19
votes
Why presheaves with transfer?
I hope someone can provide a better answer with more of an eye towards the motivic world, but for now let me outline that the exact same phenomenon exists in classical stable homotopy theory. Here the ...
17
votes
Accepted
What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?
There is no need a priori to define these categories of motives starting from correspondences. The stable homotopy theory of schemes $SH$ may be characterized by a universal property saying that it is ...
16
votes
Accepted
Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?
To my knowledge, one can only make this analogy fully consistent with Weibel's homotopy invariant $K$-theory $KH$ and $G$-theory (although the proofs of what I claim below rely heavily on our ...
14
votes
Accepted
How to think about infinite generatedness of motivic cohomology
While waiting for someone more competent than me to answer, let me turn the question right back to you. Why should motivic cohomology be finitely generated?
The answer is, of course, that there's no ...
14
votes
Who proved the motivic 6-functor formalism?
My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF. This result ...
13
votes
Accepted
Intuition behind the definition of finite correspondences
Traditionally correspondences were defined simply as cycles on the product, but then you need a moving lemma just to define composition. This limits you to working on smooth varieties. The beauty of ...
13
votes
Accepted
Which motivic cohomology groups of complex numbers are non-torsion?
This is going to be a slightly extended explanation, I apologize. The short version is basically that little is actually known (and even that is hard to prove), but conjecturally everything permitted ...
13
votes
Accepted
Motivic vs Deligne cohomology
The existence of a cycle class map from motivic cohomology is a general fact to every cohomology satisfying certain axioms. For example, to every mixed Weil theory in the terminology of Cisinski-...
12
votes
Accepted
A question on Voevodsky´s categories
One could say that the story begins with Beilinson's conjectures on the existence of a theory of motivic cohomology. In accordance with the insights of the Grothendieck school that cohomology ...
11
votes
Accepted
How to think about $\mathbf{Z}(n)_{\mathcal{M}}$
[All cohomology will be reduced cohomology for ease of notation].
There is no analog for classical homotopy theory. This is related to the fact that the Picard group of the category of spectra is $\...
10
votes
Accepted
Motivic cohomology and cohomology of Milnor K-theory sheaf
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.
...
10
votes
A question about the (motivic) integral cohomology of the Eilenberg-MacLane spectrum
For the topological question, this thread and this thread have interesting answers and references about integral (co)homology of Eilenberg-MacLane spaces and spectra. Let me add a few comments.
I'll ...
10
votes
Accepted
On the swapping map of $\mathbb{G}_m$
Recall that $S_t^1$ is the reduced motive $\tilde M(\mathbf G_m)$ of $\mathbf G_m$, obtained most explicitly as the kernel of the projector $M(\mathbf G_m) \to M(\mathbf G_m)$ given by $\operatorname{...
8
votes
Accepted
What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory
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,
...
8
votes
Accepted
Two motivic complexes, compared
They are not quasi-isomorphic: your $\check{\mathbf Z}(n)$ is concentrated in a single degree. As Denis points out in the comments, your definition of $\check{\mathbf Z}(n)$ is wrong because you ...
7
votes
Motivic vs Deligne cohomology
In addition to the papers mentioned, I would like to add Kerr, Lewis, Müller-Stach, The Abel–Jacobi map for higher Chow groups, Compositio (2006). They give an explicit construction (if you like that ...
7
votes
Idempotent completions in K-theory
Thomason discusses the construction in (A.9.1) in his paper with Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88,...
6
votes
the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme
This does not hold in general.
An explicit counterexample is given in:
Karpenko, Nikita A. Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (1998), no. 4, 305–330. (Reviewer: Jean-Pierre ...
6
votes
Motivic cohomology is universal with respect to what (co)homology theories?
I think a proof that motivic Borel-Moore homology is a universal Borel-Moore homology theory could be deduced from work of Bloch, Geisser-Levine, and Voevodsky, at least for varieties over fields. I ...
6
votes
Accepted
Is $B\mathbb{G}_m$ strongly $A^1$-invariant?
The point of the notion is that for a strongly $\mathbb{A}^1$-invariant sheaf of groups $G$ the classifying space $BG$ is $\mathbb{A}^1$-local. The characterization in terms of $H^0$ and $H^1$ is ...
6
votes
Accepted
Surjective étale morphisms étale locally split
We can work locally on $X$ and even (by standard limit arguments) assume that $X=\mathrm{Spec}(R)$ where $R$ is local and strictly henselian. Then $Y=\coprod_{i=1}^{n}Y_i$ where each $Y_i$ is local ...
6
votes
Grothendieck group and faithfully flat morpshim
It is not. For example, take a surjective morphism $f:\mathbb{A}^1\to\mathbb{P}^1$. $K^0(\mathbb{P}^1)=\mathbb{Z}\oplus\mathbb{Z}$, while $K^0(\mathbb{A}^1)=\mathbb{Z}$.
5
votes
Spectral sequences in $K$-theory
Yes. Firstly, I apologize for the self-referencing. But the references for this is a joint paper with Marc Levine (basically finishing up what he had sketched in aforementioned paper), Markus ...
5
votes
Around algebraic equivalence of cycles
A summary of some of these results is given in §19.3 of Fulton's Intersection theory [Ful98]. To address for example the questions you ask:
If $A \stackrel f\twoheadrightarrow B \stackrel g \to C$ ...
5
votes
A question about the vanishing of motivic cohomology in negative Tate twist
Here's an easy way to see this (which is more or less an elaboration of Mikhail's answer):
Suppose $i > 0$ and $n\in \mathbb Z$. We have the Thom isomorphism
$$
H^n(X, \mathbb Z(-i)) \cong H^{n+2i}...
5
votes
Representable cohomology theories in motivic homotopy theory
Recall that $\mathrm{DM}(k)$ can be described as the subcategory of $\mathrm{SH}(k)$ made of modules over the motivic cohomology. This implies that cohomologies which are representable in $\mathrm{DM}(...
5
votes
Accepted
Etale $K$ theory coincides with algebraic one in high enough degrees
To my knowledge the most general known statement has been proven by Clausen and Mathew in their paper Hyperdescent and étale K-theory as Theorem 1.2. The precise conditions on your commutative ring (...
5
votes
Support of torsion in the Borel–Moore homology
The first question at least can be done as follows. The restriction of $\alpha$ to $X^{\mathrm{sm}}$ vanishes away from a divisor $D$ by the result quoted in your second paragraph and the Poincaré ...
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