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I hope that Urs sees this, but anyone is welcome to answer, of course! I am wondering if we can recover Tate twists in, say, singular cohomology or Deligne cohomology as a special case of twisted cohomology? If so, could someone unwrap how?
We should presumably add to Tate twist first.
I am afraid that I don’t know about this. But let me check…
Yes, what I would like is to be able to say is that a Tate twist, from the ’n point of view’, is a special case of the definition at twisted cohomology, and then explain how to recover some classical definitions. For example, if $H$ at twisted cohomology is the $(\infty,1)$-topos of homotopy types, and if $A$ is the Eilenberg-MacLane space $K(\mathbb{C}, 1)$, do we recover the Tate twist of singular cohomology (i.e. the tensoring of it with copies of the abelian group $2 \pi i\mathbb{Z}$)? It seems likely.
(This was in reply to #2).
(And if so, I am interested in what happens in the motivic case. I.e. take $H$ to be the $\mathbb{A}^{1}$-localisation of the Nisnevich $(\infty,1)$-topos over some field $k$. This is no longer an $(\infty,1)$-topos, but let’s ignore that for the moment, as probably things can be made to work out anyway. Take $A$ to be the constant simplicial sheaf valued at $K(k, 1)$. Is ordinary cohomology in $H$ twisted by $A$ anything like motivic cohomology? I am not expecting anybody to answer this, but maybe somebody dropping by has some ideas.)
I second David in #2: My main problem with thinking about this right away is that I’d need to remind myself of ordinary Tate twisting. Maybe you could take the opportunity to sketch out the relevant background for your question in that stub entry?
Urs and David: I’ve now written quite a lengthy idea section at Tate twist. I originally intended to write something short, but ended up writing quite a lot. It is maybe a bit digressive, so feel free to edit as you see fit. But hopefully there’s something of use there.
With that in place, maybe you now see why I think one might be able to see Tate twists as twisted cohomology? In the latter, $Aut(A)$, or $GL_{1}$ in the stable case, plays a fundamental role, so there seems to be a great deal of formal similarity.
What excites me a bit about the story at twisted cohomology is that it is claimed that twisted cohomology can in fact be viewed as ordinary cohomology in a certain slice topos. When one takes $A$ to be an ordinary Eilenberg-MacLane space, this should mean that the resulting twisted cohomology theory vanishes in negative degree, regardless of what degree one is looking at coming from the twisting. Now, if one could see motivic cohomology as twisted cohomology in this way, this would be the Beilinson-Soulé conjecture…!
Thanks, Richard!! That helps.
I’ll think about it. But some other people here, for instance Adeel, are probably more into the relevant issues.
How broad is the definition of Tate twist? Here, even the shift in grading in the category of graded $\mathbb{k}$-vector spaces counts.
David: I believe that this is actually a pretty much direct abstraction of the definition at Tate twist. The Tate twist can be viewed as a grading (and in motivic cohomology, it is the source of the additional grading that one gets there). I actually think that this is quite under-emphasised: I think that l-adic cohomology, say, really should be viewed as bi-graded. This is part of what I was getting at with drawing the distinction between the geometric and arithmetic aspects of a cohomology theory. And I also think that we lack a really good conceptual understanding of Tate twists; this is really the kind of thing I’m asking about here. As an aside, I once emailed Olivia Caramello to ask whether she had considered Tate twists in her logical approach to motives here, because I would consider it crucial that this ingredient is there; but I never got a reply.
To be specific, I think that this abstraction of Tate twist as a grading goes back to a very influential paper Koszul duality patterns in representation theory by Beilinson, GInzburg, and Soergel.
The ’half Tate twists’ that I was referring to at the end are used for instance in On Koszul duality for Kac-Moody groups by Bezrukavnikov and Yun.
I don’t think Tate-twisted cohomology can reasonably be viewed as a form of twisted cohomology, because Tate twists are defined over the point, while the point of twisted cohomology is that it involves local systems on the object whose cohomology you’re taking. Of course, twisted cohomology is a special case of cohomology and conversely, so the answer is “yes” but not in an interesting way. And in all cases “twists” are just invertible objects in the relevant symmetric monoidal category of coefficients.
In fact, for cohomology theories that are not oriented, like Hermitian K-theory, there are twisted Tate twists: every vector bundle gives rise to a Tate twist, the usual one corresponding to the trivial line bundle.
take $H$ to be the $\mathbb{A}^{1}$-localisation of the Nisnevich $(\infty,1)$-topos over some field $k$. This is no longer an $(\infty,1)$-topos, but let’s ignore that for the moment, as probably things can be made to work out anyway. Take $A$ to be the constant simplicial sheaf valued at $K(k, 1)$. Is ordinary cohomology in $H$ twisted by $A$ anything like motivic cohomology?
No, in fact any map $X\to A$ from a smooth $k$-scheme is null-homotopic because there is no higher Nisnevich cohomology.
Thanks very much for your thoughts!
while the point of twisted cohomology is that it involves local systems on the object whose cohomology you’re taking.
Yes, so the question really is: what do $k^{\times}$-local systems look in the étale topos, where $k^{\times}$ is the multiplicative group? I.e. what do morphisms $X \rightarrow k^{\times}$ look like? (I’m aware that this does not quite parse, but I’m sure you’ll see what I mean). A few things in the literature lead me to think that such things are quite close to, maybe even exactly the same as, Tate twisted local systems, which is what we need.
Indeed, even if there is no direct relationship, I feel there must be some correlation, because the two stories are extraordinarily similar. Both involve Poincaré duality, both involve the multiplicative group of something, … all in similar ways.
No, in fact any map $X\to A$ from a smooth $k$-scheme is null-homotopic because there is no higher Nisnevich cohomology.
By $A$ here, I guess you meant the specific $A$ I defined (constant simplicial sheaf valued at $K(k,1)$)? But it is not that gadget which we are looking at morphisms into, it is rather $k^{\times}$ (again, up to the fact that this does not quite parse).
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