Cohomology of tori over \(p\)-adic curves
Let \(V\) be a regular curve over a \(p\)-adic field, \(T\) a torus over \(V\), and \(T'\) the corresponding dual torus. The authors construct a suitable étale cohomology group \(H^q_{cc}(V,T)\) together with a pairing \[ H^q_{cc}(V,T)\times H^{3-q}(V,T')\longrightarrow \mathbb Q/\mathbb Z, \] and prove various duality properties for this pairing. This pairing in closely related to three other pairings: First, the Lichtenbaum-Tate pairing \[ H^q(C,\mathbb G_m)\times H^{3-q}(C,\mathbb G_m)\longrightarrow\mathbb Q/\mathbb Z \] for regular proper curves \(C\) over \(p\)-adic fields [\textit{S. Lichtenbaum}, Invent. Math. 7, 120--136 (1969; Zbl 0186.26402)]. Second, the Poincaré duality pairing \[ H^q_c(V,{}_n T)\times H^{4-q}(V,{}_n T')\longrightarrow \mathbb Q/\mathbb Z \] on torsion points. Third, the Tate-Nakayama pairing \[ H^q(k,T)\times H^{2-q}(k,X^*(T))\longrightarrow \mathbb Q/\mathbb Z \] for tori \(T\) and their character group \(X^*(T)\) over \(p\)-adic fields \(k\) [\textit{J.-P. Serre}, Cohomologie Galoisienne, 5th ed. (1994; Zbl 0812.12002)]. The main problem to define the desired pairing is that the curve \(V\) is not necessarily compact and that the torus \(T\) does not necessarily extend over the compactification \(C\). The authors solve this in an elegant way by introducing the cohomology group \(H^q_{cc}(V,T)\), which is defined as the cohomology group of a sheaf \(j_{!!}T\) on the compactification \(C\). In the special case \(T=\mathbb G_{m,V}\), this sheaf is defined by the exact sequence \[ 0\to j_{!!}\mathbb G_{m,V}\to\mathbb G_{m,C}\to i_*\mathbb G_{m,Z}\to 0, \] where \(j:V\rightarrow C\) is the open embedding, and \(i:Z\rightarrow C\) the closed embedding of the complement \(Z=C-V\).
scientific article; zbMATH DE number 1945199