Brauer Groups of Curves

Let {C/k} be a smooth projective curve over an algebraically closed field. The main goal of today is to show that {Br(C)=0}. Both smooth and being over an algebraically closed field are crucial for this computation. The computation will run very similarly to the last post with basically one extra step.

We haven’t actually talked about the Brauer group for varieties, but there are again two definitions. One has to do with Azumaya algebras over {\mathcal{O}_C} modulo Morita equivalence. The other is the cohomological Brauer group, {Br'(C):=H^2(C, \mathbb{G}_m)}. As already stated, it is a big open problem to determine when these are the same. We’ll continue to only consider situations where they are known to be the same and hence won’t cause any problems (or even require us to define rigorously the Azumaya algebra version).

First, note that if we look at the Leray spectral sequence with the inclusion of the generic point {g:Spec(K)\hookrightarrow C} we get that {R^1g_*\mathbb{G}_m=0} by Hilbert 90 again which tells us that {0\rightarrow H^2(C, g_*\mathbb{G}_m)\hookrightarrow Br(K)}. Now {K} has transcendence degree {1} over an algebraically closed field, so by Tsen’s theorem this is {C_1}. Thus the last post tells us that {H^2(C, g_*\mathbb{G}_m)=0}.

The new step is that we need to relate {H^2(C, g_*\mathbb{G}_m)} to {Br(C)}. On the ├ętale site of {C} we have an exact sequence of sheaves

\displaystyle 0\rightarrow \mathbb{G}_m\rightarrow g_*\mathbb{G}_m\rightarrow Div_C\rightarrow 0

where {\displaystyle Div_C=\bigoplus_{v \ \text{closed}}(i_v)_*\mathbb{Z}}.
Taking the long exact sequence on cohomology we get

\displaystyle \cdots \rightarrow H^1(C, Div_C)\rightarrow Br(C)\rightarrow H^2(C, g_*\mathbb{G}_m)\rightarrow \cdots .

Thus it will complete the proof to show that {H^1(C, Div_C)=0}, since then {Br(C)} will inject into {0}. Writing {\displaystyle Div_C=\bigoplus_{v \ \text{closed}}(i_v)_*\mathbb{Z}} and using that cohomology commutes with direct sums we need only show that for some fixed closed point {(i_v): Spec(k(v))\hookrightarrow C} that {H^1(C, (i_v)_*\mathbb{Z})=0}.

We use Leray again, but this time on {i_v}. For notational convenience, we’ll abuse notation and call both the map and the point {v\in C}. The low degree terms give us {H^1(C, v_*\mathbb{Z})\hookrightarrow H^1(v, \mathbb{Z})}. Using the Galois cohomology interpretation of ├ętale cohomology of a point {H^1(v,\mathbb{Z})\simeq Hom_{cont}(G_{k(v)}, \mathbb{Z})} (the homomorphisms are not twisted since the Galois action is trivial). Since {G_{k(v)}} is profinite, the continuous image is compact and hence a finite subgroup of {\mathbb{Z}}. Thus {H^1(C, v_*\mathbb{Z})=0} which implies {H^1(C, Div_C)=0} which gives the result that {Br(C)=0}.

So again we see that even for a full curve being over an algebraically closed field is just too strong a condition to give anything interesting. This suggests that the Brauer group really is measuring some arithmetic properties of the curve. For example, we could ask whether or not good/bad reduction of the curve is related to the Brauer group, but this would require us to move into Brauer groups of surfaces (since the model will be a relative curve over a one-dimensional base).

Already for local fields or {C_1} fields the question of determining {Br(C)} is really interesting. The above argument merely tells us that {Br(C)\hookrightarrow Br(K)} where {K} is the function field, but this is true of all smooth, proper varieties and often doesn’t help much if the group is non-zero.

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