An additive version of higher Chow groups

الأصل · EN

The cosimplicial scheme Deltaᵇullet = Δ⁰ smallmatrix → smallmatrix Δ¹ smallmatrix to smallmatrix...; Δⁿ:=(k[t₀,...c,tₙ]/(∑ tᵢ -t)) was used in B to define higher Chow groups. In this note, we let t tend to 0 and replace Δ by a degenerate version Q = Q⁰ smallmatrix to smallmatrix Q¹ smallmatrix to smallmatrix...; Qⁿ:= (k[t₀,...c,tₙ]/(∑ tᵢ)) to define an additive version of the higher Chow groups. For a field k, we show the Chow group of 0-cycles on Qⁿ in this theory is isomorphic to the absolute (n-1)-Kähler forms Ωⁿ⁻¹ₖ. An analogous degeneration on the level of de Rham cohomology associated to ``constant modulus'' degenerations of varieties in various contexts is discussed.