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

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

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

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

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

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

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

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

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.