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I think I can help locating the statement in the Bass-Milnor-Serre paper. First, if I understand your question correctly, the translation between your notation and the one of Bass-Milnor-Serre is $\Ga …

Here are some relevant references. First, three papers on different methods to compute the K-theory of compact Lie groups with finite cyclic fundamental group. (The first one also discusses the projec …

The Atiyah-Hirzebruch spectral sequence does have a multiplicative structure, and I think this can be used to determine the multiplication on K-theory. From the paper of Braun, it follows that the spe …

The relation between motivic cohomology and cohomology of the Milnor K-theory sheaf is discussed in Motivic cohomology and cohomology of Milnor K-theory sheaf There is a natural comparison morphism …

The category of flat vector bundles is equivalent to the category of local systems, see for instance this MO-question, which in turn are equivalent to representations of the fundamental group, see thi …

Yes, the natural multiplication morphisms induce a morphism of Gersten complexes from Milnor to Quillen K-theory. The basic points are made in the paper M. Rost. "Chow groups with coefficients", Do …

The previous answer contained a major error/misconception, and I apologize for the dealy in correcting it. The answer to the question is "yes" locally in the Zariski topology but "no" globally. Comp …

We have a short exact sequence of sheaves of abelian groups (see e.g. Morel's book on $\mathbb{A}^1$-algebraic topology): $$ 0\to\mathbf{I}^{n+1}\to\mathbf{K}^{\rm MW}_n\to \mathbf{K}^{\rm M}_n\to 0, …

I think the answer to this question is not known. All we can say about the K-theory of $\mathbb{C}$ concerns the torsion. The trouble starts with $K_1(\mathbb{C})\cong\mathbb{C}^\times$, which is pre …

I'll draw the connection in the case of where the special fiber $\mathcal{C}_p$ is a triangle of three crossing copies of $\mathbb{P}^1$. Let $Z$ be the closed subscheme of $\mathcal{C}_p$ consisting …

Concerning the definition of symplectic K-theory: there are various possible definitions, the homotopy groups of the plus-construction of the classifying space of the infinite symplectic group is one …

Examples of smooth real varieties whose Witt group has odd torsion can be found in a paper of Jacobson: J.A. Jacobson. From the global signature to higher signatures. arXiv:1411.0993, https://arxiv …

The degrees of the attaching maps (and hence the integral chain complex) for the real Grassmannians have been determined in L. Casian and Y. Kodama. On the cohomology of real Grassmann manifolds. a …

Let me flesh out my comment, and give some more details. I will denote $C_p$ the cyclic group of order $p$. First, the structure of the group ring $\mathbb{Z}[C_p]$ is a little more complicated. If …

The definition of the class is actually given in the cited sentence. For the relevant motivic cohomology group we have $${\rm H}^{0,1}({\rm Spec} k,\mathbb{Z}/p\mathbb{Z})\cong {\rm H}^0_{\rm ét}({\rm …