The Stacks project

42.1 Introduction

In this chapter we discuss Chow homology groups and the construction of Chern classes of vector bundles as elements of operational Chow cohomology groups (everything with $\mathbf{Z}$-coefficients).

We start this chapter by giving the shortest possible algebraic proof of the Key Lemma 42.6.3. We first define the Herbrand quotient (Section 42.2) and we compute it in some cases (Section 42.3). Next, we prove some simple algebra lemmas on existence of suitable factorizations after modifications (Section 42.4). Using these we construct/define the tame symbol in Section 42.5. Only the most basic properties of the tame symbol are needed to prove the Key Lemma, which we do in Section 42.6.

Next, we introduce the basic setup we work with in the rest of this chapter in Section 42.7. To make the material a little bit more challenging we decided to treat a somewhat more general case than is usually done. Namely we assume our schemes $X$ are locally of finite type over a fixed locally Noetherian base scheme which is universally catenary and is endowed with a dimension function. These assumptions suffice to be able to define the Chow homology groups $\mathop{\mathrm{CH}}\nolimits _*(X)$ and the action of capping with Chern classes on them. This is an indication that we should be able to define these also for algebraic stacks locally of finite type over such a base.

Next, we follow the first few chapters of [F] in order to define cycles, flat pullback, proper pushforward, and rational equivalence, except that we have been less precise about the supports of the cycles involved.

We diverge from the presentation given in [F] by using the Key lemma mentioned above to prove a basic commutativity relation in Section 42.27. Using this we prove that the operation of intersecting with an invertible sheaf passes through rational equivalence and is commutative, see Section 42.28. One more application of the Key lemma proves that the Gysin map of an effective Cartier divisor passes through rational equivalence, see Section 42.30. Having proved this, it is straightforward to define Chern classes of vector bundles, prove additivity, prove the splitting principle, introduce Chern characters, Todd classes, and state the Grothendieck-Riemann-Roch theorem.

There are two appendices. In Appendix A (Section 42.68) we discuss an alternative (longer) construction of the tame symbol and corresponding proof of the Key Lemma. Finally, in Appendix B (Section 42.69) we briefly discuss the relationship with $K$-theory of coherent sheaves and we discuss some blowup lemmas. We suggest the reader look at their introductions for more information.

We will return to the Chow groups $\mathop{\mathrm{CH}}\nolimits _*(X)$ for smooth projective varieties over algebraically closed fields in the next chapter. Using a moving lemma as in [Samuel], [ChevalleyI], and [ChevalleyII] and Serre's Tor-formula (see [Serre_local_algebra] or [Serre_algebre_locale]) we will define a ring structure on $\mathop{\mathrm{CH}}\nolimits _*(X)$. See Intersection Theory, Section 43.1 ff.

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