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 =
nαZα of irreducible subvarieties of X. If all the have the same
codimension i we say that Z is a codimension i cycle. We introduce the abelian
= {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
:= rαZα | F, a codim. i cycle =
⊗Z F.
Examples 1.1.1. (1) The group of codimension 1 cycles, the divisors, is
also written Div(X).
(2) The zero-cycles
= Z0(X). These are finite formal sums Z =

nαPα where is an irreducible 0-dimensional k-variety. The de-
gree of is just the degree of the field extension [k(Pα)/k] and deg(Z) =

(3) To any subscheme Y of X with irreducible components of dimension
one can associate the class [Y ] =


Zdα (X), where is
the length of the zero-dimensional Artinian ring OY,Yα ; see [Ful, 1.5].
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