I’d like to use this post to try to understand the “theorem of the cube,” following Mumford’s “Abelian varieties.”
Theorem 1 Let be proper varieties over an algebraically closed field , and let be a connected variety. Let be a line bundle. Suppose there exist -valued points in such that is trivial when restricted to . Then is trivial.
I’ve found the proof due to Weil and Murre in the second chapter of Mumford to be quite impenetrable; the argument makes sense line by line but I have never been able to see a larger picture. Fortunately, it turns out that the third chapter has a more scheme-theoretic (nilpotents will be used!) argument which is much more transparent.
1. Infinitesimal trivializations
Consider the set of such that is trivial. Then this is a closed set: in fact, a point belongs to this set if and only if
Now, the semicontinuity theorem implies that this set is closed. So far, we know that it is nonempty: it contains . Using an infinitesimal thickening argument, we will show that it consists of all of , from which we will be able to conclude easily.
Let be an local finite-dimensional -algebra, and let be a morphism which is set-theoretically at (in other words, an “infinitesimal thickening” of the point at ). We will show that
is trivial. To see this claim, we use induction on the dimension of . When is one-dimensional, this is the assumption. In general, assume the claim for local artinian -algebras of smaller dimension. Given , there is an element annihilated by the maximal ideal of . Then we have a surjection of -algebras
and an exact sequence of sheaves (on a one-point space)
This gives an exact sequence of sheaves
Tensoring with gives an exact sequence of sheaves on ,
The goal is now to show that there exists an element which restricts to a generator at each closed point, for this will give a trivialization of . By the inductive hypothesis on the dimension, we know that there exists a section with this property. It suffices to show that lifts to a section of , as that will automatically by a generator at each closed point.
Now, to show that , we use the fact that
by the Künneth formula, since is trivial there. Consequently, to show that , it suffices to show that after replacing by and after replacing by .
But the coboundary map
is trivial because the line bundle is trivial and thus (by the Künneth formula) any section over can be lifted to . Similarly for the analogous one when is replaced by a point.
It follows now that in our original situation, the coboundary vanishes when restricted to and dually when restricted to .
2. Extending to a full neighborhood
To complete the proof of the theorem of the cube, we now need to pass from arbitrary nilpotent thickenings to the whole scheme. In other words, we need to show the following:
Proposition 2 Let be a proper -variety, and let be a connected variety. Let . Suppose there exists a closed point such that for every local finite-dimensional -algebra and map thickening , the restriction
is trivial. Then for a suitable line bundle for the projection.
Assume irreducible, without loss of generality.
We will in fact take . We observe that is a coherent sheaf on , so that is a finitely generated -module. Moreover, by the formal function theorem, we have
where the last step used the fact that is a trivial line bundle. It follows that the completion of is free of rank one, so is free of rank one.
In a neighborhood of , we find that is a line bundle. We also find from this that is surjective. In particular, the unit element of (recall triviality of here) prolongs to a section over for some open neighborhood of . Shrinking (and using properness of ), we may assume is invertible at all points of .
In any event, we find that is trivial and that the map
is an isomorphism on . In particular, is trivial for .
But the set of points such that is trivial is a closed subset of (by the reasoning earlier), and consequently is trivial for all . From this, one can appeal to the general machine “cohomology and base change” to argue that is an isomorphism in general.
3. Conclusion of the argument
Let’s now finish the proof of the theorem of the cube. Given as in the statement, and satisfying the desired conditions, we find that is trivial for all . Moreover, is the pull-back of a line bundle on . Restricting to , we find that this line bundle must be trivial, so is trivial.