A few years ago, I got curious to see how people attack the Jacobian conjecture. The Jacobian conjecture says the following:
Let be a field of characteristic zero, and let be an unramified map. Then is an isomorphism.
The conjecture is usually stated algebraically, not geometrically. That statement goes as follows: Let the map be given by .
Define , the Jacobian of , to be . Then the algebraic statement is
Let ,, …, are polynomials in variables such that is in . Then the generate the ring .
The geometric meaning of is that vanishes at the points where is ramified1.
I have noticed a pattern in many of the false proofs. Today, I’m going to tell you how to spot these proofs. The warning sign is the phrase “but has no nontrivial units.”
Disclaimer: I am not an expert on the Jacobian conjecture. I will not referee manuscripts on the subject unless they are relevant to my published work in some specific way. Please do not send me preprints on the subject (unless you are a personal friend).
It will be convenient to separate the domain and range of . Let’s write . Assume for the sake of contradiction that is not an isomorphism. The proofs I am thinking of proceed as follows:
Step 1: We can factor as where is an open inclusion, is finite, and is affine and normal.
Step 2: is ramified. (This is the part that uses characteristic zero and that uses the assumption that is not an isomorphism.)
Step 3: is not ramified, a contradiction. (This is the step that uses the assumption that the Jacobian is nonvanishing.)
Step 4: Tenure!
The factorization in step 1 can be built; the easiest way is to construct the coordinate ring as the integral closure of in . Step 2 can be made correct (although there are often errors here). The difficulty is in step 3.
Here is the argument that I think most people have in mind for step 3:
Completely Bogus Proof: Consider the Jacobian of . The restriction of to is , which is assumed to be a nonzero constant. Since is dense in , the function is also a nonzero constant on , so is unramified.
Now, this is completely bogus, because there is no such function as . The Jacobian is defined for maps from affine space to affine space, not for maps from an arbitrary variety to affine space. And no one writes this proof.
If I were trying to fix the bogus proof, I would think about how to define something like a Jacobian for a general map. Here’s how that would go:
False Proof: Let be the canonical module2 of . If is smooth, this is . Let be a nowhere vanishing section of . Define by
Now, since is unramified, the function must not vanish on . But there are no nontrivial units in the coordinate ring of , so must be a nonzero constant. Thus, is nowhere vanishing on , and the map is nowhere ramified.
The part of this proof which is true is that really is a well defined section of and this section vanihses exactly where is ramified. Can you spot the lie in this proof? Here it is: “Let be a nowhere vanishing section of .” There is no reason to believe that will be a trivial line bundle. The reader may take it as a challenge to construct an embedding where , is affine and is nontrivial. The construction in my previous post doesn’t quite work for this purpose, but it may be a good starting point.
Authors don’t explicitly write the false proof. What they do is to define a function , usually by some explicit formula. This function is often equal in many special cases to the function defined above, so in those cases vanishes precisely where is ramified; the authors tend to believe that their will always capture the ramification behavior of . But in reality, there may be no function whose vanishing encodes the ramification of . For number theorists, this is related to the possibility that the different ideal need not be principal.
Authors who take this approach will often phrase step 2 as “ vanishes somewhere on ” and step 3 as “ vanishes nowhere on .” This is why the proof of step 2 is often broken, because the definition of may make this restatement is false.
But the real problem is in step 3. The authors want to say “ is unramified, so does not vanish on , so since the coordinate ring of has no nontrivial units, does not vanish anywhere.” But, because doesn’t truly control the vanishing of , it is quite possible that does vanish on . This is why the observation about nontrivial units is a warning flag to me.
I would like to build a counter-example to defeat this line of attack once and for all. Namely, I would like to build spaces and maps such that
(2) is unramified;
(3) the coordinate ring of has no nontrivial units;
BUT (4) is ramified.
I suspect that I have such a counter-example, as described above. It’s pretty darn messy though, and I haven’t locked down the proof of (3). Better ideas are welcome!
1 When is not algebraically closed, this statement has to be interpreted with some care.
2 There are some technical issues defining when is not Cohen-Macaulay, but I don’t think that they are crucial.