The String Coffee Table

Varghese Mathai yesterday was so kind to explain to me how to push forward K-theory classes. It’s called lower shriek, and I think the name is somewhat reminiscent of the feeling I had when seeing the construction.

Very roughly, the point is this: let

be a map of (topological) spaces. Let

be a bundle on $X$. Naïvely, the push-forward of $E$ along $f$ is the bundle whose fiber over $y\in Y$ is the combined preimage $p^{-1}(f^{-1}(y))$.

The problem ist, of course, that in general $f^{-1}(y)$ contains infinitely many points, so that $p^{-1}(f^{-1}(y))$ would be something like a continuous direct sum of vector spaces.

So in order to get a sensible vector bundle on $Y$, we need to cut down drastically the size of $p^{-1}(f^{-1}(y))$.

Assume there is a ${\mathrm{Spin}}^{ℂ}$ structure around. It allows us to construct a Dirac operator on the space of sections of $E$. Somehow like in the family index theorem, we use this to pass to its kernel and cokernel, thus obtaining finite dimensional vector spaces again.

That’s very roughly one way to look at this construction. I think.

While interesting, I have the feeling there must be something more direct and conceptual than this.

If we pass from vector bundles to sheaves of modules, there is.

A sheaf has an obvious push-forward. Now it’s the pullback which is more problematic. But at least the bare-bones definition of the pullback of sheaves is pretty straightforward.

So here goes, taken from the textbook

R. Hartshorne
Algebraic Geometry
Spinger (1977) .

(p.109-110, with exercise 1.18 on p. 68 and first def on p. 65).

So let $(X,O_X)$ be a ringed space over $X$ and $(Y,O_Y)$ a ringed space over Y. Say we are handed a morphism of ringed spaces

coming with a continuous map

and a morphism ov sheaves of rings

Let $G$ be some sheaf of $O_Y$-modules on $Y$ (playing the role of a vector bundle).

We do something like a pullback from $Y$ to $X$ for $G$ as follows. The problem of course is that only preimages under $f$ of open sets are guaranteed to be open. We want to find a way to take an open set $U$ in $X$ and send it to an open set in $Y$. So we simply take $f(U)$ and look for something like the smallest open set $V$ containing $f(U)$.

More precisely, we let $f^{-1}G$ be the sheaf given by

This is a nice definition, because the functor $f^{-1}$ is simply the adjoint to the push-forward functor $f_{*}$.

But the pullback of a sheaf of modules is still a slight modification of $f^{-1}$.

Notice that, by the adjointness property just mentioned, from the given ring homomorphism $f^{♯}:O_Y\to f_{*}{O}_X$ we obtain an “inverse” morphism

This we can use to act with $O_X$ on $f^{-1}G$. So we devide out by this action and define the pullback sheaf $f^{*}G$ (of modules) to be given by

This is nice again, since it turns out that pullback $f^{*}$ and pushforward $f_{*}$ are adjoint functors between the category of $O_X$-modules to that of $O_Y$-modules.