Computing the divisor class group of toric varieties over an arbitrary field

Question

Let $k$ be an arbitrary field and let $X$ be a toric variety over $k$, coming from a fan $\Sigma$. If $k$ is algebraically closed, then theorem 4.1.3 of Cox ,Little and Schenck’s Toric Varieties book says that there is an exact sequence $M\rightarrow \mathrm{Div}_T(X)\rightarrow \mathrm{Cl}(X)\rightarrow 0$.The first map sends every $m\in M$ to $\mathrm{div}(\chi^{m})$ and the second map sends every torus-invariant Weil divisor on $X$ to its class in $\mathrm{Cl}(X)$.This exact sequence allows for the convenient computations of divisor class groups of toric varieties coming from a fan, defined over an algebraically closed field. I am wondering whether the same exact sequence works over an arbitrary base field $k$, so that similar computations work in this generality. When I look at the proof of theorem 4.1.3 in the book mentioned above, it says that $\mathbb{C}[x_1^{\pm 1},x_2^{\pm 1},…,x_n^{\pm 1}]$ is the coordinate ring of the torus $(\mathbb{C}^{*})^n$, which is isomorphic to the coordinate ring $\mathbb{C}[M]$ of the torus $T_N$. But over an arbitrary field $k$, the $n$-dimensional algebraic torus is not always isomorphic to $(\mathbb{G}_m)^n$.

Answer 1

The torus $T$ may have non-trivial Picard group. Therefore this sequence is not exact in general. One may consider for example $T: x^2 + y^2 = 1$ over $\mathbb{Q}$. The closure of this is a conic, and $T$ is isomorphic to the projective line with a closed point of degree two removed. Hence in this case $\mathrm{Pic}(T) \cong \mathbb{Z}/2\mathbb{Z}$ and the boundary divisors only generate a subgroup of index $2$ in the full Picard group.

However all is not lost and one can write down an analogous sequence using Galois cohomology. I assume for simplicity that $X$ is a smooth and proper toric variety for some torus $T$ over $k$ and that $k$ is perfect with absolute Galois group $\Gamma_k$. Then the sequence $$0 \to M\rightarrow \mathrm{Div}_T(X_{\bar{k}})\rightarrow \mathrm{Cl}(X_{\bar{k}})\rightarrow 0$$ is exact. Applying Galois cohomology, one obtains $$0 \to M^{\Gamma_k} \rightarrow \mathrm{Div}_T(X_{\bar{k}})^{\Gamma_k} \rightarrow \mathrm{Cl}(X_{\bar{k}})^{\Gamma_k} \rightarrow H^1(k,M) \rightarrow H^1(k,\mathrm{Div}_T(X_{\bar{k}})).$$ We have $\mathrm{Cl}(X_{\bar{k}})^{\Gamma_k} = \mathrm{Cl}(X)$ since $X(k) \neq \emptyset$. Moreover by Shapiro's Lemma and $H^1(k,\mathbb{Z}) = 0$, one sees that $H^1(k,\mathrm{Div}_T(X_{\bar{k}})) = 0$ since $\mathrm{Div}_T(X_{\bar{k}})$ is a finitely generated free $\mathbb{Z}$-module which is a permutation module for $\Gamma_k$. Moreover an application of the Hochshild-Serre spectral sequence shows that $H^1(k,M) = \mathrm{Pic}(T)$ (see Lemma 6.3 of [1]). Altogether one obtains the sequence $$0 \to M^{\Gamma_k} \rightarrow \mathrm{Div}_T(X_{\bar{k}})^{\Gamma_k} \rightarrow \mathrm{Cl}(X) \rightarrow \mathrm{Pic}(T) \rightarrow 0,$$ so the failure of exactness comes precisely from $\mathrm{Pic}(T)$.

[1] Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. reine angew. Math. 327 (1981), 12--80.