Suppose I have a (homotopy) fiber sequence of spaces $\Omega A \to B \to C$, can I always rotate this to the right once ? i.e., can I extend it to a fiber sequence $\Omega A \to B \to C \to A$ ?
Oh, I'm dumb, I guess yes iff the map on the left $\Omega C \to \Omega A$ is a loop map, is that right ? Is there an "easy" way to check that this map is indeed a loop map, given just the initial fiber sequence $\Omega A \to B \to C$ ?
To everyone: suppose I have a cosimplicial $E_\infty$-algebra and I understand it completely. I want to take the limit and compute its homotopy groups: those are given by the standard Bousfield-Kan spectral sequence, so no problem there. Now I want to understand the action of the Dyer-Lashof algebra over those (say that my E_\infty-algebra is over F_2). What can I do? That is, does anyone have a nice example where this stuff is worked out?
Thanks! I'll try to have a look at that, though it talks about spaces rather than E_\infty-rings. However I'm more looking for examples of computations rather than theoretical framework
ah right yeah, I actually just got curious by your question cuz I'm also using the BK spectral sequence so I googled a few minutes. Do the differentials respect the action by Dyer-Lashof algebra ?
@DenisNardin I was going to suggest Phils stuff as well.
The Hinfty book has computations but not of the kind you seem to be asking about. I would try using Phil's thesis. What he does is work out how the operations and the differentials interact. I think this shouldn't be so special as to not work for $E_{\infty}$ ring spectra. They are algebras over the same monad. I guess you might have to mess around a bit with the universal example stuff, but really it should all be fine if you stick to working with ordinary homology.
Oh, he has another paper where he looks at the infinite loop space case, which is closer to what you want.
Uh no sorry. I haven't had time to work on it these days (stuff has come up), but I promise I want to see this through. I think that the multicategory I was describing may exists by chance in the general case as the multiassociativity is trickier than I expected but the proof in Saul's paper still looks good to me, so there must be some kind of trick
This may be a long shot, but do ppl know a reference for the following fact:
Let $f : X \to Y$ be a morphism of schemes, locally of finite presentation, finite and flat. Then the locus $\{y \in Y: X_y\to y is \'{e}tale\}$ is open in $Y$.
I looked at Milne's etale cohomology property 2.14 but that proposition is about an open set in $X$.
Ok, that explains how your previous life is affecting your current life :D
@DenisNardin Actually we don't even need loc. finite presentation
All I used was the fact that the morphism is finite, which makes sense given that the relevant exercise in Olsson doesn't assume loc. finite presentation
Then we are looking at a ring map $A \to B$ that is finite flat
and then $\Omega^1_{B/A}$ is finite as an $A$-module. Its support as an $A$-module is closed, and we now just take the complement of that in Spec A @Adeel
anyway, I think in the derived setting, where the cotangent complex is not concentrated in degree zero, you have to use this Thomason-style result, or the analogue of it. at least that's the only proof i know
Staying in the topic of perfect complexes, does anyone knows when the restriction of a perfect complex to a closed subscheme (say an effective Cartier divisor) is perfect?
@BenLim since you are working with M_g you don't need to bother about non-noetherian things. It's a general fact that, for X locally of finite type over some base S, the R-points of X, for arbitrary R, are determined by the A-points of X, for A of finite type over S.
@AndrewSenger for what? I think if he wants to prove something for M_g over Z it doesn't matter as the integers are noetherian. (but I think you are right, I should be saying finite presentation, I think I was confusing myself with the case of modules)
@bananastack For the general statement of the result that you quoted above. And I'm not sure it make sense to show that the diagonal of M_g is representable using the fact that M_g is locally of finite type over Z: how do you make sense of local finite type-ness of a stack without already knowing that its diagonal is representable?
Instead, one should reduce to the Noetherian case using the finite presentedness of genus g curves. (I guess that one could define locally of finite presentation for stacks as a 2-version of the definition for contravariant functors on the category of schemes and declare that a stack over a Noetherian base is locally of finite type iff it is locally of finite presentation, but this seems backwards...)
@DenisNardin Well, what is the $E_{\infty}$ complex orientation of $ku$ then? It isn't one that kills all the higher $x_i$'s, that would violate Johnson-Noel.
@Sean What complex orientation of ku are you talking about? The one induced by the complex orientation of KU?
If that's the one it is given by the multiplicative formal group law, so in principle you should be able to explicitely compute the image of all x_i's (but it may be a bit painful)
@DenisNardin There must be multiple orientations that induce the multiplicative formal group. I am curious about the one that is $E_{\infty}$. Maybe this is in something of @TylerLawson's
okay. well, certainly Ab-enrichment is enough. if m is the corresponding idempotent, then m and 1 - m respectively are the projections onto the direct summand and its complement
If A-->B-->A is your retraction, does saying the A splits off imply that the other summand is fib(B--->A)? Is this property equivalent to the existent of fibers for the morphisms of interest?
the main reason i think commutative monoids is a counterexample is that i bet you can find lots of commutative monoids with idempotent endomorphisms that don't have complements
like endomorphism semirings of commutative monoids should be as badly behaved as arbitrary semirings
e.g. in the above example, the projection {0, 1, 2} -> {0, 1} (given by rounding down) has fiber {0}
anyway, other than not being Ab-enriched I bet CMon has a lot of lovely properties as a category (e.g. it's certainly complete and cocomplete) so I don't think completeness / cocompleteness-type conditions are the way to go