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Differential algebraic K-theory is the differential cohomology-refinement of algebraic K-theory.
In (Bunke-Tamme 12) this is realized effectively as the differential cohomology in a cohesive topos of the tangent (∞,1)-topos of the cohesive (∞,1)-topos
of ∞-stacks on the site of smooth manifolds with values in turn in ∞-stacks over a site of arithmetic schemes, hence by smooth ∞-groupoids but over a base (∞,1)-topos
of algebraic ∞-stacks.
This may be regarded as sitting inside the smooth E-∞-groupoids.
It is observed in this context that the Beilinson regulator in algebraic K-theory is naturally understood as a Chern character in this perspective of differential cohomology (Bunke-Tamme 12), which helps with studying it.
Write
for the chain complex of abelian sheaves (regarded as a sheaf of spectra under the stable Dold-Kan correspondence) which computes absolute Hodge cohomology of complex varieties.
Write
for the inverse image of $\Omega^\bullet_{\mathbb{C}}$ under the base change given by
There is a resolution of $\Omega^\bullet_{\mathbb{Z}} \in Stab(\mathbf{B}) \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H})$ by a sheaf of complexes of differential forms on smooth manifolds tensored with $\Omega^\bullet_{\mathbb{Z}}$
While $\Omega^\bullet \simeq \Omega^\bullet_{\mathbb{Z}}$, below we use the chain-level truncation $\Omega^{\bullet \geq 0}$ which is no longer in the image of $Disc$, hence no longer a flat modality-modal type.
Write
for the stack which to $X\times S \in SmthMfd \times Sch_{\mathbb{Z}}$ assigns the groupoid of locally free and locally finitely generated $pr_S^\ast \mathcal{O}_S$-modules (modules over the inverse image of the structure sheaf of $S$ under the projection map $X \times S \to S$).
This is a commutative monoid object with respect to direct sum. Write
for the stackification of the objectwise K-theory of a symmetric monoidal (∞,1)-category-construction.
This is the ordinary algebraic K-theory of schemes, as in (Thomason-Trobaugh 90) (Bunke-Tamme 12, section 3.3), see at algebraic K-theory – as the K-theory of algebraic vector bundles.
There is a refinement of the Beilinson regulator to a smoothly parameterized version $\mathbf{K}$ of algebraic K-theory:
(…)
As a homomorphism of spectrum objects this is (Bunke-Tamme 12, def. 4.26). As a homomorphism of E-∞ ring objects, this is (Bunke-Tamme 13, def. 2.18).
Recall the discussion at differential cohomology hexagon.
Differential algebraic K-theory is the homotopy fiber product
of the inclusion of the non-negative degree truncation of the de Rham resolution of the absolute Hodge complex, def. 3, with the refined Beilinson regulator, def. 4
(Bunke-Tamme 12, def. 5.1, Bunke-Tamme 13, def. 3.1).
Differential algebraic K-theory as above is introduced and studied in
Ulrich Bunke, Georg Tamme, Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)
Ulrich Bunke, David Gepner, Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory (arXiv:1306.0247)
Ulrich Bunke, Georg Tamme, Multiplicative differential algebraic K-theory and applications (arXiv:1311.1421)
Relevant references in ordinary algebraic K-theory include
Last revised on December 30, 2016 at 16:19:16. See the history of this page for a list of all contributions to it.