Journal of Pure and Applied Algebra 18 (1980) 233-252 @ North-Holland Publishing Company SOME EXACT SEQUENCES IN THE THEORY OF HERMITIAN FORMS Gunnar CARLSSON zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC Unicersity of Chicago, Chicago, IL 60637, USA R. James MILGRAM* Stanford University, Stanford, CA 94305, USA Communicated by P.J. Freyd Revised version received 24 September 1979 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML Introduction In this note we construct exact sequences valid for Witt groups over a fairly large class of rings A, which specialize to give a long exact sequence for the surgery Wall groups, and the classical sequence o-, W(A)+ W(A 0 Q)+ Ll W(A/p)-+ZO2/2-+0 p prime where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA W denotes Witt ring, .I a Dedekind domain, and Z its ideal class group. W. Pardon has also obtained this sequence, at least in the case of surgery Wall groups, originally by geometric considerations and at about the same time as we did, also algebraically, M. Karoubi has also obtained such a sequence. Our sequence is constructed for Grothendieck groups of Hermitian forms rather than quadratic forms, as in Pardon’s or Karoubi’s work, although the same tech- niques yield the sequence for quadratic forms as well. An indication of the details is given in the appendix. We also point out that we do not require that the localized ring should be Artinian, as in Pardon’s work. This permits, e.g., localizations of the form 2~ --, &,)~, which is not Artinian. This exact sequence provides a computational tool of considerable power for studying these Witt and Wall groups, though we only make partial remarks on the structure of the various groups here. In particular, in view of Appendix 4 of [l], it provides a generalization of the classical theory of quadratic forms and gives the proper basis for generalizations of Gauss-Hilbert-Weyl reciprocity. A crucial step is due to C.T.C. Wall in his basic paper [6] for the case A = Z. We are also grateful to A. Ranicki who pointed out several errors and omissions in a preliminary version of this paper. * This research wassupported in part by the National Science Foundation grant GP MPS 7407491AOl. 233