothendieck in 1968 (see ). Grothendieck’s motivation for them was that the celebrated Weil conjectures were a consequence of the standard conjectures. For an account of these, see the articles  and  by Kleiman. For an account of motives, see . 3 Motives Let be a smooth complex projective variety. We then have the following relationship between the classical Weil cohomology theories: dR sing and sing sing et This suggests that there should exist a

universal cohomology theory of which these cohomology theories are realizations. This is the basis of Grothendieck’s conjectural theory of motives The category of pure motives ), according to Grothendieck, is supposed to be a -linear abelian category admitting a functor from SmProj such that any Weil cohomology theory factors through ). SmProj Graded -algebras
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The category SmProj itself is not additive as it does not have enough morphisms. An attempt to linearize it by adding a few more morphisms can be made by regarding a morphism as a cycle on of dimension dim( ) given by

the graph of Deﬁnition 3.1. correspondence of degree from to is deﬁned to be a cycle on of dimension dim( . We write Corr dim( For a morphism , its graph Corr ). Any Corr ) deﬁnes a homomorphism ) by 7 )) Let be an adequate equivalence relation and deﬁne Corr ) : Corr Let Corr Corr . We can now deﬁne a category Corr( ) by taking an ob- jects ) for each SmProj , and deﬁning Hom Corr( )) : Corr . The cor- respondences are composed as follows: if Corr ) and Corr ), then XZ XY YZ )). This gives us a preadditive category. We now take the pseudo-abelian

envelope of this. Deﬁne a category e f f ) by taking for objects ), where SmProj( ) and Corr ) is an idempotent, that is, . Deﬁne the morphisms by setting Hom e f f )) : Corr This makes e f f ) a pseudo-abelian category. Deﬁnition 3.2. e f f ) is called the category of ective motives for over The category of pure motives for is now obtained by adding twists . This will make every object in the category to have a dual. Deﬁnition 3.3 (Pure motives) The category ) whose objects are triples ), where ) is as above, , and the morphisms are deﬁned by Hom )) : Corr

is a rigid pseudo-abelian category. It is called the category of pure motives for over We end by enlisting some of the known properties of ). ) has direct sums and tensor products. If Num, then Hom-sets in ) ae ﬁnite dimensional -vector spaces. A remarkable theorem Jannsen asserts that ) is a semisimple abelian category if and only if Num. (Thus, ) is the only possible candidate for motives. But there are no realization functors if the Standard Conjecture D does not hold!)
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We end by indicating how one can get the realization functors if is ﬁner than or equal to

homological equivalence. Fix a Weil cohomology theory . We indicate how to construct a functor Graded -algebras Let ) be a pure motive. Note that Corr CH dim( ) is an idempotent. Its cycle class ) is an element of the group 2 dim( ' 2 dim( ' ' Hom( )) by Poincar e duality and the K unneth isomorphism. We thus obtain the induced maps Deﬁne ) to be the image of inside ) and set ) : References  Fulton, W.: Intersection theory . Second edition. Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2. Springer-Verlag, Berlin, 1998. 

Grothendieck, A.: Standard conjectures on algebraic cycles . Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford University Press, p. 193 – 199.  Jannsen, U.: Motives, numerical equivalence, and semi-simplicity . Invent. Math. 197 (1992), p. 447 – 452.  Kleiman, S.: Algebraic cycles and the Weil conjectures . In: Dix expos es sur la cohomologie des sch emas, North-Holland 1968, p. 359 – 386.  Kleiman, S.: Motives . Algebraic Geometry, Oslo 1970 (edited by F. Oort), Walters-Noordho Groningen, 1972, p. 53 – 82.  Kleiman, S.: The standard conjectures .

Motives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics 55, American Mathematical Society, p. 3 – 20.