Lecture Notes On Motivic Cohomology

By C. Mazza, Vladimir Voevodsky, Charles A. Weibel. 2006

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A DG Guide to Voevodsky’s Motives (2006, background): “eﬀ −−→M tri →∼ DPSh tr/I ∆Zar ←∼ (I∆Zar)⊥, (2.3.4) Deﬀ M tri →∼ (DPSh tr/I ∆Zar tr ) perf. 16At the moment, we do not care about the group structure on Gm. Our notation is not consistent with that of [MVW] and [Vo2]: in loc. cit., Gm denotes the pointed variety (A1r{0},1). 21 The right I∆Zar tr-loca”
Galois equivariance and stable motivic homotopy theory (2014, background): “We then reinterpret the Beilinson-Lichtenbaum conjectures and establish an equivariant version of Suslin-Voevodsky’s theorem [46] on Suslin homology.”
The norm residue isomorphism theorem (2009, methodology): “We will be also be interested in the Lefschetz motives LR = Rtr(A /A − 0), which represent integral motivic cohomology in the triangulated categoryDM(R) of effective motives or its full subcategoryDM (see [23], [24] or [11].”
The generalized slices of Hermitian K‐theory (2016, methodology): “…one we have outlined above: Voevodsky has deﬁned motivic cohomology via a category DM ( k ) which is reasonably computable [ 11 ], and in fact motivic cohomology in this sense coincides with motivic cohomology in the sense of higher Chow groups [ 11 , Theorem 19.1] which is certainly very explicit.”
Support theory via actions of tensor triangulated categories (2011, background): “Their utility has been demonstrated in algebraic geometry, motivic theory, homotopy theory, modular representation theory, and noncommutative geometry: the theory of Grothendieck duality ([24], [27], [33], [35], [37]), Voevodsky’s motivic category ([32], [2]), Devinatz, Hopkins, and Smith’s work on ”
Milnor -theory of local rings with finite residue fields (2010, background): “Furthermore let Hm mot(Spec(A),Z(n)) be the hypercohomology of the complex of Zariski sheaves Z(n) constructed in [11], see also the remark following the corollary.”