nLab differential algebraic K-theory

under construction

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

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

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

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

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

$$\Omega^\bullet_{\mathbb{C}} \in Stab(Sh_\infty(Sch_{\mathbb{C}}))$$

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

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

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

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

(Bunke-Tamme 12, section 3.2)

There is a resolution of $\Omega^\bullet_{\mathbb{Z}} \in Stab(\mathbf{B}) \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H})$ by a sheaf of complexes of differential forms on smooth manifolds tensored with $\Omega^\bullet_{\mathbb{Z}}$

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

(Bunke-Tamme 12, (47)).

Definition

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.

Algebraic K-theory sheaf of spectra

Write

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

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

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

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 $\mathbf{K}$ of algebraic K-theory:

Definition

(…)

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

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

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

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.

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

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

Applications

• differential refinement of Becker-Gottlieb transfer and its image under Borel regulators. See at transfer index conjecture.

Related concepts

• differential K-theory

• differential form with logarithmic singularities

References

Differential algebraic K-theory as above is introduced with application to higher Beilinson regulators in:

• Ulrich Bunke, Georg Tamme, Regulators and cycle maps in higher-dimensional differential algebraic K-theory, Advances in Mathematics Volume 285, 5 November 2015, Pages 1853-1969

  (arXiv:1209.6451, doi:10.1016/j.aim.2015.08.004)

• Ulrich Bunke, David Gepner, Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory, Memoirs of the American Mathematical Society 2021 (arXiv:1306.0247, ISBN:978-1-4704-4685-7)

• Ulrich Bunke, Georg Tamme, Multiplicative differential algebraic K-theory and applications, Ann. K-Theory 1(3): 227-258 (2016) (arXiv:1311.1421, doi:10.2140/akt.2016.1.227)

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)