Last week I started looking at the chapter on chow homology. When I first wrote it in 2009, it was a bit of an experiment. As a consequence, various proofs used three different approaches: one using K-groups, one using blow-up lemmas as in Fulton, and one via the key lemma (see below). Today I rearranged the whole thing to simplify the exposition and stick with the approach using the key lemma.
How much intersection theory does the chapter cover? We work consistently with schemes locally of finite type over a fixed universally catenary, locally Noetherian base scheme. We introduce cycle groups, flat pullback, proper pushforward, and rational equivalence. After proving some basic properties, we introduce the operation c_1(L) ∩ – where L is an invertible sheaf and the Gysin map for an effective Cartier divisor. Having proved the basic properties of these operations, we have enough to introduce chern classes of locally free sheaves and prove their basic properties. The chapter ends with stating the Grothendieck-Riemann-Roch theorem (without proof).
The key lemma is a statement about tame symbols over a Noetherian local domain of dimension 2. I think of it as the statement that the secondary ramifications add up to zero. I’d love it if you could tell me a reference for this lemma in the literature (I assume there is some paper on K-theory that contains this result).
The key lemma implies our key formula. I am sure I discussed this statement with somebody in my office at some point, but I cannot remember who; if it was you, please email me! Anyway, the key formula quickly implies that c_1(L) ∩ – passes through rational equivalence (Section Tag 02TG), the fact that c_1(L) ∩ c_1(N) ∩ – = c_1(N) ∩ c_1(L) ∩ -, and that the Gysin map for an effective Cartier divisor passes through rational equivalence (Section Tag 02TK).