A coherent spectral realization of derived abelian complexes

Assume the Axiom of Choice and the fixed compatible universes of the derived and spectral interfaces. Let $\mathcal{D}(\mathbb{Z})$ be the stable infinity-localization of unbounded cochain complexes of abelian groups at quasi-isomorphisms; its homotopy category is $D(\mathbb{Z})$. Let $H\mathbb{Z}$ denote the discrete Eilenberg–Mac Lane ring spectrum of the integers. Fix the unit-preserving equivalence $\mathrm{\Phi }:{\mathrm{Mod}}_{H\mathbb{Z}}\simeq \mathcal{D}(\mathbb{Z})$ and an inverse with its equivalence data. Here a module spectrum is a spectrum with an $H\mathbb{Z}$ action including its associative and unital coherences; morphisms are coherent linear maps. Write $U$ for the forgetful functor to spectra and define $\mathcal{H}:=U{\mathrm{\Phi }}^{-1}:\mathcal{D}(\mathbb{Z})⟶\mathrm{S}\mathrm{p}\mathrm{.}$ This is a coherent functor preserving small limits, zero objects, fibers and integer shifts, with natural identifications ${\pi }_i\mathcal{H}(C)\cong H^{-i}(C),\mathcal{H}(C[r])\simeq \mathcal{H}(C)[r]$ for every derived complex $C$ and all integers $i,r$. Different inverse data for the fixed equivalence give naturally equivalent functors. Thus composing any supplied coherent complex-valued diagram with $\mathcal{H}$ gives a coherent spectral diagram, including its restriction arrows, limits and fibers, independent up to equivalence of those inverse data. No lift of an arbitrary ordinary $D(\mathbb{Z})$-valued diagram is asserted.

External input on derived enhancements

External stable mapping and derived linear target interfaces

Stable spectra and exact triangles

The Axiom of Choice

Reference 0 imports the normalized derived/module equivalence, the module limit criterion and the unit mapping formula used in [F2]. Their hypotheses are checked with $R=\mathbb{Z}$ and spectra as ambient category. These established inputs are not proved here. Their composition into $\mathcal{H}$ and the indexing calculation are the local argument below.