Today, we’re going to construct a ring that encodes quite a lot of intersection data (though not terribly transparently) as well as some special combinations of Chern classes.  A lot of modern intersection theory and enumerative geometry takes place in the K-theory ring of a scheme X.

So first we need to figure out what is K-Theory.  There are a few approaches we can take, and I’m going to start with the most basic.  Let K(X) be the free abelian group on the vector bundles (of any rank) on X, modded out by the relation that E=E'+E'' if there exists a short exact sequence 0\to E'\to E\to E''\to 0.  So K-theory will ignore all extension problems, and just say that an extension is equal to the sum of the two factors.

This makes K-Theory into an abelian group, and a useful one.  What makes it into a ring, is that we have a second operation on vector bundles.  We can take the tensor product.  So we define the multiplication that way, and then extend it by linearity.

Now, the Whitney Sum formula implies that the total Chern classes multiply in short exact sequence, which means that we have c(E+F)=c(E)c(F) as a map from K-theory to, well…something.  The what we won’t worry about just yet.  The problem comes when we try to take c(E\otimes F), which isn’t as simple.  However, we can define ch(E), the Chern character of E, to be ch(E)=\sum_{i=1}^r \mathrm{exp}(\alpha_i), where \alpha_i are Chern roots of E.  Working things out, the first few terms we get ch(E)=r+c_1+\frac{1}{2}(c_1^2-2c_2)+\frac{1}{6}(c_1^3-3c_1c_2+3c_3)+\ldots in terms of the Chern classes.

Here’s what’s wonderful about the Chern character: it’s a homomorphism.  That is, ch(E+F)=ch(E)+ch(F) and ch(E\otimes F)=ch(E)ch(F).  Now, we haven’t quite nailed down what it’s a homomorphism into, but it’s something like the dual space of A_*(X), or traditionally it is H^*(X) with appropriate coefficients.  So in answer to “Why does K-Theory tell us something about intersections?” we have “It has a natural homomorphism into an intersection ring.”

But it gets better! K-Theory isn’t REALLY about vector bundles.  It’s just a trick of the light, really.  We look for vector bundles, so we see them.  But really, K-Theory is something we do MUCH more generally: with complexes of coherent sheaves.  So yes, we’re sitting on the Derived Category here, though for the moment, we’ll just say that if there is a map from one complex to another that induces isomorphisms on the cohomology, they’re the same.  In this case, we take K-Theory to be a free abelian group on the coherent sheaves, with the same relation by short exact sequences (which implies relations for long exact sequences).  Now, the trick for showing that they’re the same theory is that under mild hypotheses (I’m not sure exactly what hypotheses are necessary, but I know smooth is more than sufficient…is it maybe Cohen-Macaulay? Anyone out there know?) Every coherent sheaf has a locally free resolution.

This means that every sheaf is quasi-isomorphic to a complex of vector bundles.  The relations in K-Theory then say that if E^\cdot \to \mathscr{F} is the locally free resolution, we have \mathscr{F}=\sum (-1)^i E^i.  So we get the same K-Theory.  Even better, using these resolutions, we can define the Chern classes of complexes of coherent sheaves, via the Chern character homomorphism!

Now, this version of K-Theory is much more clearly intersection theoretic: Let V,W\subset X be subschemes.  Then \mathscr{O}_V,\mathscr{O}_W are their structure sheaves, and their scheme theoretic intersection is given by…the tensor product! There’s some interesting work being done on making these products explicit in K-theory, working out generalizations of the classical Schubert calculus  for homogeneous spaces.  Among those working on this are Buch, Kresch, Tamvakis, Mihalcea, and others (those are just the ones that I’m familiar with).

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