By Ding S.S., Gai Y.Y.

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**Additional info for Ar-Weighted Poincare-Type Inequalities for Differential Forms in Some Domains**

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Then the differential symbol dF : Kn (F )/p → ΩnF , {a1 , . . 1). In other words, the sequence d ℘ 1 0 −−−−→ Kn (F )/p −−−F−→ ΩnF −−−−→ ΩnF /dΩn− F is exact. This theorem relates the Milnor K -group modulo p of a field of characteristic p with a submodule of the differential module whose structure is easier to understand. The theorem is important for Kato’s approach to higher local class field theory. For a sketch of its proof see subsection A2 in the appendix to this section. There exists a natural generalization of the above theorem for the quotient groups Kn (F )/pi by using De Rham–Witt complex ([ BK, Cor.

Hence C(K) → C(L)G is an injection. The claim is proved. 4. Step IV. We use the Hochschild–Serre spectral sequence H r (Gk , hq (Kur )) =⇒ hq+r (K). For any q , Ωqksep Ωqk ⊗k k sep , Z1 Ωqksep Z1 Ωqk ⊗kp (k sep )p . Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 50 J. Nakamura Thus, grm hq (Kur ) grm hq (K) ⊗kp (k sep )p for 1 copies of k sep , hence we have m < e . This is a direct sum of H 0 (Gk , U1 hq (Kur )) U1 hq (K)/Ue hq (K), H r (Gk , U1 hq (Kur )) = 0 for r 1 because H r (Gk , k sep ) = 0 for r the following two exact sequences 1 .

Therefore the group Hpn+1 (F ) is naturally identified with the quotient group F ⊗F ∗ ⊗ · · · ⊗F ∗ /J . It is not difficult to show that the subgroup J is generated by the following elements: (ap − a) ⊗ b1 ⊗ · · · ⊗ bn , a ⊗ a ⊗ b2 ⊗ · · · ⊗ bn , a ⊗ b1 ⊗ · · · ⊗ bn , where bi = bj for some i = j . This description of the group Hpn+1 (F ) can be easily generalized to define Hpni+1 (F ) for an arbitrary i 1 . Namely, we define the group Hpni+1 (F ) as the quotient group Wi (F ) ⊗ F ∗ ⊗ · · · ⊗ F ∗ /J n where Wi (F ) is the group of Witt vectors of length i and J is the subgroup of Wi (F ) ⊗ F ∗ ⊗ · · · ⊗ F ∗ generated by the following elements: (F(w) − w) ⊗ b1 ⊗ · · · ⊗ bn , (a, 0, .

### Ar-Weighted Poincare-Type Inequalities for Differential Forms in Some Domains by Ding S.S., Gai Y.Y.

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