Exact sequence for smooth Deligne cohomology

Put ${\mathcal{D}}_k=\mathrm{Cone}(f)[-1]$, where $f:\mathbb{Z}\oplus {\sigma }_{\ge k}{\mathrm{\Omega }}^{\bullet }\to {\mathrm{\Omega }}^{\bullet }$ sends $(z,\alpha )$ to $z-\alpha$, as in [F1]. The cone long exact sequence [F4], applied after the Cech-to-sheaf comparison [F2], gives a long exact hypercohomology sequence. Its relevant terms are ${\mathbb{H}}^{k-1}(\mathbb{Z}\oplus {\sigma }_{\ge k}{\mathrm{\Omega }}^{\bullet })\to {\mathbb{H}}^{k-1}({\mathrm{\Omega }}^{\bullet })\to H^k({\mathcal{D}}_k)\to {\mathbb{H}}^k(\mathbb{Z}\oplus {\sigma }_{\ge k}{\mathrm{\Omega }}^{\bullet })\to {\mathbb{H}}^k({\mathrm{\Omega }}^{\bullet })$.

For each form degree $j$, the Cech row ${\check{C}}^{\bullet }(\mathcal{U},{\mathrm{\Omega }}^j)$ has contraction in positive Cech degrees $(hc{)}_{i_0\dots i_{p-1}}={\sum }_i{\rho }_i{c}_{i{i}_0\dots i_{p-1}},$ where $({\rho }_i)$ is the subordinate partition of unity in [F1]. Local finiteness makes the sum smooth and the support condition lets each summand extend by zero; the alternating Cech formula gives $\delta h+h\delta =1$ for $p>0$. Consequently the Cech total complex for the full ${\mathrm{\Omega }}^{\bullet }$ has the same cohomology as its global-section de Rham complex, namely $H_{\mathrm{d}\mathrm{R}}^{\ast }(M;\mathbb{R})$. For ${\sigma }_{\ge k}{\mathrm{\Omega }}^{\bullet }$, all total terms below degree $k$ vanish; in degree $k$ the only term is ${\check{C}}^0(\mathcal{U},{\mathrm{\Omega }}^k)$, whose cocycles are global closed $k$-forms and which has no degree-$(k-1)$ boundaries. Its hypercohomology is therefore ${\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^k(M)$ in degree $k$ and zero below.

The comparison input [F5] identifies the constant-integer Cech groups in step 1.1 with ordinary singular cohomology; [F3] identifies the form-complex groups with real singular cohomology. The map $H^{k-1}(M;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^{k-1}(M;\mathbb{R})$ is coefficient inclusion followed by de Rham, and the final map sends $(c,\omega )$ to $c_{\mathbb{R}}-[\omega ]$. Thus the long exact sequence becomes $H^{k-1}(M;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^{k-1}(M;\mathbb{R})\stackrel{a}{\to }{\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^k(M;\mathbb{Z})\stackrel{(I,R)}{\to }{H}^k(M;\mathbb{Z})\oplus {\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^k(M)\stackrel{(c,\omega )\mapsto c_{\mathbb{R}}-[\omega ]}{\to }{H}^k(M;\mathbb{R})\mathrm{.}$ The middle maps are induced by the cone projections, with the sign convention of [F1]. Exactness at every displayed term follows from the long exact sequence.

The kernel at $(I,R)$ in step 2.1 is exactly the fiber product in the Statement; exactness of that long exact sequence makes $(I,R)$ surjective onto it. If $(c,\omega )$ lies in the fiber product, then $[\omega ]=c_{\mathbb{R}}$; for every smooth integral $k$-cycle $z$, de Rham evaluation gives ${\int }_z\omega =⟨c,z⟩\in \mathbb{Z}$. Hence $\omega \in {\mathrm{\Omega }}_{\mathbb{Z}}^k(M)$, so the fiber product is unchanged when the last factor is written with integral-period forms. Exactness at the earlier terms, including the kernel of $a$, was already established in step 2.1.