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Spectral sequences in smooth generalized cohomology. (English) Zbl 1375.55002
Let $$\text{Sh}_\infty({\mathcal C}\text{art}{\mathcal S}\text{p})_+$$ be the category of pointed smooth stacks over $${\mathcal C}\text{art}{\mathcal S}\text{p}$$, the category of Cartesian spaces, and let $${\mathcal A}\text{b}_{\text{gr}}$$ be the category of graded abelian groups. A smooth cohomology theory is a functor ${\mathcal E}^\ast : \text{Sh}_\infty({\mathcal C}\text{art}{\mathcal S}\text{p})_+^{\text{op}} \to {\mathcal A}\text{b}_{\text{gr}}$ satisfying four axioms: invariance, additivity, Mayer-Vietoris and suspension. Herein the homotopy axiom is dropped since it does not necessarily yield an equivalence of stacks, while on the other hand an equivalence of stacks induces a weak homotopy equivalence of geometric realizations. Let $${\mathcal E}$$ denote a smooth spectrum representing $${\mathcal E}^\ast$$ which consists of pointed stacks. Then we can define $${\mathcal E}^\ast(M)$$ for any manifold $$M$$ by considering the category of manifolds as an $$\infty$$-subcategory of $$\text{Sh}_\infty({\mathcal C}\text{art}{\mathcal S}\text{p})$$. The authors prove (Theorem 25) that there is a spectral sequence with $E_2^{p, q}=H^p(M, {\mathcal E}^q) \Rightarrow {\mathcal E}^{p+q}(M)$ where $$H^p$$ denotes the $$p$$th Čech cohomology with coefficients in the presheaf $${\mathcal E}^q$$. The proof can be done in the usual way using a filtration by the Čech resolution with respect to an open covering of $$M$$. In this paper the authors mainly consider the spectral sequence for the differential refinement $${\hat{\mathcal E}}^\ast$$ of $${\mathcal E}^\ast$$ with the intention of comparing with that for $${\mathcal E}^\ast$$. The differential function spectrum representing $${\hat{\mathcal E}}^\ast$$ is written here $$\text{diff}({\mathcal E}, \text{ch})$$. Then we have a natural map $$I : \text{diff}({\mathcal E}, \text{ch}) \to \underline{\mathcal E}$$ where $$\underline{\mathcal E}$$ denotes the presheaf of spectra $${\mathcal E}$$ which is equivalent to the underlying theory of $${\mathcal E}$$ as smooth spectra. It is shown that this map $$I$$ induces a morphism between the corresponding spectral sequences. The use of this morphism enables us to compare the differentials in those two spectral sequences; however, as a result, it turns out that this does not work well. For this reason the authors try to look for a different map and obtain (Theorem 34) that under certain conditions there is a map between the lower quadrants of the two spectral sequences corresponding to $$\text{diff}({\mathcal E}, \text{ch})$$ and $${\mathcal E}$$. This morphism allows us to carry out what we want to do above. The final section (Section 4) is devoted to examining the spectral sequences for the following three differential cohomology theories: smooth Deligne cohomology, differential complex $$K$$-theory and differential Morava $$K$$-theory where the last one is a smooth extension of integral Morava $$K$$-theory which has been introduced by the authors.

##### MSC:
 55N15 Topological $$K$$-theory 55T10 Serre spectral sequences 55T25 Generalized cohomology and spectral sequences in algebraic topology 53C05 Connections (general theory) 55S05 Primary cohomology operations in algebraic topology 55S35 Obstruction theory in algebraic topology
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