nLab higher regulator

Idea

In the context of arithmetic geometry a higher regulator, or just regulator for short, is a homomorphism from algebraic K-theory to a suitable ordinary cohomology theory. (It makes sense to think that it “regulates” cocycles in algebraic K-theory, which tend to be hard to analyze, to become cocycles in ordinary cohomology, about which typically more may be said.) This generalizes the original concept of a regulator of a number field which is a measure for the group of units of the ring of integers of the number field, in view of the fact that the determinant function provides a canonical homomorphism $K_1 \to GL_1$ from the first algebraic K-theory to the group of units.

Regulators are used in the study of L-functions for instance in the context of the Beilinson conjectures and the Bloch-Kato conjecture.

The simplest example of such regulators are

• the analytic class number formula;

• the Dedekind zeta-function of $K$;

• the Dirichlet regulator map, discussed below.

These regulators may be understood as essentially being the Chern characters in algebraic K-theory (Deninger Scholl (2.6), Tamme 12). Based on this observation they serve to define differential cohomology-refinements of algebraic K-theory, namely differential algebraic K-theory (Bunke-Tamme 12).

we usually call transformations from K-theory to cyclic homology Chern characters, and transformations from K-theory to differential forms regulators. There is one exception, namely the usual Chern character from topological K-theory to cohomology with complex coefficients calculated by the de Rham cohomology. (Bunke 14, remark 2.20)

Definition

Dirichlet and Borel regulator

For $R$ a ring, $K R$ denoting its algebraic K-theory spectrum and $\Sigma^n H \mathbb{R}$ a suspension of the Eilenberg-MacLane spectrum of the real numbers, then a regulator of $K R$-cohomology theory is a homomorphism of spectra

$$r_{\sigma,p} \;\colon\; K R \longrightarrow \Sigma^p H \mathbb{R}$$

or equivalently is the induced cohomology operations. e.g. (Bunke-Tamme 12, 1.2)

If here $R$ is the ring of integers of a number field and $\sigma \colon R \hookrightarrow \mathbb{C}$ is a choice of embedding into the complex numbers, then the Borel regulator (Borel 74) is of this form, for odd $p$, such that its induced cohomology operation

$$r_{\sigma,p} \;\colon\; K_1(R) \longrightarrow \mathbb{R}$$

is the Dirichlet regulator given by

$$u \mapsto log {\vert \sigma(u) \vert} \,.$$

A description of this in differential algebraic K-theory is in (Bunke-Tamme 12, 1.2):

for $X$ a smooth manifold, then a class in $K R^0(X)$ is represented by a finitely generated projective $R$-module bundle $V \to X$. Write

$$cycl(V) \in K R^0(X)$$

for this class.

Under the chosen embedding $\sigma$ we have the complexification $V \otimes_\sigma \mathbb{C}$ of this module bundle, which is a complex vector bundle with (because $R$ is geometrically discrete) a flat connection $\nabla_{V_\sigma}$.

The choice of hermitean structure on $V_\sigma$, hence a reduction of the structure group to the unitary group induces an adjoint connection $\nabla_{V_\sigma}^\ast$. Write then

$$CS_p(\nabla_{V_\sigma}^\ast, \nabla_{V_\sigma})$$

for the relative Chern-Simons form between these two connections, hence for the transgression of the relative Chern character in degree $p+1$.

This is a closed differential form, (the Kamber-Tondeur form, see Bismut-Lott 95).

This differential form represents, via the de Rham theorem isomorphism, the Dirichlet regulator above

$$r_{\sigma,p}(cycl(V)) \simeq [CS_p(\nabla_{V_\sigma}^\ast, \nabla_{V_\sigma})] \in H_{dR}^p(X) \simeq H^p(X,\mathbb{R}) \,.$$

In differential algebraic K-theory this construction can be refined from landing in de Rham cohomology to landing in genuine ordinary differential cohomology (higher prequantization), hence with $CS_p(\nabla_{V_\sigma}^\ast, \nabla_{V_\sigma})$ itself realized as the curvature of a circle (p-1)-bundle with connection.

Regulators of generalized algebraic K-theories

Based on the above abstract formulation of the classical Beilinson- and Borel-regulators, the following general definition suggests itsef:

Definition

Consider the following data:

1. For $\mathcal{C}^\otimes$ a symmetric monoidal (∞,1)-category write $\mathcal{K}(\mathcal{C})$ for its algebraic K-theory of a symmetric monoidal (∞,1)-category.

2. For $A \in Ch_\bullet(Ab)$ any chain complex write $H A$ for its Eilenberg-MacLane spectrum given by the stable Dold-Kan correspondence.

Then a regulator with coefficients in $A$ of the algebraic K-theory represented by $\mathcal{K}(\mathcal{C})$ is a homomorphism of spectra (hence a cohomology operation)

$$r \;\colon\; \mathcal{K}(\mathcal{C}) \longrightarrow H A \,.$$

Accordingly the spectrum of $A$-regulators of $\mathcal{K}(\mathcal{C})$ is the mapping spectrum $[\mathcal{K}(\mathcal{C}), H A]$.

(Bunke-Tamme 12, def. 23).

Beilinson regulators

(…)

Examples

Constructions in terms of line $n$-bundles ($n-1$-bundle gerbes)

Some special cases of Beilinson regulators have known “geometric” constructions in terms of maps relating holomorphic line n-bundles for various $n$.

The regulator

$$c_{2,2} \colon K_2(X) \longrightarrow H^2(X, \mathbb{Z}(2)_{\mathcal{D}})$$

is given by sending pairs of non-vanishing holomorphic functions to the holomorphic line bundle which is their Beilinson-Deligne cup product (the “Deligne line bundle”) (Bloch 78).

Moreover, the regulator

$$c_{1,2} \colon K_1(X) \longrightarrow H^3(X, \mathbb{Z}(2)_{\mathcal{D}})$$

or rather its component

$$c_{1,2} \colon H^1(X, \mathbf{K}_2) \longrightarrow H^3(X, \mathbb{Z}(2)_{\mathcal{D}})$$

is given by sending functions constituing a cocycle in the relevant Gersten complex to a bundle gerbe whose transition line bundles are Deligne line bundles built from these functions (Brylinski 94, theorem 3.3).

Notice that $c_{2,1}$ is the regulator that interpolates the string 2-group/universal Chern-Simons line 3-bundle for a reductive algebraic group from the algebraic to the complex-analytic realm, see at universal Chern-Simons line 3-bundles – For reductive algebraic groups.

Properties

Becker-Gottlieb transfer and GRR for algebraic K-theory

For $\pi \colon X \to B$ a proper submersion of smooth manifolds, there is a variant of fiber integration in generalized cohomology given by the Becker-Gottlieb transfer in some $E$-cohomology theory

$$tr_\pi \;\colon\; E^\ast(X) \longrightarrow E^\ast(B) \,.$$

Moreover, for the above sheaves of $R$-modules $cycl(V)$ we have the direct image sheaves $\pi_\ast V$ and there is an identity

$$\underset{i \geq 0}{\sum} (-1)^i cycl(R^i \pi_\ast(V)) \simeq tr_\pi(cycl(V))$$

in $K R(B)$. The above Borel-Dirichlet regulator $r_{\sigma,p}$ is such that it preseves this as an identity in $H(X,\mathbb{R})$. Hence it plays a role here analogous that of the Chern character in the Grothendieck-Riemann-Roch theorem.

The Becker-Gottlieb transfer refines in turn to differential cohomology, hence differential algebraic K-theory mapping to ordinary differential cohomology, according to (Bunke-Gepner 13).

However, the above relation between direct image of sheaves and push-forward in cohomology receives a correction when refined to differential algebraic K-theory, a correction by a term in the image of the inclusion $a(-)$ of differential forms into differential cohomology, by the transfer index conjecture one has

$$\underset{i \geq 0}{\sum} (-1)^i \widehat{cycl}(R^i \pi_\ast(V)) + a(something) \simeq \widehat{tr}_\pi(\widehat{cycl}(V))$$

where the hats denote the differential cohomology refinement.

See at Becker-Gottlieb transfer.

Relation to complex volumes and Bloch group

There is some relation between the Borel regulators and complex volumes of hyperbolic manifolds via maps out of the Bloch group (Suslin 90, Neumann-Yang 97, p. 17, Zickert 07, p. 3, Zickert 09).

For $k$ an algebraic number field and $\sigma_1, \cdots, \sigma_{r_2}\colon k \to \mathbb{C}$ its complex embeddings up to conjugation, then write

$$vol_j \coloneqq vol \circ (\sigma_j)\colon H_3(PSL(2,k), \mathbb{Z}) \to \mathbb{R}$$

Then then map

$$(vol_1, \cdots, vol_{r_2}) \colon H_3(PSL(2,k),\mathbb{Z}) \longrightarrow \mathbb{R}^{r_2}$$

is the Borel regulator (Neumann 11, p. 6).

For the moment, see at Bloch group for more details.

Related concepts

• Beilinson conjecture

References

The Borel regulator is due to

• Armand Borel, Cohomologie de $SL_n$ et valeurs de fonctions de zeta, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 417 (1971), 613–636.

• Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 235-272 (1975). MR 0387496

The Beilinson regulator with values in Deligne cohomology is due to

• Spencer Bloch, Applications of the dilogarithm function in algebraic K-theory and algebraic geometry, in: Proc. of the International Symp. on Alg. Geometry, Kinokuniya, Tokyo, 1978

• Spencer Bloch, The dilogarithm and extensions of Lie algebras, Algebraic K-Theory Evanston 1980, Lecture Notes in Mathematics Volume 854, 1981, pp 1-23

• Alexander BeilinsonHigher regulators and values of L-functions, Journal of Soviet Mathematics 30 (1985), 2036-2070, (mathnet (Russian), DOI)

• Alexander Beilinson, Higher regulators of curves, Funct. Anal. Appl. 14 (1980), 116-118, mathnet (Russian).

• Alexander Beilinson, Height pairing between algebraic cycles, in K-Theory, Arithmetic and Geometry, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, DOI.

reviewed in

• Christophe Soulé, Régulateurs Séminaire Bourbaki, 27 (1984-1985), Exp. No. 644, 17 p. (Numdam)

• Hélène Esnault, Eckart Viehweg, Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)

• Alexander Goncharov, Regulators, Algebraic K-theory Handbook (arXiv:0407308)

A discussion of the Beilinson regulator on $K_2$ in terms of bundle gerbes is in

• Jean-Luc Brylinski, Holomorphic gerbes and the Beilinson regulator, Astérisque 226 (1994): 145-174 (pdf)

See also

• Christophe Soulé, On higher p-adic regulators, Algebraic K-Theory Evanston 1980 Lecture Notes in Mathematics Volume 854, 1981, pp 372-401_ (publisher page)

The relation to Chern characters is made very explicit in

• Georg Tamme, Karoubi’s relative Chern character and Beilinson’s regulator, Ann. Sci. Ec. Norm. Super. (4) 45 (2012), no. 4, 601-636. (pdf)

see also

• Christopher Deninger, Anthony Scholl, section 2.6 of The Beilinson conjectures (pdf)

The interpretation of these regulator Chern characters in differential algebraic K-theory is due to

• 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, Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)

• Ulrich Bunke, Thomas Nikolaus, Georg Tamme, The Beilinson regulator is a map of ring spectra (arXivL1509.05667)

• Ulrich Bunke, A regulator for smooth manifolds and an index theorem (arXiv:1407.1379)

based on

• Jean-Michel Bismut, John Lott, Flat vector bundles, direct images and

  higher real analytic torsion_, J. Amer. Math. Soc. 8 (1995), no. 2, 291-363.

See also

• Charles Weibel, Algebraic K-theory and the Adams e-invariant, in Algebraic K-theory, number theory, geometry and analysis(Bielefeld, 1982), volume 1046 of Lecture Notes in Math., pages 442-450. Springer, Berlin, 1984. 61 pdf

• A.A. Suslin, On the K-theory of local fields, In Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), volume 34, pages 301-318, 1984.

• Wikipedia, Beilinson regulator

For more references see also at Beilinson conjecture.

Relation of the Borel regulator to the Bloch group, the Cheeger-Simons class/complex volumes of hyperbolic manifolds is discussed in

• Andrei Suslin. $K_3$ of a field, and the Bloch group. Trudy Mat. Inst. Steklov., 183:180–199, 229, 1990. Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239, Galois theory, rings, algebraic groups and their applications (Russian).

• Walter Neumann, Jun Yang, Bloch invariants of hyperbolic 3-manifolds, Duke Math. J. Volume 96, Number 1 (1999), 29-59. (arXiv:math/9712224, Euclid)

• Christian Zickert, The volume and Chern-Simons invariant of a representation, Duke Math. J., 150 (3):489-532, 2009 (arXiv:0710.2049, Euclid)

• Christian Zickert, The extended Bloch group and algebraic K-theory (arXiv:0910.4005)

• Walter Neumann, Realizing arithmetic invariants of hyperbolic 3-manifolds, Contemporary Math 541 (Amer. Math. Soc. 2011), 233–246 (arXiv:1108.0062)