By I. Fesenko, M. Kurihara

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174, Amer. Math. , Providence, RI, 1996. [I3] O. ps [K1] K. Kato, Galois cohomology of complete discrete valuation fields, In Algebraic Lect. Notes in Math. 967, Springer-Verlag, Berlin, 1982, 215–238. [K2] K. Kato, Symmetric bilinear forms, quadratic forms and Milnor istic two, Invent. Math. 66(1982), 493–510. group of K2 (F ), preprint, K -theory, K -theory in character- [MS1] A. S. Merkur’ev and A. A. Suslin, K -cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad.

Then, k X i=1 = = (xai k X i=1 k X i=1 (xai (xai ai ) ; Xl i=1 (xbi xai ) ; i=1 x = y + z for some y , z 2 A bi ) xai ; xai x) ; Xl bi ) 2 J (xbi Xl i=1 (xbi xbi ; xbi x) xbi ) 2 J: Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields , so we Part I. Appendix to Section 2. 33 Thus, J is an A -module, and M = (A A )=J is also an A -module. In order to show the bijectivity of M ! Ω1A , we construct the inverse map Ω1A ! M . By definition of the differential module (see the property after the definition), it is enough to check that the map ': A ;!

2. ΩnB=A = Then, d naturally defines an complex ^ B is an k X i=1 ai = A -algebra. Xl i=1 bi . For a positive Ω1B=A : +1 , A -homomorphism d: ΩnB=A ! ΩnB=A and we have a ;1 ;! ΩnB=A ;! ΩnB=A +1 ::: ;! ΩnB=A ;! ::: which we call the de Rham complex. Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields 34 M. Kurihara and I. Fesenko For a commutative ring A , which we regard as a Z -module, we simply write Ω nA n for ΩnA=Z. 1. Therefore we obtain V Lemma. If A is a local ring, we have a surjective homomorphism A (A ) n ;!