@JosephVictor I am somewhat familiar

Let $X$ be a qcqs scheme. Which limits in D(X) (the derived ∞-category of X) can be computed on affine patches? It can't be all of them (because restriction to an open does not have a left adjoint), but if, e.g., fixed points for a finite group were good I'd be very happy

Hi, I have a question about higher algebra Definition 1.1.2.11: why we have to define in this way using W instead of just requiring the existence of pushouts X\to Y\to Z\to X[1]? Isn't this equivalent or am I confused? (I asked this in Math StackExchange math.stackexchange.com/questions/3262566/… but got no answer for a week...)

@NarukiMasuda I'm pretty sure it is equivalent. I don't know why Lurie defines it that way

@DenisNardin Thanks, that makes my mind peaceful (I was anxious if I was misunderstanding something fundamental).

@Naruki Actually there's a little subtlety though: you need to impose that the external square is the canonical one, i.e that the endomorphism of \Sigma X us the identity

I

@DenisNardin stability is going to give you finite pullbacks and products, since they agree with the corresponding colimits (by stability) and restriction does have a right adjoint. for the finite group case, I think the statement is equivalent to base change for square which is the product of the inclusion U into X and the map BG -> pt. I'd expect some trouble for this approach if the order of G isn't invertible, but I don't actually know.

@pupshaw If 2 is invertible, the statement is automatically true, since the fixed points coincide with the orbits, but unfortunately I am especially interested in characteristic 2 and mixed char

I'm sure you've already thought of this but in the affine case at least, restricting to a standard open (inverting an element) seems to go alright by computing the localization as a a telescope. but that doesn't get you all the way there either.