Deligne cup-product input data

Assume full AC and let $M$ have the good cover and comparison input required for the Deligne exact-sequence lemma. For each $r\ge 1$, use the cone-model projections $I_r:{\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^r(M;\mathbb{Z})⟶H^r(M;\mathbb{Z}),R_r:{\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^r(M;\mathbb{Z})⟶{\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^r(M),$ where $I_r$ is projection to integer-sheaf hypercohomology followed by the given integral comparison, and $R_r$ is projection to the truncated de Rham component, identified with global closed forms. Define $a_r:{\mathrm{\Omega }}^{r-1}(M)⟶{\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^r(M;\mathbb{Z}),a_r(\eta )=[(0,d\eta ,-\eta )]$ in $\mathrm{Cone}(\mathbb{Z}\oplus {\sigma }_{\ge r}{\mathrm{\Omega }}^{\bullet }\to {\mathrm{\Omega }}^{\bullet })[-1]$. Its degree-$r$ coordinates are respectively the integer component, the truncated-form component and the degree-$(r-1)$ full-form component. The cone differential on these global coordinates is $(z,\alpha ,b)\mapsto (dz,d\alpha ,-db+z-\alpha )$; hence this is a cocycle, $I_r{a}_r=0$, and $R_r{a}_r=d$.

A Deligne cup-product input is a family of bilinear maps ${⌣}_{\mathrm{D}\mathrm{e}\mathrm{l}}:{\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^p(M;\mathbb{Z})\times {\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^q(M;\mathbb{Z})\to {\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^{p+q}(M;\mathbb{Z})$ for all $p,q\ge 1$ satisfying the following conditions: it is natural under smooth pullback, associative and graded-commutative; $I_{p+q}(x{⌣}_{\mathrm{D}\mathrm{e}\mathrm{l}}y)=I_p(x)⌣I_q(y)$ for the integral singular cup product and $R_{p+q}(x{⌣}_{\mathrm{D}\mathrm{e}\mathrm{l}}y)=R_p(x)\wedge R_q(y)$; and $a_p(\eta ){⌣}_{\mathrm{D}\mathrm{e}\mathrm{l}}x=a_{p+q}(\eta \wedge R_q(x))$ for $\eta \in {\mathrm{\Omega }}^{p-1}(M)$ and $x\in {\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^q(M;\mathbb{Z})$. This definition names conditional data and does not assert that such a family exists.

Exact sequence for smooth Deligne cohomology

Singular cohomology ring

The Axiom of Choice

Smooth Deligne cohomology model

Exponential-sequence equivalence of Deligne models

Bunke states existence and uniqueness of this structure and gives a truncated-complex product formula. In v6, Problems 3.96–3.97 are followed by chain-map, associativity and homotopy-commutativity verifications; compatibility with $a$ and the factorization used for uniqueness are posed in Problems 3.98–3.99. This item defines conditional product data and does not reconstruct that existence-and-uniqueness argument locally.