Splitting Principle – Origins and Generalizations

Question

Basically, I'm aware of "splitting principles" for the following three objects (which are all isomorphic modulo torsion).

1. The Chow group a la Fulton.

2. The classical Grothendieck group of vector bundles or coherent sheaves.

3. The $\gamma$-graded Grothendieck group.

I was just wondering where the idea of "the splitting principle" comes from. I'm guessing somewhere in topology when one wanted to define Chern classes and show some properties. But I don't know.

And above that, is there some more general way of looking at this? I know there is a theorem that connects higher K-groups with Chow groups in a sense. So I ask, is there a way of deducing the splitting principle for one of the above objects from the other? (It's easy if we want to do this modulo torsion, of course.)

Answer 1

We can think of the splitting principle as a condition on a "cohomology theory" (of some sort) $E^*$, coming about when working with Chern classes for instance, and then ask: When does $E^*$ satisfy this condition? First, let's make the condition more precise and reformulate it:

Condition 1: Given $X$ and a vector bundle $V$ on $X$, there exists $f: X' \to X$ such that $f^* V'$ has a filtration with subquotients line bundles, and $f^*: E^*(X) \to E^*(X')$ is injective.

But there is a universal choice for $X'$, namely the flag variety of $V$: $p: Fl(V) \to X$. Any $f: X' \to X$ with $f^* V'$ filtered with line bundle subquotients will factor through $p$, and so we're really just asking if $p^*: E^*(X) \to E^*(Fl(V))$ is injective.

Condition 1': For all $X$ and $V$, $p^*: E^*(X) \to E^*(Fl(V))$ is injective.

At this point there are two ways this answer can go, depending on ones tastes:

1. $Fl(V)$ is a very geometric object over $X$, so we might as well ask that we actually have a formula for $E^*(Fl(V))$ in terms of $E^*(X)$. If $E^*$ is "reasonable" (i.e., has Chern classes giving rise to a "projective bundle formula") then iteratively applying the projective bundle formula will give such a thing, and in fact show that $E^*(X)$ is a direct summand of $E^*(Fl(V))$.
2. (My favorite:) There's a nice way of strengthening Condition 1' that also holds in all reasonable cases, and that looks rather natural. You can ask that $Fl(V) \to X$ behave like a "covering", i.e. that (Condition 2:) $$ E^*(X) \to E^*(Fl(V)) \to E^*\left(Fl(V) \times_X Fl(V)\right) $$ is an equalizer diagram. (So not only is pullback injective, but you can identify its image...) (In fact, in reasonable cases it'll be a split equalizer diagram, related to the direct summand thing above.)

If your question is one of proof + generalization (which I think it is), rather than vague motivation, then I haven't addressed it yet:

In topology. one can show that any complex-oriented cohomology theory (i.e., one with Chern classes for line bundles) $E^*$ has a projective bundle formula, satisfies all the conditions, etc.

In more-algebro-geometric contexts, you could deduce the Chow + K-theory (I don't know anything about the $\gamma$-filtration) statements by either

1. Constructing $c_1$ + proving a projective bundle formula, and then feeding this into a general argument using these to prove the rest.
2. Going to the universal example of algebraic cobordism and then deducing the results for Chow + K-theory from the known relationships between them and algebraic cobordism. (Though this second approach is not so great, since those relationships hold under much more stringent hypotheses than are necessary to run the argument.)

One could also ask to generalize this in another direction, replacing vector bundles and $Fl(V)$ by more general $G$-bundles and their associated $G/B$-bundles. In general, that's a more complicated story...