No Theorem B for henselian affine schemes

This is just to record informally a counter example which we found in March 2017. Please let me know if such an example is in the literature and I will add a reference.

Let (A, I) be a henselian pair. On Z = Spec(A/I) with the Zariski topology consider the presheaf O^h which associates to the open V = D(f) \cap Z of Z the ring A_f^h where (A_f^h, I_f^h) is the henselization of the pair (A_f, I_f) = (A_f, IA_f). It is easy to see that A_f^h only depends on the Zariski open V of Z.

Then O^h is a sheaf (on the basis of standard opens of Z), but it may have nonvanishing higher cohomology.

You can deduce the sheaf property from Tag 09ZH if you think about it right.

To get an example of where the cohomology is nonzero, start with Z_0 : xy(x+ y – 1) = 0 in the usual affine plane over the complex numbers. Let R be the henselization of C[x, y] at the ideal of Z_0. Let A be the integral closure of R in the algebraic closure K of C(x, y). Then A is a domain and (A, I) is henselian where I is the ideal generated by xy(x + y – 1) in A. Denote Z = V(I).

The reason for going all the way up to A is that A is a normal domain whose fraction field K is algebraically closed. Hence all local rings of A are strictly henselian and moreover affine schemes etale over Spec(A) are just disjoint unions of opens of Spec(A). See Tag 0EZN. Thus O^h = O_{Spec(A)}|_Z in this case (restriction is usual restriction of sheaves in Zariski topology). In particular, the map O_{Spec(A)} —> (constant sheaf value K on Spec(A)) induces a map O^h —> (constant sheaf value K on Z).

Observe that since Z_0 is a triangle, we have H^1(Z_0, Z) = Z. Let g be a generator of this cohomology group. Then you check that g|_Z is still non-torsion. I do this using a limit argument and trace maps for finite maps between normal surfaces. If you have a clever short argument, let me know. You do have to use/prove something because we can “unwind” the triangle topologically, so your argument has to show that this doesn’t happen (in some sense) for the map Z —> Z_0 of topological spaces.

Next we consider the maps of sheaves

Z —> O^h —> (constant sheaf with value K on Z)

Since g|_Z is nontorsion we see that its image in the first cohomology of the last sheaf is nonzero and we conclude H^1(Z, O^h) is nonzero.

Conclusion: no theorem B for henselian affine schemes. Enjoy!

Leave a Reply

Your email address will not be published. Required fields are marked *