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An additive version of higher Chow groups. (English) Zbl 1100.14014
The cosimplicial scheme \[ \Delta^n = \text{Spec} \,k[t_0, t_1, \dots, t_n]/(\sum_{i=0}^n t_i - t) \] was used by S. Bloch [Adv. Math. 61, 267–304 (1986; Zbl 0608.14004)] to define higher Chow groups. In this note, we let \(t\) tend to \(0\) and replace \(\Delta^\bullet\) by a degenerate version \[ Q^n = \text{Spec} \,k[t_0, t_1, \dots, t_n]/(\sum_{i=0}^n t_i) \]
to define an additive version of the higher Chow groups. For a field \(k\), we show the Chow group of 0-cycles on \(Q^n\) in this theory is isomorphic to the group of absolute \((n-1)\)-Kähler forms \(\Omega^{n-1}_k\). An analogous degeneration on the level of de Rham cohomology associated to “constant modulus” degenerations of varieties in various contexts is discussed.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19D45 Higher symbols, Milnor \(K\)-theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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