process

428 C. WeibelGiven (0.4) and axiom 0.3 (b), triangle (0.5) is equivalent to the duality assertionthat D ˝ L d b Š D; (0.5) is isomorphic to L d times the dual of triangle (0.4).Remark 0.6. In the language of Chow motives, any direct summand M D .X; e/ ofX has a dual motive M D .X; e t ;d/and a transpose summand M 0 D .X; e t / of X.Thus the summand M 0 D M ˝ L d of X is a priori different from M . Axiom 0.3 (b)says that M 0 ! X ! M is an isomorphism. (The differences between these languagesis partly due to the fact that the category of Chow motives embeds contravariantly intoDM eff .)Although we only use X-duals fleetingly in Lemma 3.5 below, they fit into a largercontext, which we shall now quickly describe. For this, we need to assume that kadmits resolution of singularities, so that A exists by [2, 20.3].DM X and X-duals 0.7. Let DM X denote the full subcategory of DM consisting ofobjects M isomorphic to A ˝ X for a geometric motive A over k. For this, we assumethat k admits resolution of singularities, so that A exists by [2, 20.3]. As pointed outin [11, 6.11], Hom.M; B/ Š Hom.M; B ˝ X/ for all B in DM eff . ThusHom.U ˝X;A ˝X/ D Hom.U ˝X;A / D Hom.U ˝X˝A; Z/ D Hom.U ˝M; X/:That is, Hom X .M; X/ D A ˝ X is an internal Hom object from M to X in the tensorcategory DM X . As such, Hom X .M; X/ is unique up to canonical isomorphism (cf. [11,8.1]). Any map M 1 ! M 2 induces a map Hom X .M 2 ; X/ ! Hom X .M 1 ; X/ viaHom.A 1 ˝ X;A 2 ˝ X/ Š Hom.A 1 ˝ X ˝ A 2 ; X/ Š Hom.A 2 ˝ X;A 1 ˝ X/:It is easy to check that Hom X . ; X/ is a contravariant functor, and that it is a dualityin the sense that Hom X .Hom X .M; X/; X/ Š M .In this paper we consider the X-dual DM. / D Hom X . ; X/ ˝ L d , which alsosatisfies D 2 .M / Š M . Our choice of the twist is such that D.X/ Š X, D.M / is thetranspose M 0 of Remark 0.6, and Dy is given by (0.2).This paper is an attempt to clarify the ending of the Voevodsky–Rost programto prove that the norm residue map is an isomorphism, and specifically the proof aspresented in Voevodsky’s 2003 preprint [8]. One important feature of our Theorem 0.1is that only published results (and [3]) are used.When ` D 2, triangle (0.4) was constructed in [10, 4.4] using D D X, and (0.5) isits X-dual. Axioms 0.3 (a)–(b) hold by a result of Rost (cited as [10, 4.3]).When `>2, there are two different programs for constructing M . Both start byusing the norm residue symbol of a to construct a nonzero element of H 2bC1;b .X/.The construction of is given in [10, p. 95] and [8, (5.2)] (see 2.5 below).In Voevodsky’s approach, the triangles (0.4) and (0.5) are constructed in (5.5) and(5.6) of [8] with M D M` 1 and D D M` 2 , starting with 2 H 2bC1;b .X/. Theduality axiom 0.3 (b) for M and D is given by [8, 5.7].Unfortunately, there is a gap in the proof of Lemma [8, 5.15], which asserts thatAxiom 0.3 (a) holds, i. e., that M` 1 is a summand of X. That proof depends via