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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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##### References:
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