Exponential-sequence equivalence of Deligne models

On every sufficiently small open set $V$, a smooth map $g:V\to U(1)$ has a smooth real logarithm because its image is contained locally in an arc on which the exponential map is a diffeomorphism. Thus $0\to \mathbb{Z}\to {\mathrm{\Omega }}^0\to \mathcal{U}(1)\to 0$ is exact as a sequence of sheaves: the kernel consists exactly of locally constant integer functions and the last map is locally surjective.

Define the morphism from the integer-form complex to the shifted circle-valued complex by sending $\mathbb{Z}$ in degree zero to zero, sending $f\in {\mathrm{\Omega }}^0$ to $e^{2\pi if}\in \mathcal{U}(1)$ in degree one, and sending $\alpha \in {\mathrm{\Omega }}^j$ to $(-1{)}^j\alpha$ for $1\le j<k$. Here the circle complex has $\mathcal{U}(1)$ in degree zero and ${\mathrm{\Omega }}^j$ in degree $j$, and its shift $[-1]$ has differential $-d_{\mathcal{T}}$ or $-d$. Thus the degree-one identity is $-d_{\mathcal{T}}(e^{2\pi if})=-df$, equal to the image of $df$ in the ${\mathrm{\Omega }}^1$ term; in each higher degree, $-d((-1{)}^j\alpha )=(-1{)}^{j+1}d\alpha$. The integer inclusion maps to the circle identity, so the degree-zero square also commutes. For $k=1$ there is no outgoing circle differential. At each stalk, degree-zero cohomology of the integer-form complex is zero because $\mathbb{Z}\to {\mathrm{\Omega }}^0$ is injective. For $k=1$, both complexes terminate in degree one, and their degree-one cohomology groups are respectively all smooth real germs modulo integer germs and all smooth circle-valued germs; exponentiation identifies them by step 1.1. For $k\ge 2$, degree-one cohomology instead consists of locally constant real germs modulo integer germs, identified with locally constant $U(1)$ germs. In degree two, when present, the kernels of the outgoing differentials agree (the outgoing differential is zero at the last term), and the incoming image sheaves agree by the local logarithms of step 1.1. In every later degree the alternating-sign isomorphisms identify the kernels and images, since each differential differs by exactly the sign just checked. Thus the map induces an isomorphism on every cohomology sheaf and is a quasi-isomorphism by [F3].

Choose Cartan–Eilenberg resolutions $I_{\mathcal{D}}$ and $I_{\mathcal{S}}$ by [F6]–[F7], and retain them as data. Their total augmentations $j_{\mathcal{D}}:\mathcal{D}\to J_{\mathcal{D}}$ and $j_{\mathcal{S}}:\mathcal{S}\to J_{\mathcal{S}}$ satisfy [F8]. Thus ${\mathbb{H}}^n(M,\mathcal{D})=H^n(\mathrm{\Gamma }(M,J_{\mathcal{D}}))$ and similarly for $\mathcal{S}$ by [F5]. Write $q:\mathcal{D}\to \mathcal{S}$ for step 2.1. Applying the precomposition bijection of [F8] to $j_{\mathcal{D}}$ and target $J_{\mathcal{S}}$ supplies a unique homotopy class $\bar{q}:J_{\mathcal{D}}\to J_{\mathcal{S}}$ with $\bar{q}{j}_{\mathcal{D}}\simeq j_{\mathcal{S}}q$.

In the derived category, $Q(\bar{q})=Q(j_{\mathcal{S}})Q(q)Q(j_{\mathcal{D}}{)}^{-1}$ is invertible by [F4]. Its inverse has a representative $r:J_{\mathcal{S}}\to J_{\mathcal{D}}$ by [F8]. The uniqueness in [F8] gives $r\bar{q}\simeq 1$ and $\bar{q}r\simeq 1$. Applying the additive global-sections functor preserves these homotopy equations, so $H^n(\mathrm{\Gamma }\bar{q})$ is an isomorphism in every degree. Replacing either chosen resolution and using the same construction over its identity gives a unique comparison homotopy class; compositions agree by uniqueness. Hence these isomorphisms are independent of the resolutions under precisely these canonical identifications. Since $\mathcal{S}=\mathcal{T}[-1]$, they identify ${\mathbb{H}}^n(M,\mathcal{D})$ with ${\mathbb{H}}^{n-1}(M,\mathcal{T})$.

For a smooth map $f:N\to M$, pullback of functions and forms defines cochain maps $u_K:K_M\to f_{\ast }{K}_N$, for $K=\mathcal{D},\mathcal{S}$, and $u_{\mathcal{S}}{q}_M=(f_{\ast }{q}_N)u_{\mathcal{D}}$ because exponentiation and exterior derivative commute with pullback. By [F9], $f_{\ast }{J}_{K_N}$ is bounded below and termwise injective, hence K-injective by [F8]'s bounded-below injective criterion. The precomposition bijection of [F8] therefore gives a unique homotopy class $b_K:J_{K_M}\to f_{\ast }{J}_{K_N}$ with $b_K{j}_{K_M}\simeq (f_{\ast }{j}_{K_N})u_K$. On global sections this defines pullback. The two maps $b_{\mathcal{S}}{\bar{q}}_M$ and $(f_{\ast }{\bar{q}}_N)b_{\mathcal{D}}$ have homotopic composites with $j_{{\mathcal{D}}_M}$ by the displayed identities; [F8] makes them homotopic. Their maps on cohomology agree. The same uniqueness proves compatibility with identity maps, composition and changes of resolution. This does not assume that $f_{\ast }{j}_{K_N}$ is a quasi-isomorphism.

Cover refinement commutes with $q$ on the model's Cech total complexes, since its components commute with restrictions and alternating reindexing. One can check invertibility there directly: on each nonempty good-cover intersection $W$, any $g:W\to U(1)$ has a smooth logarithm. Indeed, for a supplied smooth contraction $H:[0,1]\times W\to W$ from $x_0$ to the identity and $e^{2\pi ia}=g(x_0)$, set $\mathrm{\ell }(x)=a+{\int }_0^1(2\pi i{)}^{-1}(g\circ H{)}^{-1}{\mathrm{\partial }}_t(g\circ H)dt$. Differentiating $e^{-2\pi i{\int }_0^t(2\pi i{)}^{-1}(g\circ H{)}^{-1}{\mathrm{\partial }}_s(g\circ H)ds}g(H(t,x))$ in $t$ proves $e^{2\pi i\mathrm{\ell }}=g$. Thus the Cech map induced by $q$ is degreewise surjective; AC permits the simultaneous component lifts. Its kernel is the Cech total complex of $\mathbb{Z}\stackrel{1}{\to }\mathbb{Z}$ in sheaf degrees zero and one. The vertical contraction from degree one to zero, multiplied by $(-1{)}^p$ on Cech degree $p$, satisfies $Dh+hD=1$, so the kernel is contractible and the quotient map induces an isomorphism on cohomology. Therefore the refinement squares and, by invertibility, their inverse squares commute. Together with step 5.1 this proves both claimed naturalities.