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.<\/p>\n
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.<\/p>\n
Then O^h is a sheaf (on the basis of standard opens of Z), but it may have nonvanishing higher cohomology.<\/p>\n
You can deduce the sheaf property from Tag 09ZH<\/a> if you think about it right.<\/p>\n 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).<\/p>\n 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<\/a>. 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).<\/p>\n Observe that since Z_0 is a triangle, we have H^1(Z_0, Z<\/b>) = Z<\/b>. 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. <\/p>\n Next we consider the maps of sheaves<\/p>\n Z<\/b> —> O^h —> (constant sheaf with value K on Z)<\/p>\n 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.<\/p>\n Conclusion: no theorem B for henselian affine schemes. Enjoy!<\/p>\n","protected":false},"excerpt":{"rendered":" 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. … Continue reading