group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 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 dierential 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)
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
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
is the Dirichlet regulator given by
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
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
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
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.
Based on the above abstract formulation of the classical Beilinson- and Borel-regulators, the following general definition suggests itsef:
Consider the following data:
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.
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)
Accordingly the spectrum of $A$-regulators of $\mathcal{K}(\mathcal{C})$ is the mapping spectrum $[\mathcal{K}(\mathcal{C}), H A]$.
(…)
Some special cases of Beilinson regulators have known “geometric” constructions in terms of maps relating holomorphic line n-bundles for various $n$.
The regulator
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
or rather its component
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.
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
Moreover, for the above sheaves of $R$-modules $cycl(V)$ we have the direct image sheaves $\pi_\ast V$ and there is an identity
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
where the hats denote the differential cohomology refinement.
See at Becker-Gottlieb transfer.
There is some relation betwee 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
Then then map
is the Borel regulator (Neumann 11, p. 6).
For the moment, see at Bloch group for more details.
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
See also
The relation to Chern characters is made very explicit in
see also
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
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
Formore 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)
Last revised on November 7, 2015 at 05:17:17. See the history of this page for a list of all contributions to it.