This has probably been asked a bunch, but I'm having trouble making clear what is the right derived setting for localization. In particular, Lurie defines in his thesis infinity ringed topoi and uses a universal property to characterize the ones that are `affine'.

Toen and many others have a natural definition of derived scheme as a locally ringed space (|X|, O_X) equipped with a sheaf of simplicial commutative rings, where the underlying classical sheaf of rings defines a scheme structure and pi_n O_X are quasi-coherent pi_0 O_X-modules

Now, one is motivated to ask for a simplicial commutative ring R_, how to make sense of Spec R_ as an affine derived scheme

in particular, | Spec R_* | = | Spec pi_0 R_*| but how does one define the sheaf? This comes down to handling localization.....which seems dicey in the (infinity) category of simplicial commutative rings

I've seen in Higher algebra, that one can generalize the Ore localization if one is willing to work with E_infty ring spectra, but if one is working specifically in the category of simplicial commutative rings, what is the right way to define a localization? For f in pi_0 R_* I see two definitions:

R_* otimes^L pi_0 R[x]/(xf -1) and hocolim( R_* -> R_* -> R_* -> ....)

are these the same up to weak equivalence?

@lemiller If $R$ is a simplical commutative ring, and $f\in R_0$ is an element in degree $0$, isn't $R[x]/(xf-1)$ isomorphic (in simplicial abelian groups) to the direct limit?

@Charles

@CharlesRezk, that is exactly my hope

I wanted a quick verification. But I'm also curious if this is the accepted right way to localize

and if, its enough to define the structure sheaf for Spec R_* as a derived affine scheme

It should be.

It's basically the motivation for how to handle it $\infty$-categorically.

also, one really does need the derived tensor product above to define R[x]/(xf -1) right? That is, if we work with the underived tensor product, then we might not get the right homotopy groups. I get the philosophy of why this motivates to consider the infinity categories as a good model (if we want sheaves targed in such) and I'm happy with generalizating to infinity topoi to accommodate, but one wants to know what about alg. geom. fails, or

how to say explicitly compute Spec R_* for given R_*

but thanks for the quick verification!!!

No, for simplicial commutative rings the point-set level construction works.

Because $R[f^{-1}]:= R[x]/(xf-1)$ is flat over $R$.

ah you mean there is a weak equivalence R_* otimes^L pi_0 R[x]/(xf -1) and R_* otimes pi_0 R[x]/(xf -1) since f has degree 0?

Acually, I don't understand what your tensor product means. I am literally talking about $R[x]/(xf-1)$, on the level of simplicial commutative rings. It is true that $\pi_*( R[x]/(xf-1) ) = (\pi_*R)[x]/(xf-1)$.

the tensor product was how I was handling polynomial adjunction, instead of doing it term wise. This is similar to how one defines derived completion (in the stacks project or in Bhatt/Scholze's prismatic paper, an A-module M is derived I-complete if M -> Rlim( M otimes^L_{Z[x]} Z[x]/x^n) is a weak equivalence where one uses restriction of scalars Z[x] -> A sending x to f

So what are you taking the tensor product over?

Z[x]

or above?

Above

pi_0 R_* is what I want to use

but maybe that is bogus

What's the map $\pi_0 R \to R$?

(hence the questions, and thanks in advance!)

You can do $R\otimes_{Z[x]} Z[x,y]/(xy-1)$.

ah yes, and that should also be giving the same thing correct?

It should give the correct thing.

how are you defining R_*[x]/(xf -1) termwise?

(sorry if that is basic)

It is exactly that in each degree, where $f\in R_n$ is just the unique image of $f\in R_0$ under degeneracy operators.

ah great

again, thanks a bunch!

👍

@CharlesRezk I finally checked what it was I'd convinced myself about Quillen's $S^{-1} X$ construction. What I believe is that the $\infty$-categorical analog of the construction $\langle S, X \rangle$ is simply to view the $S$-action on $X$ as a fibration over the 1-object $(\infty,2)$-category $BS$ corresponding to $S$ (if $S$ is a monoidal groupoid or $\infty$-groupoid, this is just an $(\infty,1)$-category) and then take the total space.

So by Gepner-Haugseng-Nikolaus, it's the oplax colimit of the corresponding functor $BS \to Cat_\infty$ picking out the action on $X$. Upon geometric realization, this become an $(\infty,1)$-categorical colimit, and then you're in business to get a universal property of $S^{-1} S$. But if you want to stay at the level of $\infty$-categories without geometrically realizing, then it doesn't seem to be the right construction for what you wanted the other day.

So sorry, I guess it was a bit of a red herring

Thanks. Maybe it's not what I want, but I'm glad to learn it, because I learned something interesting from it. Namely that the Stiefel category is an example of an $\langle S,S\rangle$.