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 ...
Denis Nardin's user avatar
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17 votes
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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 ...
D.-C. Cisinski's user avatar
16 votes
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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 ...
D.-C. Cisinski's user avatar
14 votes
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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 ...
Denis Nardin's user avatar
  • 16.2k
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My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF.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 ...
Marc Hoyois's user avatar
  • 8,672
13 votes
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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 ...
Donu Arapura's user avatar
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13 votes
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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 ...
Matthias Wendt's user avatar
13 votes
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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-...
Tintin's user avatar
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12 votes
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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 ...
AAK's user avatar
  • 5,841
11 votes
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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 $\...
Denis Nardin's user avatar
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10 votes
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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. ...
Matthias Wendt's user avatar
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 ...
Martin Frankland's user avatar
10 votes
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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{...
R. van Dobben de Bruyn's user avatar
8 votes
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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, ...
Matthias Wendt's user avatar
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 ...
Marc Hoyois's user avatar
  • 8,672
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 ...
Donu Arapura's user avatar
  • 34.2k
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,...
Dan Grayson's user avatar
  • 1,403
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 ...
Eoin's user avatar
  • 342
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 ...
Jesse Silliman's user avatar
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 ...
Matthias Wendt's user avatar
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 ...
Laurent Moret-Bailly's user avatar
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 $K^0(\mathbb{A}^1)=\mathbb{Z}$.
Mohan's user avatar
  • 6,117
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 ...
Elden Elmanto's user avatar
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 $A \stackrel f\twoheadrightarrow B \stackrel g \to C$ ...
R. van Dobben de Bruyn's user avatar
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}...
Marc Hoyois's user avatar
  • 8,672
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}(...
Tintin's user avatar
  • 2,741
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 (...
Lennart Meier's user avatar
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 ...
Dan Petersen's user avatar
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