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 algebraic 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 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 Z d−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 | F, 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 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) = deg(Pα). (3) To any subscheme Y of X with irreducible components of dimension one can associate the class [Y ] = α nαYα α Z (X), where is the length of the zero-dimensional Artinian ring OY,Y α see [Ful, 1.5]. 1
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