A pointed topological space is a pair [math]\left( {X,x} \right)[/math] consisting of a topological space [math]X[/math] and a choice of basepoint [math]x \in X[/math].
A morphism of pointed spaces is a continuous map preserving basepoints.
If [math]\operatorname{Top} [/math] is the usual category of topological spaces, then the category of pointed spaces (denoted [math]{\operatorname{Top} _*}[/math]) can be constructed as the undercategory [math]{}^{*/}\operatorname{Top} [/math] of the single-point topological space.
The category of pointed spaces has products and coproducts as follows...
The product is the cartesian product given in the "obvious" way.
The coproduct is the wedge sum [math]X \vee Y[/math] given as a pushout [math]\begin{array}{*{20}{c}} *& \to &X \\ \downarrow &{}& \downarrow \\ Y& \to &{X \vee Y} \end{array}[/math]
There is also a functor [math] \wedge :{\operatorname{Top} _*} \times {\operatorname{Top} _*} \to {\operatorname{Top} _*}[/math] called the smash product.
It is given as a pushout [math]\begin{array}{*{20}{c}} {X\coprod Y}& \to &{X \times Y} \\ \downarrow &{}& \downarrow \\ *& \to &{X \wedge Y} \end{array}[/math]
If we restrict to a "nice" category of topological spaces, the smash product will provide a symmetric monoidal structure.
Their is a pair of functors [math]\Sigma :{\operatorname{Top} _*} \rightleftarrows {\operatorname{Top} _*}:\Omega [/math] inducing an adjunction on homotopy classes of morphisms.
For simplicity, we define the reduced suspension functor by [math]\Sigma X = X \wedge {S^1}[/math].
The loop space functor is defined as basepoint preserving morphisms [math]\Omega X = {\operatorname{Hom} _{{{\operatorname{Top} }_*}}}\left( {{S^1},X} \right)[/math] for some choice of basepoints.
A cohomology theory [math]{E^*}[/math] is a collection of functors [math]{\left\{ {{E^n}:{\mathcal{C}^{op}} \to \operatorname{Ab} } \right\}_{n \in \mathbb{Z}}}[/math] , from the category [math]\mathcal{C}[/math] consisting of pairs of topological spaces [math]\left( {X,Y} \right)[/math] s.t. [math]Y \subset X[/math] , along with natural transformations [math]{\delta ^n}:{E^n}\left( {Y,Z} \right) \to {E^{n + 1}}\left( {X,Y} \right)[/math] functorial in triples [math]Z \subset Y \subset X[/math] . Required to satisfy the follow axioms...
(i) Weak equivalences of topological spaces (i.e. continuous maps inducing isomorphisms on all homotopy groups) induce isomorphisms on cohomology.
(ii)The cohomology of a disjoint union is isomorphic to the product of the cohomology of each element of the union
(iii) Each triple [math]Z \subset Y \subset X[/math] induces a long exact sequence [math]... \to {E^n}\left( {X,Y} \right) \to {E^n}\left( {X,Z} \right) \to {E^n}\left( {Y,Z} \right)\mathop \to \limits^{{\delta ^n}} {E^{n + 1}}\left( {X,Y} \right) \to ...[/math]
(iv) If [math]Y \subset X[/math] and [math]\mathcal{U}[/math] open subset with closure contained in the interior of [math]Y[/math] , then the induced maps [math]{E^n}\left( {X,Y} \right) \to {E^n}\left( {X\backslash \mathcal{U},Y\backslash \mathcal{U}} \right)[/math] are isomorphisms. We denote [math]{E^n}\left( X \right) = {E^n}\left( {X,\emptyset } \right)[/math].
A motivating example is singular cohomology [math]H{A^*}[/math] associated to an abelian group [math]A[/math]. i.e. [math]H{A^n}\left( {X,Y} \right) = {H^n}\left( {X,Y;A} \right)[/math]
Singular cohomology of [math]A[/math] is the unique cohomology theory such that [math]H{A^n}\left( * \right) = \left\{ \begin{gathered} A,n = 0 \hfill \\ 0,otherwise \hfill \\ \end{gathered} \right.[/math]
Michael Taylor
We define an infinity category of pointed spaces, [math]{S_*}[/math], via infinity categorical localization of [math]{{{\operatorname{Top} }_*}}[/math] at weak homotopy equivalences.
The (stable) infinity category of Spectra, [math]\operatorname{Sp} [/math], is the defined as the limit (in the infinity category of infinity categories) of the following diagram [math]... \to {S_*}\mathop \to \limits^\Omega {S_*}\mathop \to \limits^\Omega {S_*} \to ...[/math] .
Explicitly, a spectrum [math]E \in \operatorname{Sp} [/math] is a sequence [math] {\left\{ {E\left( n \right)} \right\}_{n \in \mathbb{Z}}} [/math] of pointed spaces together with weak equivalences [math]E\left( n \right) \to \Omega E\left( {n + 1} \right)[/math] .
First examples consist of the suspension spectrum of an arbitrary space, [math]{\Sigma ^\infty }X = {\left\{ {{\Sigma ^n}X} \right\}_{n \in \mathbb{Z}}}[/math], and the sphere spectrum [math]\mathbb{S} = {\Sigma ^\infty }{S^0}[/math] .
The nth homotopy group of a spectrum is given by [math]{\pi _n}\left( E \right) = {\operatorname{colim} _k}{\pi _{n + k}}\left( {E\left( k \right)} \right)[/math].
The homotopy category of the infinity category of spectra, [math]\operatorname{Ho} \left( {\operatorname{Sp} } \right)[/math] , is a triangulated category with shift functor given by [math]E\left[ n \right]\left( j \right) = E\left( {j - n} \right)[/math] .
The Brown Representability Theorem states the spectra and cohomology theories are essentially the same thing. The spectrum --> cohomology theory direction given explicitly by [math]{E^n}\left( X \right) \equiv {\operatorname{Hom} _{\operatorname{Ho} \left( {\operatorname{Sp} } \right)}}\left( {{\Sigma ^\infty }X,E\left[ n \right]} \right)[/math] .
As a first example, note that Eilenberg-MacLane spaces [math]{K\left( {A,n} \right)}[/math] give a spectrum via the natural isomorphisms [math]{\pi _n}\left( {K\left( {A,n} \right)} \right) = A = {\pi _{n + 1}}\left( {K\left( {A,n + 1} \right)} \right) = {\pi _n}\left( {\Omega K\left( {A,n + 1} \right)} \right)[/math] .
The cohomology theory associated to the spectrum [math]{\left\{ {K\left( {A,n} \right)} \right\}_{n \in \mathbb{Z}}}[/math] is singular cohomology [math]H{A^*}[/math] .
Dominic Jones
The infinity category of spectra has a (unique) symmetric monoidal structure give by...
The smash product [math] \wedge :\operatorname{Sp} \times \operatorname{Sp} \to \operatorname{Sp} [/math] as the monoidal product preserving colimits in each variable.
And the sphere spectrum [math]\mathbb{S}[/math] as the unit.
Want to define the notion of ring spectra, i.e. [math]{\mathbb{E}_\infty }[/math] rings.
An [math]{\mathbb{E}_\infty }[/math] ring can be defined a commutative monoid object in the infinity category [math]\operatorname{Sp} [/math], the same way a commutative ring can be defined as commutative monoid object in [math]\operatorname{Ab} [/math] .
This means it is commutative up to higher coherent homotopy.
An occasionally more efficient definition is as symmetric monoidal functor [math]A:\mathbb{F} \to \operatorname{Sp} [/math] from the category [math]\mathbb{F}[/math] of finite sets with monoidal structure given via disjoint unit.
That way we can define the infinity category of [math]{\mathbb{E}_\infty }[/math] rings, denoted [math]\operatorname{Rg} \operatorname{Sp} [/math] , as the sub-infinity category of [math]Fu{n^\infty }\left( {\mathbb{F},\operatorname{Sp} } \right)[/math] consisting of symmetric monoidal functors.
A first example would be the Eilenburg-Maclane spectrum associated to a usual commutative ring.
Another example would be the sphere spectrum.
A third example would the spectrum associated to cohomology theory, Complex K-theory.
Benjamin White
where are you going with this m8?
Brandon Sanders
Idk. It is a Saturday and I'm reviewing recent notes while practicing my latex. Maybe Veeky Forums can learn something.
Jonathan Allen
Then you'd have to start with a problem you want to solve with those tools
Owen Hill
We'll I'm building to Topological Algebraic Geometry.
Hudson Wilson
Let [math]A[/math] be an [math]{\mathbb{E}_\infty }[/math]-ring.
We define the category associated to [math]A[/math] as the category with [math]\operatorname{Ob} A = \bullet [/math] and [math]{\operatorname{Hom} _A}\left( { \bullet , \bullet } \right) = A[/math].
Using this we can define the infinity category of [math]A[/math]-Modules, [math]\operatorname{Mod} \left( A \right)[/math] , as the sub-infinity category of [math]Fu{n^\infty }\left( {A,\operatorname{Sp} } \right)[/math] consisting of [math]\operatorname{Sp}[/math] enriched functors.
This is analogous to the fact for a usual commutative ring R, an R-module is equivalent to additive functor for the category associated to R to the category of abelian groups.
This category can be given a symmetric monoidal structure [math]{ \wedge _A}:\operatorname{Mod} \left( A \right) \times \operatorname{Mod} \left( A \right) \to \operatorname{Mod} \left( A \right) [/math] induced by the symmetric monoidal structure on [math]\operatorname{Sp}[/math] and the [math]{\mathbb{E}_\infty }[/math] structure of [math]A[/math].
This is analogous to commutative algebra where you can define the tensor product of R-modules (considered as functors like above) via the tensor product of abelian groups. i.e. [math]M{ \otimes _R}N = \int_{}^R {M{ \otimes _\mathbb{Z}}N} [/math]
An [math]A[/math]-Module [math]N[/math] is discrete if all higher homotopy groups are 0.
[math]M[/math] is said to have Tor-amplitude n of [math]M{ \wedge _A}N[/math] are 0.
[math]M[/math] is pseudocoherent if it is the colimit of a simplicial [math]A[/math]-Module. It is coherent if the simplicial model is cohomologically bounded.
[math]M[/math] is perfect if it is coherent and has finite Tor amplitude.
[math]A[/math] is regular if every [math]A[/math]-Module is perfect.
Bentley Stewart
That last statement should be every coherent module is perfect.
Ian Reyes
Which year are you in? Postgrad? I'm in year two and you lost me 1 1/2 posts in
Isaac Edwards
How is anyone going to learn something from straight definitions? Is this supposed to be impressing someone or something?
Kayden Hill
To add: this is the style of mathematics education I absolutely can't stand. Just a bunch of unmotivated definitions for pages and pages interspersed with a few sparsely detailed proofs of even more unmotivated theorems. Seen it in so many upper-level textbooks it has put me off a lot of subjects.
John Morales
Motivation for what part? Pointed spaces should be motivated if you have ever seen even a little algebraic topology.
Generalized cohomology theories can be motivated by specific examples like singular cohomology and complex k-theory you will see in a typical intro algebraic topology course.
Spectra can be motivated by the study of cohomology theories and/or stable homotopy theory. The higher categorical framework used so everything behaves correctly from a homotopical point of view.
Motivation for building a theory of commutative algebra from Spectra is more complicated.
Robert Miller
How much Alg Top and related areas have you studied to get to this point?
Lincoln Gray
I'm pretty new to the more complicated algebraic topology (i.e. the theory of spectra).
I have more of a background in algebraic geometry and am trying to piece together background to get into derived algebraic geometry.
Luke Lee
Define "background" please. I am interested in going into academia to do algebraic topology/geometry/category theory so I appreciate any advice.
James Carter
Why would you want to get into spectra if you plan on doing derived alggeom? Just read Huybrechts' book mate.
Juan Bailey
Standard background in algebraic topology and algebraic geometry can be obtained via the books by JP May and Hartshorne respectively.
For the theory of Spectra I've been learning from the papers cited on nlab.
There are various levels of derived geometry. By Huybrechts book I assume you mean his one on Fourier-Mukai transforms. That is probably the most naive form of derived algebraic geometry, i.e. studying derived categories of sheaves on a scheme.
However the bigger idea is to build spaces, analogous to schemes, that have derived (i.e. cohomological or homotopical) information built into them.
Schemes are built over commutative rings. Derived Schemes are built over various enhancements of this notion.
You can have derived schemes built over dg-algebras, which is one of the older notions and the most obviously cohomological. Nowadays these are good for motivation but not really still used in current research as much.
A better notion of derived schemes is on built of simplicial commutative rings. This is where algebraic topology really starts becoming important, in the form of simplicial homotopy theory. These are discussed in Lurie's thesis and also used in Toën's work. They work great in characteristic 0, but don't behave well in more general circumstances.
Then we have derived schemes built over E_infinity rings, also called spectral schemes. In characteristic 0 they agree with derived schemes over simplicial rings, but in higher characteristic they behave much better. This is where all the heavy algebraic topology comes in and is what I'm currently trying to learn.
David Bailey
Judging from this I reckon you are close to getting a PhD
Christian Evans
An [math]{\mathbb{E}_\infty }[/math] - ring [math]A[/math] is called connective if the homotopy groups vanish for negative degree.
(note all categorical things discussed in the following are infinity-categorical)
We define the category of affine spectral schemes [math]\operatorname{SAff} [/math] as the opposite category of the category [math]\operatorname{Rg} \operatorname{Sp} [/math] of [math]{\mathbb{E}_\infty }[/math] - rings.
[math]{\operatorname{SAff} _{ \geqslant 0}}[/math] , the category of connective affine spectral schemes, is defined as the opposite category of connective [math]{\mathbb{E}_\infty }[/math] - rings.
We denote the functor by [math]{\mathbf{Spec}}:\operatorname{Rg} {\operatorname{Sp} ^{op}} \to \operatorname{SAff} [/math] .
Explicitly, an affine spectral scheme [math]{\mathbf{Spec}}\left( A \right)[/math] is a pair consisting of an affine scheme [math]X = \operatorname{Spec} \left( {{\pi _0}\left( A \right)} \right)[/math] and a sheaf [math]{\mathcal{O}_X}:\operatorname{Open} {\left( X \right)^{op}} \to \operatorname{Rg} \operatorname{Sp} [/math] of [math]{\mathbb{E}_\infty }[/math] - rings.
Lincoln Hall
A morphism [math]f:A \to B [/math] of [math]{\mathbb{E}_\infty }[/math] -rings is called etale if the induced morphism of commutative rings, [math]{\pi _0}f:{\pi _0}A \to {\pi _0}B [/math] , is etale.
A morphism [math]f:A \to B [/math] of [math]{\mathbb{E}_\infty }[/math] -rings is said to exhibit [math]B[/math] as a localization of [math]A[/math] at [math]a \in {\pi _0}A[/math] if
(i) [math]f[/math] is etale
(ii) [math]f[/math] induces an isomorphism [math]\left( {{\pi _0}A} \right)\left[ {{a^{ - 1}}} \right] \cong {\pi _0}B [/math]
In this case we denote [math]B \equiv A\left[ {{a^{ - 1}}} \right][/math] .
Let [math]A[/math] be an [math]{\mathbb{E}_\infty }[/math] -ring. Then the structure sheaf [math]{\mathcal{O}_X}[/math] of [math]X = {\mathbf{Spec}}\left( A \right)[/math] satisfies the following conditions:
(a) For all [math]a \in {\pi _0}A[/math] define [math]{\operatorname{U} _a} = \left\{ {p \in \operatorname{Spec} \left( {{\pi _0}\left( A \right)} \right)|a \notin p} \right\}[/math] , the usual basis for the topology of affine scheme. Then there is an equivalence of an [math]{\mathbb{E}_\infty }[/math] -rings, [math]{\mathcal{O}_X}\left( {{\operatorname{U} _a}} \right) \cong A\left[ {{a^{ - 1}}} \right][/math] .
(b) [math]{\pi _0}{\mathcal{O}_X}[/math] is the usual structure sheaf of the affine scheme [math]\operatorname{Spec} \left( {{\pi _0}\left( A \right)} \right)[/math] .
Lucas Robinson
To clarify, a Sheaf of [math]{\mathbb{E}_\infty }[/math] -rings is a contravariant (infinity) functor [math]\mathcal{F}:\operatorname{Open} {\left( X \right)^{op}} \to \operatorname{Rg} \operatorname{Sp} [/math] from the category of open sets on the topological space [math]X[/math] to the category of [math]{\mathbb{E}_\infty }[/math] -rings, such that the functor satisfies a descent condition for the topology on [math]X[/math].
The derived structure of these spectral schemes is coming from the fact the structure sheaves have nontrivial higher homotopy.
The nth homotopy sheaf of [math]\mathbb{F}[/math] is the sheafification of the presheaf [math]\left( {{\pi _n}\mathcal{F}} \right)\left( \operatorname{U} \right) = {\pi _n}\left( {\mathcal{F}\left( \operatorname{U} \right)} \right)[/math] .
For an affine spectral scheme (and in fact a general spectral scheme that I won't define) the higher homotopy sheaves [math]{\pi _n}{\mathcal{O}_X}[/math] are quasi-coherent [math]{\pi _0}{\mathcal{O}_X}[/math] -modules.
Moreover we have isomorphisms [math]\left( {{\pi _n}{\mathcal{O}_X}} \right)\left( X \right) \cong {H^{ - n}}\left( {X,{\pi _0}{\mathcal{O}_X}} \right)[/math] .
Isaiah Miller
>F should be [math]\mathcal{F}[/math]
Camden Fisher
This is a nigger thread now post niggers or gtfo
niggers
Matthew Long
Now I'm going to focus on examples.
Let [math]X = {\mathbf{Spec}}\left( \mathbb{S} \right)[/math] . This is the terminal object in the category of spectral schemes.
Thus the underlying scheme of [math]{\mathbf{Spec}}\left( \mathbb{S} \right)[/math] is just [math]\operatorname{Spec} \mathbb{Z}[/math] .
To see interesting arithmetic properties, will study coverings of the form [math]{\mathbf{Spec}}\left( {H\mathbb{Q}} \right) \to {\mathbf{Spec}}\left( \mathbb{S} \right)[/math] , where [math]{H\mathbb{Q}}[/math] is the Eilenberg-MacLane spectrum (i.e. the singular cohomology spectrum) as before.
Caleb Bell
Have you transcended?
Parker Clark
>unmotivated his posts are full of examples and motivation... it's just that his style is super concise
Jayden Cook
Which ones of the /mg/ regulars are you? Hopefully not one of the anime posters.
Ryan Barnes
I think the lack of anime avatars on every post answers that question.
Alexander Torres
>I'm in year two and you lost me 1 1/2 posts in Not him, but I'm about 6 months in and I can see you seem to be a retard.
David Ward
>unmotivated definitions If you need any further motivation for this, you should be studying "engineering".
Julian Taylor
Abandon ship everyone. This is a Spurdo thread.
Noah Powell
>Hopefully not one of the anime posters. >anime ""avatars""
Ayden Campbell
...
Andrew Baker
What is the best introduction to infinity categories?
Cooper Taylor
Lurie's book is the obvious answer. But it is very long. This paper is a decent overview: arxiv.org/pdf/1007.2925v2.pdf
If you notice in these posts I have just been trying to avoid to many higher categorical details. Every infinity category I have defined is implicitly.
i.e. Pointed spaces as a localization, Spectra as limit, and the rest as subcategories of various functor categories
Jackson Morgan
This next stuff is hard. Not gonna type up something before I even understand it. It will be awhile before I continue this.
Nicholas Bell
Ok so before I get to the interesting arithmetic, need to understand more of the theory around [math]{\mathbf{Spec}}\left( \mathbb{S} \right)[/math] .
A pointed space [math]X \in {S_*}[/math] is said to be an infinite loop space if [math]X \cong {\Omega ^\infty }E[/math] for some spectrum [math]E \in \operatorname{Sp} [/math].
An infinite loop space structure on [math]X[/math] is essentially equivalent to an abelian group structure on [math]X[/math] (up to coherent homotopy).
For an infinite loop space [math]X[/math] , can form an [math]{\mathbf{S}}\left[ X \right] \equiv {\Sigma ^\infty }{X_ + }[/math] . This called the "group ring spectrum" and inherits an [math]{\mathbb{E}_\infty }[/math] -ring structure from the infinite loop structure of [math]X[/math] .
The diagonal [math] X \to X \times X [/math] induces a coproduct [math]{\mathbf{S}}\left[ X \right] \to {\mathbf{S}}\left[ X \right] \wedge {\mathbf{S}}\left[ X \right][/math] that is coassociative and counital (up to coherent homotopy).
So [math]{\mathbf{S}}\left[ X \right][/math] is an [math]{\mathbb{E}_\infty }[/math] version of a Hopf Algebra and [math]{\mathbf{G}}\left( X \right) = {\mathbf{Spec}}\left( {{\mathbf{S}}\left[ X \right]} \right)[/math] is spectral version of a proalgebraic group.
Leo Turner
An [math]{\mathbb{E}_\infty }[/math] -Space is defined as a commutative monoid in the category of pointed spaces (w.r.t smash product).
The natural adjunction [math]{\Sigma ^\infty }:{S_*} \rightleftarrows \operatorname{Sp} :{\Omega ^\infty }[/math] can be lifted to an adjunction [math]{\Sigma ^\infty }:\left\{ {{\mathbb{E}_\infty } - \operatorname{Spaces} } \right\} \rightleftarrows \operatorname{Rg} \operatorname{Sp} :{\Omega ^\infty }[/math] .
i.e. If [math] A [/math] is an [math]{\mathbb{E}_\infty }[/math] -ring then [math]{\Omega ^\infty }A[/math] is an [math]{\mathbb{E}_\infty }[/math] -space.
In particular, [math]{\pi _0}{\Omega ^\infty }A[/math] is an abelian monoid.
We define the "General Linear Group of [math]A[/math] " via an (infinity-)pullback [math]\begin{array}{*{20}{c}} {{{\operatorname{GL} }_1}\left( A \right)}& \to &{{\Omega ^\infty }A} \\ \downarrow &{}& \downarrow \\ {{{\left( {{\pi _0}{\Omega ^\infty }A} \right)}^ \times }}& \to &{{\pi _0}{\Omega ^\infty }A} \end{array}[/math]
This pullback removes the components of [math]{{\Omega ^\infty }A}[/math] corresponding to invertible elements of [math]{{\pi _0}{\Omega ^\infty }A}[/math] . Therefore making [math]{\pi _0}{\operatorname{GL} _1}\left( A \right)[/math] an abelian group.
Should note there is an adjunction [math]\left\{ {\infty \operatorname{Loop} - \operatorname{Spaces} } \right\} \rightleftarrows \left\{ {{\mathbb{E}_\infty } - \operatorname{Spaces} } \right\}[/math] where the right adjoint is viewed as removing invertible elements.
The functor [math] {\operatorname{GL} _1}:\operatorname{Rg} \operatorname{Sp} \to \left\{ {\infty \operatorname{Loop} - \operatorname{Spaces} } \right\} [/math] can be viewed as a composition of the right adjoints of the two adjunctions described above.
Jaxon Ramirez
A Spherical Fibration over a space [math]X[/math] is a fibration [math]E\mathop \to \limits^\pi X[/math] such that each fiber is (homotopically) a sphere of some dimension.
For any two spherical fibrations [math]{E_1},{E_2} \to X[/math] , the relative smash product fibration [math]{E_1}{ \wedge _X}{E_2} \to X[/math] is given by taking fiberwise smash products of spheres.
The classifying space for spherical fibrations is [math]{\mathbf{B}}{\operatorname{GL} _1}\left( \mathbb{S} \right)[/math] .
i.e. For every spherical fibration [math]E\mathop \to \limits^\pi X[/math] , there exists a unique map [math]{\varsigma _\pi }:X \to {\mathbf{B}}{\operatorname{GL} _1}\left( \mathbb{S} \right)[/math] such that [math]E[/math] is the pullback (via [math]{\varsigma _\pi }[/math] ) of the universal [math]{\operatorname{GL} _1}\left( \mathbb{S} \right)[/math] -Bundle.
The Thom Space of a spherical fibration [math]E\mathop \to \limits^\pi X[/math] is defined as the (infinity-)pushout [math]\begin{array}{*{20}{c}} E& \to &* \\ \downarrow &{}& \downarrow \\ X& \to &{\operatorname{Th} \left( {X,{\varsigma _\pi }} \right)} \end{array} [/math] in the (infinity-)category of pointed spaces.
To any vector bundle [math]V\mathop \to \limits^\pi X[/math] we associated the unit-sphere bundle [math]S\left( V \right)\mathop \to \limits^{S\left( \pi \right)} X[/math] by restricting fiberwise to unit spheres (i.e. all points in the fiber spaces of norm 1).
The Thom Space of the vector bundle is then defined as Thom Space of the correspond unit-sphere bundle.
Example:
Let [math]{V_n}\mathop \to \limits^{{\pi _n}} {\mathbf{B}}\operatorname{U} \left( n \right)[/math] be the vector bundle associated to the universal [math]\operatorname{U} \left( n \right)[/math] -Bundle. Then we define [math]\operatorname{MU} \left( n \right) \equiv \operatorname{Th} \left( {{\mathbf{B}}\operatorname{U} \left( n \right),{\varsigma _{{\pi _n}}}} \right)[/math].
Kayden Davis
Let [math]E\mathop \to \limits^\pi X[/math] be a spherical fibration. We define a spectrum associated to this fibration, [math] {X^{{\varsigma _\pi }}} [/math] called the Thom Spectrum.
Explicitly, it is given by [math]{X^{{\varsigma _\pi }}}\left( n \right) \equiv \operatorname{Th} \left( {X,{\varsigma _\pi } \wedge {\varsigma _{{s^n}}}} \right)[/math] .
Where [math] {\varsigma _\pi } \wedge {\varsigma _{{s^n}}}:X \to {\mathbf{B}}{\operatorname{GL} _1}\left( \mathbb{S} \right) [/math] is the classifying map for the smash product of the given spherical bundle and the trivial bundle [math]{s^n}:X \times {S^n} \to X[/math] .
The structure maps on the spectrum are induced by the natural maps [math] \Sigma \operatorname{Th} \left( {X,{\varsigma _\pi } \wedge {\varsigma _{{s^n}}}} \right) \to \operatorname{Th} \left( {X,{\varsigma _\pi } \wedge {\varsigma _{{s^{n + 1}}}}} \right) [/math] .
The topologist will know there is a deep connection between Thom Spectra and (co)bordism theory. For instance, the Thom Spectrum association to the Thom spaces [math] \operatorname{MU} \left( n \right) [/math] define the generalized cohomology theory of Complex Cobordism. However an explanation would be far to long a digression from my current focus.
Daniel Wilson
If [math]{\varsigma _\pi }:X \to {\mathbf{B}}{\operatorname{GL} _1}\left( \mathbb{S} \right)[/math] is a map of infinite loop spaces, then the Thom Spectrum [math]{X^{{\varsigma _\pi }}}[/math] is an [math]{\mathbb{E}_\infty }[/math] -ring.
In this case there is a canonical morphism [math] {X^{{\varsigma _\pi }}} \to {X^{{\varsigma _\pi }}} \wedge {X_ + }[/math] called the Thom Diagonal.
It induces an equivalence [math] {X^{{\varsigma _\pi }}} \wedge {X^{{\varsigma _\pi }}} \to {X^{{\varsigma _\pi }}} \wedge {X_ + }[/math].
This has a geometric interpretation. Recall [math] {\mathbf{G}}\left( X \right) = {\mathbf{Spec}}\left( {{\Sigma ^\infty }{X_ + }} \right) [/math] .
Then the Thom diagonal gives a group action [math] {\mathbf{G}}\left( X \right){ \times _{{\mathbf{Spec}}\left( \mathbb{S} \right)}}{M^{{\varsigma _\pi }}}\left( X \right) \to {M^{{\varsigma _\pi }}}\left( X \right) [/math] of this spectral proalgebraic group associated to [math]X[/math] on the spectral scheme associated to the Thom spectrum.
It induces an equivalence in the category of affine spectral schemes, [math]{\mathbf{G}}\left( X \right){ \times _{{\mathbf{Spec}}\left( \mathbb{S} \right)}}{M^{{\varsigma _\pi }}}\left( X \right) \to {M^{{\varsigma _\pi }}}\left( X \right){ \times _{{\mathbf{Spec}}\left( \mathbb{S} \right)}}{M^{{\varsigma _\pi }}}\left( X \right) [/math] .
Eli Jenkins
Y-Yeah interesting OP, understood every word
Aaron Gomez
Let [math]A \to \operatorname{B} [/math] map of [math]{\mathbb{E}_\infty }[/math] -rings.
The Amitsur Complex is defined as the cosimplicial object [math]{\mathcal{A}^ \bullet }\left( {\operatorname{B} |A} \right):\Delta \to \operatorname{Rg} \operatorname{Sp} [/math] such that the n-simplicies are [math]{\mathcal{A}^n}\left( {\operatorname{B} |A} \right) \equiv \operatorname{B} { \wedge _A}....{ \wedge _A}\operatorname{B} [/math]
There is an canonical augmentation map [math]A \to \lim {\mathcal{A}^n}\left( {\operatorname{B} |A} \right)[/math] . When this map is an equivalence, we say [math]{\mathbf{Spec}}\left( A \right)[/math] is complete along [math]{\mathbf{Spec}}\left( \operatorname{B} \right)[/math] .
The Adams-Novikov Theorem states [math]{\mathbf{Spec}}\left( \mathbb{S} \right)[/math] is complete along [math]{M^{{\varsigma _\pi }}}\left( X \right)[/math] for any infinite loop map [math]{\varsigma _\pi }:X \to {\mathbf{B}}{\operatorname{GL} _1}\left( \mathbb{S} \right)[/math] .
The Thom diagonal gives [math]{M^{{\varsigma _\pi }}}\left( X \right)[/math] a [math]{\mathbf{G}}\left( X \right)[/math] -Torsor structure over [math]{\mathbf{Spec}}\left( \mathbb{S} \right)[/math] .
Parker Hall
>Someone posting actual math on Veeky Forums It has been a good day, thanks user
Gabriel Diaz
Recall [math]{\operatorname{MU} }[/math] is defined as the Thom spectrum given by the vector bundles associated to the universal [math]\operatorname{U} \left( n \right)[/math] -Bundles for each n.
It can be shown that [math]{\pi _*}\left( {\operatorname{MU} } \right) \cong \mathbb{Z}\left[ {{x_1},{x_2},...} \right][/math] where [math]{x_j} \in {\pi _{2j}}\left( {\operatorname{MU} } \right)[/math] .
We denote by [math]{v_n} = {x_{{p^n} - 1}}[/math] there generators obtained from [math]{\pi _{2\left( {{p^n} - 1} \right)}}\left( {\operatorname{MU} } \right)[/math] for each prime p and n>=0.
Want to consider a spectrum, [math]\operatorname{BP} [/math] , obtained via some type of localization of [math]{\operatorname{MU} }[/math] at p.
Not getting into details, I will simply say there exists a spectrum [math]\operatorname{BP} [/math] obtained by from [math]{\operatorname{MU} }[/math] such that [math] {\pi _*}\left( {\operatorname{BP} } \right) \cong {\mathbb{Z}_{\left( p \right)}}\left[ {{v_1},{v_2},...} \right] [/math] . Called the Brown-Peterson Spectrum.
We can take this idea further and obtain spectra [math] E\left( n \right) [/math] and [math]K\left( n \right)[/math] , called Johnson-Wilson Spectra and Morava K-theories respectively, from [math]\operatorname{BP} [/math] .
Johnson-Wilson Spectra satisfy [math]{\pi _*}\left( {E\left( n \right)} \right) \cong {\mathbb{Z}_{\left( p \right)}}\left[ {{v_1},...,{v_n},{v_n}^{ - 1}} \right] [/math] with the convention [math] E\left( 0 \right) = H\mathbb{Q} [/math] .
The point of these notes, which I am still getting to, is not to prove a bunch of theorems but to outline an interesting connection between Algebraic Topology and Arithmetic Geometry.
Jeremiah Rogers
Let [math]A[/math] be an [math]{\mathbb{E}_\infty }[/math] -Ring and [math]M,N[/math] be [math]A[/math] -Modules.
[math]N[/math] is called [math]M[/math] -Acyclic if [math]M{ \wedge _A}N \cong 0[/math] .
An [math]A[/math] -Module [math]P[/math] is called [math]M[/math] -Local if [math]N{ \wedge _A}P = P{ \wedge _A}N[/math] for every [math]M[/math] -Acyclic module [math]N[/math] .
For fixed [math]M[/math] , define [math]\operatorname{Mod} {\left( A \right)_M}[/math] the full sub-(infinity-)category of [math]\operatorname{Mod} \left( A \right)[/math] spanned by [math]M[/math] -local modules.
The inclusion [math]\operatorname{Mod} {\left( A \right)_M} \subset \operatorname{Mod} \left( A \right)[/math] has a left-adjoint, denoted [math]{L_M}:\operatorname{Mod} \left( A \right) \to \operatorname{Mod} {\left( A \right)_M}[/math] , called the Bousfield localization.
The Bousfield localization is called "smashing" if there is a natural equivalence [math]{L_M}\left( - \right) \cong \left( - \right){ \wedge _A}{L_M}A[/math] .
In the case it is smashing, we have an equivalence of (infinity-)categories [math]\operatorname{Mod} {\left( A \right)_M} \cong \operatorname{Mod} \left( {{L_M}A} \right)[/math] .
Moreover, in this case [math]{\mathbf{Spec}}\left( {{L_M}A} \right) \to {\mathbf{Spec}}\left( A \right)[/math] is a Zariski-open immersion. (i.e. the underlying map of affine schemes is an open immersion)
Daniel Ramirez
Consider the Johnson-Wilson spectrum [math]E\left( n \right)[/math] as an [math]\mathbb{S}[/math] -Module.
Then the Bousfield localization [math]{L_{E\left( n \right)}}:\operatorname{Mod} \left( \mathbb{S} \right) \to \operatorname{Mod} {\left( \mathbb{S} \right)_{E\left( n \right)}} [/math] is smashing and induces a sequence....
called the Chromatic Covering of [math]{\mathbf{Spec}}\left( \mathbb{S} \right)[/math] .
Camden Evans
>connection between Algebraic Topology and Arithmetic Geometry. perkele
Joseph Kelly
>mfw i find out i'm on the Johnson-Wilson spectrum >mfw i have no face
Evan Robinson
you're typing this in latex for your own benefit to right? this board will eventually be pruned. saged
Austin Ramirez
All the latex is saved in another document.
Bentley Flores
>An AA -Module PP is called MM -Local if N∧AP=P∧ANN∧AP=P∧AN for every MM -Acyclic module NN .
This is flat out wrong. I misinterpreted notation and just realized it.
An [math]A[/math] -Module [math]P[/math] is called [math]M[/math] -local if the space of maps in the homotopy category, [math]\left[ {N,P} \right][/math] , is [math]0[/math] for all [math]M[/math] -acyclic modules [math]N[/math] .
Tyler Reed
A morphism of spectral schemes [math]{\mathbf{Spec}}\left( \operatorname{B} \right) \to {\mathbf{Spec}}\left( A \right)[/math] is called a Zariski-Open immersion if the pushfoward [math]\operatorname{Mod} \left( \operatorname{B} \right) \to \operatorname{Mod} \left( A \right)[/math] is fully faithful.
This is an equivalent condition to requiring the morphism of schemes [math]\operatorname{Spec} \left( {{\pi _0}\operatorname{B} } \right) \to \operatorname{Spec} \left( {{\pi _0}A} \right)[/math] is an open immersion in the usual sense.
A sequence, [math]{\left\{ {{\mathbf{Spec}}\left( {{\operatorname{B} _\alpha }} \right) \to {\mathbf{Spec}}\left( A \right)} \right\}_\alpha }[/math] of Zariski open immersions, is called a Zariski open cover if the induced functor [math]\operatorname{Perf} \left( A \right) \to \prod\limits_\alpha {\operatorname{Perf} \left( {{\operatorname{B} _\alpha }} \right)} [/math] on perfect modules is conservative (i.e. it reflects equivalences).
Suppose [math]M[/math] is a perfect [math]\mathbb{S}[/math] -Module.
Then [math]M \cong \lim \left[ {... \to {L_{E\left( n \right)}}M \to {L_{E\left( {n - 1} \right)}}M \to ... \to {L_{E\left( 0 \right)}}M} \right][/math] and moreover [math]{\pi _*}\left( M \right) \cong \lim {\pi _*}\left( {{L_{E\left( n \right)}}M} \right)[/math] .
It follows that the functor [math]\operatorname{Perf} \left( \mathbb{S} \right) \to \prod\limits_n {\operatorname{Perf} \left( {{L_{E\left( n \right)}}\mathbb{S}} \right)} [/math] is conservative.
And thus the Chromatic covering [math]{\mathbf{Spec}}\left( {H\mathbb{Q}} \right) \cong {\mathbf{Spec}}\left( {{L_{E\left( 0 \right)}}\mathbb{S}} \right) \to ... \to {\mathbf{Spec}}\left( {{L_{E\left( n \right)}}\mathbb{S}} \right) \to ... \to {\mathbf{Spec}}\left( \mathbb{S} \right)[/math] is a Zariski open covering.
Jordan Russell
This Chromatic covering should be though of in the following way...
The affine scheme [math]\operatorname{Spec} \mathbb{Z}[/math] is generated by open sets [math]\operatorname{D} \left( p \right) \cong \operatorname{Spec} \mathbb{Z}\left[ {{p^{ - 1}}} \right][/math] obtained by inverting a single prime.
However in affine spectral scheme, [math]{\mathbf{Spec}}\left( \mathbb{S} \right)[/math] ,each prime corresponds to an infinite family of localizations. All of these together than generate the Zariski topology.
A "chromatic prime" is defined as a pair [math]\left( {p,n} \right)[/math] corresponding to a prime [math]p \in {\mathbf{Spec}}\left( \mathbb{S} \right)[/math] and a "level" [math]n[/math] corresponding to the choice of open immersion in the chromatic covering.
Austin Watson
Let [math]A[/math] be an [math]{\mathbb{E}_\infty }[/math] -ring and [math]I \triangleleft {\pi _0}A[/math] an ideal.
An [math]A[/math] -Module [math]M[/math] is called [math]I[/math] -complete if, for every [math]I[/math] -local module [math]N[/math] , the mapping space [math]{\operatorname{Hom} _{\operatorname{Mod} \left( A \right)}}\left( {M,N} \right)[/math] is contractible (i.e. The mapping space in the homotopy category is 0).
Every [math]A[/math] -Module [math]M[/math] fits into a unique fiber sequence [math]M' \to M \to M_I^ \wedge [/math] such that [math]M'[/math] is [math]I[/math] -local and [math]M_I^ \wedge [/math] is [math]I[/math] -complete.
In this case, we call [math]M_I^ \wedge [/math] the [math]I[/math] -completion of [math]M[/math].
When [math]M[/math] is discrete, and we assume some finiteness conditions, this is equivalent to the classical [math]I[/math] -adic completion.
An ideal [math]I[/math] of [math]A[/math] corresponds to a closed subset of [math]{\mathbf{Spec}}\left( A \right)[/math] .
Like in classical algebraic geometry, want to define a notion of completion along a closed subset.
Thus we define the formal spectrum, [math]{\mathbf{Spf}}\left( A \right)[/math] ,to have the same underlying topological space [math]\operatorname{Spec} \left( {{\pi _0}A} \right)[/math] but with sheaf given by [math]{\mathcal{O}_{{\mathbf{Spf}}\left( A \right)}}\left( \operatorname{U} \right) \equiv {\mathcal{O}_{{\mathbf{Spec}}\left( A \right)}}\left( \operatorname{U} \right)_I^ \wedge [/math] .
Formal spectral schemes will help in understanding the chromatic covering.
Alexander Adams
how does this stuff apply to the real physical world
seems to me u found a way to safeguard ur virginity
Ryder Nguyen
It has applications to topological modular forms, which have applications to string theory. But thats about as close to the real world as possible.
But spectral algebraic geometry is pretty damn new, so can't expect it to have many applications yet.
Jackson Lewis
Is this the "Sokal affair" of Veeky Forums? If not, okay cool comments OP.
Blake Evans
Don't make the mistake of accepting the frame that mathematics must apply to the physical sciences or else be "applied", generally, in order to be real. You give them too much ground on that point, which they emphatically don't deserve.
Nicholas Sanders
The early posts and the stuff on Thom spectra are standard Algebraic Topology. The other stuff is spectral algebraic geometry its connections to chromatic homotopy theory. Most of the information can probably be verified in some form through math.harvard.edu/~lurie/252x.html or math.harvard.edu/~lurie/papers/SAG-rootfile.pdf
Kayden Baker
>apply >physical world Why should mathematicians care about that? There are already groups of subhumans who "research" "applications", leave it to them.
Aiden Brown
>string theory >close to the real world
Jacob Torres
I didn't say it was close to the real world, I said it is as close to the real world as possible.
Camden Lee
then what are the discrepancies between real world and string theory
Levi Davis
Idk. My knowledge of string theory consists of general facts on TFT, particularly the topological A/B-models, and it’s connections to derived geometry via mirror symmetry and the geometric langlands.
I have never read a physicists textbook on string theory.
Xavier Ross
To further understand the geometry contained in the sequence [math]{\mathbf{Spec}}\left( {H\mathbb{Q}} \right) \cong {\mathbf{Spec}}\left( {{L_{E\left( 0 \right)}}\mathbb{S}} \right) \to ... \to {\mathbf{Spec}}\left( {{L_{E\left( n \right)}}\mathbb{S}} \right) \to ... \to {\mathbf{Spec}}\left( \mathbb{S} \right)[/math] we need to understand the difference between each of the localizations.
i.e. We want to understand the "complement" [math]{\mathbf{Spec}}\left( {{L_{E\left( n \right)}}\mathbb{S}} \right)\backslash {\mathbf{Spec}}\left( {{L_{E\left( {n - 1} \right)}}\mathbb{S}} \right)[/math] as a "closed subscheme" of [math]{\mathbf{Spec}}\left( \mathbb{S} \right)[/math] .
Turns out this can be done by studying the "monochromatic sphere specturm" [math]{L_{K\left( n \right)}}\mathbb{S}[/math] .
The localization [math]{L_{K\left( n \right)}}[/math] is not smashing , so studying [math]\operatorname{Mod} \left( {{L_{K\left( n \right)}}\mathbb{S}} \right)[/math] is insufficient.
Instead we just study the localization of the entire category of spectra, [math]{L_{K\left( n \right)}}\operatorname{Sp} [/math] . This has a symmetric monoidal structure given by [math]{L_{K\left( n \right)}}\left( { - \wedge - } \right)[/math] as the monoidal product and [math]{L_{K\left( n \right)}}\mathbb{S}[/math] as the unit.
This monoidal product can be vaguely thought of as a "completed tensor product".
Geometrically [math]{L_{K\left( n \right)}}\operatorname{Sp} [/math] is not regarded as the category of modules over [math]{\mathbf{Spec}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)[/math], but instead regarded as the category of modules over [math]{\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)[/math] .
However formal schemes are understood as completions of a scheme along a closed subscheme. So the question is, what closed subscheme do we complete along to get [math]{\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)[/math] ?
Jace Richardson
Is there a reason for you vomiting your undergrad homework over this board?
Nolan Reyes
>undergrad homework lol
>over this board? This is a Science&Math board constantly spammed with IQ threads, race threads, academia shitposting, and pseudoscience. Since I was typing up notes anyway, why not post some actual math on this board?
Xavier Fisher
In the earlier post you made at 9 you typed [math]X[/math] where presumably meant [math]E[/math], no?
Brandon Nguyen
which one?
Cooper Long
>not undergrad Hows grad(e) school at brainlet U (daycare)
Noah Scott
but how does this apply to PDEs?
Caleb Powell
It doesn't. However there are attempts at derived differential geometry, where moduli spaces of solutions to some nonlinear elliptic equations over manifolds with fixed topological invariants can be realized as "derived manifolds" .
Christopher Cruz
We can form a diagram of [math]{\mathbb{E}_\infty }[/math] -rings, [math]\begin{array}{*{20}{c}} {{L_{E\left( n \right)}}\mathbb{S}}& \to &{{L_{E\left( {n - 1} \right)}}\mathbb{S}} \\ \downarrow &{}& \downarrow \\ {{L_{K\left( n \right)}}\mathbb{S}}& \to &{{L_{E\left( {n - 1} \right)}}{L_{K\left( n \right)}}\mathbb{S}} \end{array}[/math] , which is both a pushout and a pullback.
The dual geometric picture is[math]\begin{array}{*{20}{c}} {{\mathbf{Spf}}\left( {{L_{E\left( {n - 1} \right)}}{L_{K\left( n \right)}}\mathbb{S}} \right)}& \to &{{\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)} \\ \downarrow &{}& \downarrow \\ {{\mathbf{Spec}}\left( {{L_{E\left( {n - 1} \right)}}\mathbb{S}} \right)}& \to &{{\mathbf{Spec}}\left( {{L_{E\left( n \right)}}\mathbb{S}} \right)} \end{array}[/math]
The geometric diagram lets us interpret [math]{{\mathbf{Spec}}\left( {{L_{E\left( n \right)}}\mathbb{S}} \right)}[/math] as the result of gluing together [math]{{\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)}[/math] and [math]{{\mathbf{Spec}}\left( {{L_{E\left( {n - 1} \right)}}\mathbb{S}} \right)}[/math] in the "appropriate" way.
And thus we achieve our goal of formally understanding the "difference" between two levels in the chromatic cover, in the sense it is realized as the "closed subscheme" which determines the completion [math]{{\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)}[/math] .
Xavier Wright
Now for a digression to standard algebra.
Let [math]R[/math] commutative ring and fix a prime number [math]p[/math] .
A Witt vector over [math]R[/math] is simply a sequence [math]X = \left( {{X_0},{X_1},..} \right) \in \prod\limits_{n \geqslant 0} R [/math] .
The nth Witt polynomial associated to a Witt vector [math]X[/math] is defined as [math]{W_n}\left( X \right) = \sum\limits_{i = 0}^n {{p^i}X_i^{{p^{n - i}}}} [/math] .
There exists a unique ring, [math]W\left( R \right)[/math] , of Witt vectors over [math]R[/math] (called the Witt ring of [math]R[/math] ) such that:
(i) [math]{\left( {X + Y} \right)_i}[/math] and [math]{\left( {X \cdot Y} \right)_i}[/math] are polynomials over [math]\mathbb{Z}[/math] in the components of the vectors , independent of [math]R[/math] .
(ii) [math]{W_n}\left( X \right) + {W_n}\left( Y \right) = {W_n}\left( {X + Y} \right)[/math] and [math]{W_n}\left( X \right) \cdot {W_n}\left( Y \right) = {W_n}\left( {X \cdot Y} \right)[/math]
[math]W\left( {{\mathbb{F}_{{p^n}}}} \right)[/math] the uniqued degree=n unramified extension of [math]{\mathbb{Z}_p}[/math]
Luke Cox
keep going. more interesting than anything else on this board.
Austin Mitchell
I remember when I first learned topology. It's much more basic than this stuff but it has the same kind of elegance.
You define something, you define another thing, you relate the definitions, maybe it's useful and maybe it's not
Aiden Murphy
Let [math]R[/math] commutative ring.
An n-dimensional formal group law over [math]R[/math] is a choice of group structure on [math]\operatorname{Spf} \left( {R\left[\kern-0.15em\left[ {{x_1},...,{x_n}} \right]\kern-0.15em\right]} \right)[/math] .
Or equivalently, a choice of [math]F \in \prod\limits_{i = 1}^n {R\left[\kern-0.15em\left[ {{x_1},...,{x_n},{y_1},...,{y_n}} \right]\kern-0.15em\right]}[/math] such that
Interesting examples, which I won't discuss, arise in the theory of elliptic cohomology.
Let [math]F \in R\left[\kern-0.15em\left[ {x,y} \right]\kern-0.15em\right][/math] a 1 dimensional formal group law. The n-series of [math]F[/math] , [math]{n_F}\left( t \right) \in R\left[\kern-0.15em\left[ t \right]\kern-0.15em\right][/math] , is defined recursively by [math]{0_F}\left( t \right) = 0[/math] and [math]{n_F}\left( t \right) = F\left( {{{\left( {n - 1} \right)}_F}\left( t \right),t} \right)[/math] .
For a fixed prime p, let [math]{c_i}[/math] the coefficient of [math]{t^{{p^i}}}[/math] in [math]{p_F}\left( t \right)[/math] .
[math]F[/math] is said to have height = n if [math]{c_i} = 0[/math] for all i
Mason Miller
I'm almost done. These last bits of algebraic theory will be put towards building a notion of Galois extensions of [math]{L_{K\left( n \right)}}\mathbb{S}[/math] .
But before I can do that, have to define the etale topology for a spectral affine scheme. To be continued...
Sebastian Ross
A morphism of spectral schemes [math]{\mathbf{Spec}}\left( \operatorname{B} \right) \to {\mathbf{Spec}}\left( A \right)[/math] is called flat, faithfully flat, etale if...
(i) [math]\operatorname{Spec} \left( {{\pi _0}\operatorname{B} } \right) \to \operatorname{Spec} \left( {{\pi _0}A} \right)[/math] is a flat, faithfully flat, etale morphism of schemes
(ii) [math]{\pi _j}A{ \otimes _{{\pi _0}A}}{\pi _0}\operatorname{B} \to {\pi _j}\operatorname{B} [/math] is an isomorphism of [math]{\pi _0}\operatorname{B} [/math] -modules for all j.
Now this should not really be the definition of etale, the "right" definition is in terms of the cotangent complex as it extends easier to general spectral schemes (and ultimately spectral stacks), but in the affine case this is equivalent.
A family [math]{\left\{ {{V_i} \to \operatorname{U} } \right\}_{i \in I}}[/math] in [math]{\operatorname{Aff} _{/{\mathbf{Spec}}\left( A \right)}}[/math] is called an etale covering if for, finite [math]I' \subset I[/math] , each [math]{{V_i} \to \operatorname{U} }[/math] is etale for all [math]i \in I'[/math] and [math]\coprod\limits_{i \in I'} {{V_i}} \to \operatorname{U} [/math] is faithfully flat.
Elijah Scott
For a perfect field [math]k[/math] and prime p, define the Lubin-Tate ring [math]R = W\left( k \right)\left[\kern-0.15em\left[ {{u_1},...,{u_{n - 1}}} \right]\kern-0.15em\right][/math] .
It comes with a canonical map [math]{\ell _p}:R \to k[/math] such that [math]\ker \left( {{\ell _p}} \right) = \left\langle {p,{u_1},...,{u_{n - 1}}} \right\rangle [/math] .
For every formal group law [math]F[/math] over [math]k[/math] , the map [math]{\ell _p}[/math] induces a universal deformation [math]{\bar F}[/math] over the Lubin-Tate ring.
We can define a spectrum [math]{E_n}[/math] , via [math]K\left( n \right)[/math] , such that [math]{\pi _*}\left( {{E_n}} \right) \cong W\left( {{\mathbb{F}_{{p^n}}}} \right)\left[\kern-0.15em\left[ {{u_1},...,{u_{n - 1}}} \right]\kern-0.15em\right]\left[ {{u^ \pm }} \right][/math] where [math]{u_i} \in {\pi _0}{E_n}[/math] and [math]{u^ \pm } \in {\pi _2}{E_n}[/math] .
It should be thought of as a homology theory associated to the Lubin-Tate universal deformation of the formal group [math]{H_n}[/math] of height n over [math]{{\mathbb{F}_{{p^n}}}}[/math] .
[math]{E_n}[/math] is an [math]{\mathbb{E}_\infty }[/math] -ring.
Brayden Price
[math]{\mathbf{Spf}}\left( {{E_n}} \right)[/math] can be thought of as a formal affine over [math]{\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)[/math] .
Moreover, we have the following theorem....
The morphism [math]{\mathbf{Spf}}\left( {{E_n}} \right) \to {\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)[/math] is an etale covering and its group of automorphisms is the Morava stabilizer group [math]{{\mathbf{G}}_n} \equiv \operatorname{Gal} \left( {{\mathbb{F}_{{p^n}}}/{\mathbb{F}_p}} \right) \ltimes \operatorname{Aut} \left( {{H_n}} \right)[/math] .
Geometrically, this suggests to think of [math]{\mathbf{Spf}}\left( {{L_{K\left( n \right)}}\mathbb{S}} \right)[/math] as a type of quotient stack [math]\left[ {{\mathbf{Spf}}\left( {{E_n}} \right)/{{\mathbf{G}}_n}} \right][/math] .
In particular, you can think of the adelic sphere spectrum [math]{\mathbb{A}_\mathbb{S}} \equiv H\mathbb{Q} \vee {L_{K\left( 1 \right)}}\mathbb{S} \vee {L_{K\left( 2 \right)}}\mathbb{S} \vee ...[/math] as associated to the geometric object [math]{\mathbf{Spec}}\left( {H\mathbb{Q}} \right)\coprod \left[ {{\mathbf{Spf}}\left( {{E_1}} \right)/{{\mathbf{G}}_1}} \right]\coprod \left[ {{\mathbf{Spf}}\left( {{E_2}} \right)/{{\mathbf{G}}_2}} \right]\coprod ....[/math] .
/thread
Michael Johnson
Whelp I guess I'm a dirty peasant brainlet. I'm still in calc 2
