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differential algebraic K-theory

under construction

Context

Differential cohomology

Ingredients

Connections on bundles

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Higher abelian differential cohomology

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Higher nonabelian differential cohomology

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Fiber integration

Application to gauge theory

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Arithmetic geometry

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Cohesive \infty-Toposes

Contents

Idea

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

HSh (SmthMfd,B)coDiscΓDiscΠB \mathbf{H} \coloneqq Sh_\infty\left(SmthMfd, \mathbf{B} \right) \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \mathbf{B}

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

BSh (Sch ) \mathbf{B} \coloneqq Sh_\infty\left(Sch_{\mathbb{Z}}\right)

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.

Definition

Absolute Hodge cohomology

Definition

Write

Ω Stab(Sh (Sch )) \Omega^\bullet_{\mathbb{C}} \in Stab(Sh_\infty(Sch_{\mathbb{C}}))

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.

(Bunke-Tamme 12, section 3.1)

Definition

Write

Ω compl *Ω Stab(B) \Omega^\bullet_{\mathbb{Z}} \coloneqq compl^\ast \Omega^\bullet_{\mathbb{C}} \in Stab(\mathbf{B})

for the inverse image of Ω \Omega^\bullet_{\mathbb{C}} under the base change given by

compl()× Spec():Sch Sch . compl \coloneqq (-)\times_{\mathbb{Z}}Spec(\mathbb{C}) \;\colon\; Sch_{\mathbb{Z}}\longrightarrow Sch_{\mathbb{C}} \,.

(Bunke-Tamme 12, section 3.2)

There is a resolution of Ω Stab(B)DiscStab(H)\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 Ω \Omega^\bullet_{\mathbb{Z}}

Ω Stab(H) \Omega^\bullet \in Stab(\mathbf{H})

(Bunke-Tamme 12, (47)).

Definition

While Ω Ω \Omega^\bullet \simeq \Omega^\bullet_{\mathbb{Z}}, below we use the chain-level truncation Ω 0\Omega^{\bullet \geq 0} which is no longer in the image of DiscDisc, hence no longer a flat modality-modal type.

Algebraic K-theory sheaf of spectra

Write

Vect lcH \mathbf{Vect}_{lc} \in \mathbf{H}

for the stack which to X×SSmthMfd×Sch X\times S \in SmthMfd \times Sch_{\mathbb{Z}} assigns the groupoid of locally free and locally finitely generated pr S *𝒪 Spr_S^\ast \mathcal{O}_S-modules (modules over the inverse image of the structure sheaf of SS under the projection map X×SSX \times S \to S).

This is a commutative monoid object with respect to direct sum. Write

K𝒦(Vect lc ) K \coloneqq \mathcal{K}(\mathbf{Vect}_{lc}^{\oplus})

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.

The refined Beilinson regulator

There is a refinement of the Beilinson regulator to a smoothly parameterized version K\mathbf{K} of algebraic K-theory:

Definition

(…)

r Beil:KΩ r^{Beil} \;\colon\; K \longrightarrow \Omega^\bullet_{\mathbb{Z}}

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).

Differential algebraic K-theory

Recall the discussion at differential cohomology hexagon.

Definition

Differential algebraic K-theory is the homotopy fiber product

K^K×Ω Ω 0Stab(H) \hat K \coloneqq K \underset{\Omega^\bullet}{\times} \Omega^{\bullet \geq 0} \in Stab(\mathbf{H})

of the inclusion of the non-negative degree truncation of the de Rham resolution of the absolute Hodge complex, def. , with the refined Beilinson regulator, def.

K^ Ω 0 (pb) K r Beil Ω . \array{ \hat K &\longrightarrow& \Omega^{\bullet \geq 0} \\ \downarrow &(pb)& \downarrow \\ K &\underset{r^{Beil}}{\longrightarrow}& \Omega^\bullet } \,.

(Bunke-Tamme 12, def. 5.1, Bunke-Tamme 13, def. 3.1).

Applications

References

Differential algebraic K-theory as above is introduced and studied in

Relevant references in ordinary algebraic K-theory include

  • Robert Thomason and Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhauser Boston, Boston, MA, 1990, pp. 247-435. MR 1106918 (92f:19001)

Last revised on December 30, 2016 at 16:19:16. See the history of this page for a list of all contributions to it.