Graduiertenkolleg 1821 Cohomological Methods in Geometry
Summer School 2013
Regulators and Differential Algebraic K-Theory
What the summer school is about
The aim of the summer school is to give a new explicit description of the
Beilinson regulator from algebraic K-theory to Deligne cohomology, also known
as absolute Hodge cohomology. This new description is well-behaved with
respect to multiplication.
Algebraic K-theory is a fine invariant of a ring/an algebraic variety/a scheme
built from the category of vector bundles on the space. It turns out to be
very hard to compute. A first step is the introduction of invariants in
cohomology: to every vector bundle we attach its Chern class in say singular
cohomology. This is still much to coarse. It factors over a refined Chern
class with values in Deligne cohomology. Deligne cohomology is attached to
complex algebraic varieties and uses the information coming from the
corresponding analytic space, most importantly the Hodge decomposition theorem. The definition of Deligne cohomology can be
rewritten in terms of the Hodge structure on singular cohomology and then goes by
the name of absolute Hodge cohomology. The Chern class is also called
Beilinson regulator because it was introduced by Beilinson.
History of regulators
The Beilinson regulator is a generalization of a construction that
was known before in special cases. It turns out that the information in
the Beilinson regulator is enough to understand algebraic K-theory of number
fields - finite extensions of the field of rational numbers.
The first case is the Dirichlet regulator used to study the units of the ring
of integers of a number field, i.e., its K_1. A unit u is mapped to the tuple of the logarithms
of the absolute values of u under all embeddings of the field into the complex
numbers. The famous Dirichlet unit theorem states that the image of this
map is a lattice, in particular that the units form a finitely generated
abelian group. Borel in his seminal paper was able to prove the analogous
result for higher algebraic K-groups of these rings of integers. His adhoc
regulator was translated by Beilinson and later Burgos into the general
Why this is important
As described above, explicit computations of higher algebraic K-theory are
very scarce. A better understanding of the Beilinson regulator should
eventually lead to a better understanding of algebraic K-groups.
On the topological side, the K-theory spectra of varieties can be
used to define cohomology theories. Refinements of these cohomology
theories to manifolds should be useful to analyze differentiable
stuctures of fibre bundles.
There are also (at least) two number theoretic applications. The Beilinson
regulator is a main ingredient of the Beilinson conjecture on special values
of L-functions, which generalizes the class number formula (the case of the
units of ring of integers of a number field) and the conjecture of Birch and
It also occurs in the definition of arithmetic K-groups generalizing the
theory of arithmetic Chow groups in Arakelov theory.
Scope of the summer school
During the summer school, there will be foundational lecture series
introducing higher algebraic K-theory, absolute Hodge cohomology and
the Beilinson regulator. In the second half of the week, the aim will be
to give a new conceptual description of the Beilinson regulator.
This construction uses the language of infinity categories to be introduced
in a third series of foundational lectures. The new construction of the
regulator will be phrased in the language of differential cohomology.
The theory will be illustrated by presenting certain constructions
in arithmetic geometry (Heights, Bloch group).
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