In mathematics, especially in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, the Beilinson regulator is the

Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...

map from

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

Deligne cohomology In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordin ...

: :

K_n (X) \rightarrow \oplus_ H_D^ (X, \mathbf Q(p)).

Here, ''X'' is a complex smooth

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...

, for example. It is named after

Alexander Beilinson Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1 ...

. The Beilinson regulator features in Beilinson's conjecture on

special values of L-functions In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely :1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;=\; \frac,\! ...

. The ''Dirichlet regulator'' map (used in the proof of

Dirichlet's unit theorem In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a posi ...

) for the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

\mathcal O_F

of a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

''F'' :

\mathcal O_F^\times \rightarrow \mathbf R^, \ \ x \mapsto (\log , \sigma (x), )_\sigma

is a particular case of the Beilinson regulator. (As usual,

\sigma: F \subset \mathbf C

runs over all

complex embedding In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfi ...

s of ''F'', where conjugate embeddings are considered equivalent.) Up to a factor 2, the Beilinson regulator is also generalization of the Borel regulator.

References

* {{cite book, title=Beilinson's conjectures on special values of L-functions, year=1988, publisher=Academic Press, isbn=0-12-581120-9, editor=M. Rapoport, N. Schappacher and P. Schneider Algebraic geometry Algebraic K-theory