##  [Beilinson Regulator](/beilinson-regulator-0) 

 Definition

A homomorphism (or family of homomorphisms) from higher algebraic K-theory or motivic cohomology groups of a variety to Deligne cohomology, real cohomology, or other analytic/cohomological realizations, encoding arithmetic and period information of algebraic cycles.

 

 

 

 

 

 





## Principle

Principle

Bridge algebraic and arithmetic information (K-theory or motivic classes) with analytic invariants (periods, regulators) by extracting real or mixed-Hodge-theoretic cohomology classes that reflect special values of L-functions and height pairings.

 

 

 

 

 





## Demonstration

Demonstration

For a smooth projective curve, the Beilinson regulator on K_2 or motivic H^2 produces elements in Deligne cohomology whose real periods are the classical elliptic or Bloch regulators; these numbers appear in formulas relating regulators to L'(E,0) or other special values conjecturally.

 

 

 

 

## Misapplication

Misapplication

Assuming the map is injective or surjective in general, or confusing it with the Chern character: the Beilinson regulator is subtle, depends on mixed Hodge or motivic structures, and can have large kernels or cokernels; p-adic variants behave differently.

 

 

 

 

 





## Consequence

Consequence

Provides arithmetic invariants linking algebraic cycles and motives to analytic quantities; under conjectures it explains relations between K-theory, values of L-functions, and heights, and it supplies concrete regulators used in explicit computations of arithmetic invariants.

 

 

 

 

## Reversal

Reversal

There is no canonical algebraic inverse that recovers motivic classes from regulator images: analytic realizations forget integral and motivic extension data, so reversing the regulator is typically impossible without extra structure or conjectural identifications.

 

 

 

 

 





## Boundary

Boundary

Defined where mixed Hodge structures, Deligne cohomology, or appropriate realizations exist: it requires careful treatment for singular varieties, nonprojective schemes, or p-adic settings where one uses syntomic or p-adic regulators with different formal properties.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes conceptually with other regulator maps (Borel, Soulé, syntomic) and with cycle class maps; the tension arises between analytic/period descriptions and purely algebraic/motivic descriptions of the same arithmetic phenomena.

 

 

 

 

 





## Synthesis

Synthesis

The Beilinson regulator is the period map from higher algebraic invariants to analytic cohomology: it extracts real or mixed-Hodge theoretic data from motivic classes, producing regulator numbers that connect the algebraic geometry of cycles to special-value and height phenomena, while losing integral and extension data.