Duality for \(\mathbb Z\)-constructible sheaves on curves over finite fields (Q1946058)

Let \(X\) be a curve over a finite field and let \(\mathcal F\) be a \(\mathbb Z\)-constructible sheaf. The author defines Weil-etale Borel-Moore homology groups \(H_i^c(X_{\mathrm{ar}},\mathcal F)\) and constructs a pairing between \(H_i^c(X_{\mathrm{ar}},\mathcal F)\) and the Weil-etale cohomology with compact support \(H_c^i(X_W,\mathcal F)\). For \(0\leq i\leq 2\), we have pairings of free groups of finite rank \[ H_i^c(X_{\mathrm{ar}},\mathcal F)/\mathrm{tor}\times H_c^i(X_W,\mathcal F)/\mathrm{tor}\to \mathbb Z, \] and for \(0\leq i\leq 3\), we have pairings of finite groups \[ H^c_{i-1}(X_{\mathrm{ar}},\mathcal F)_{\mathrm{tor}}\times H_c^{i}(X_W,\mathcal F)_{\mathrm{tor}}\to\mathbb Q/\mathbb Z. \] All other cohomology and homology groups vanish. When \(X\) is smooth, the groups \(H_i^c(X_{\mathrm{ar}},\mathbb Z)\) are isomorphic to Weil-etale cohomology group \(H^i(X_W,\mathbb G_m)\). The pairing between \(H_c^i(X_W,\mathbb Z)\) and \(H^i(X_W,\mathbb G_m)\) is constructed by Lichtenbaum.