CHAPTER 1

Algebraic Cycles and Equivalence Relations

In this chapter k is an arbitrary field (possibly not algebraically closed) and

SmProj(k) denotes the category of smooth projective varieties over k. A variety is

a reduced scheme, not necessarily irreducible; indeed it is crucial to allow (finite)

disjoint unions of irreducible ones. If we write Xd, we are dealing with an irreducible

variety of dimension d.

Remark. The results in this Chapter are of course well known. They are col-

lected at this place for the reader’s convenience and, at the same time, to give us

the opportunity to introduce some notation we use in the remainder of this book.

We would like to mention the following sources where the reader can find further

details. For a general introduction to algebraic cycles, Chow groups and intersection

theory: see [Harts, Appendix A], [Voi03, Part III]. For a more elaborate study

we refer to [Ful], especially [Ful, Chapters 1, 6, 7, 8, 16]. An introduction to ´etale

cohomology can be found in [Harts, Appendix] or [Mil98], while [Mil80] provides

more details.

1.1. Algebraic Cycles

An cycle on a variety X is a formal finite integral linear combination

Z =

∑algebraic

nαZα of irreducible subvarieties Zα of X. If all the Zα have the same

codimension i we say that Z is a codimension i cycle. We introduce the abelian

group

Zi(X)

= {codim i cycles on X} .

If we prefer to think in terms of dimension and X = Xd is pure dimensional, we

write Zd−i(X) = Zi(X). If, instead of integral linear combinations we work with

coeﬃcients in a field F (mostly, we use F = Q), we write

Zi(X)F

:= rαZα | rα ∈ F, Zα a codim. i cycle =

Zi(X)

⊗Z F.

Examples 1.1.1. (1) The group of codimension 1 cycles, the divisors, is

also written Div(X).

(2) The zero-cycles

Zd(X)

= Z0(X). These are finite formal sums Z =

∑

α

nαPα where Pα is an irreducible 0-dimensional k-variety. The de-

gree of Pα is just the degree of the field extension [k(Pα)/k] and deg(Z) =

∑

nα deg(Pα).

(3) To any subscheme Y of X with irreducible components Yα of dimension

dα one can associate the class [Y ] =

∑

α

nαYα ∈

∑

α

Zdα (X), where nα is

the length of the zero-dimensional Artinian ring OY,Yα ; see [Ful, 1.5].

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http://dx.doi.org/10.1090/ulect/061/01