# 2021-06-06

## 12:12

Refs: A1 Homotopy

• Context: the infinity category of spaces, i.e. homotopy types

• Take smooth manifolds, take the Yoneda embedding to $$\underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Mfd}})$$: these satisfy a Mayer-Vietoris gluing property, and homotopy invariance in the sense that \begin{align*} \mathop{\mathrm{Hom}}({-}, X) \cong \mathop{\mathrm{Hom}}({-}\times I, X) .\end{align*}

• Why the first argument: homotopy invariance as a presheaf
• AG setting: $$\underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Sch}}_{/k})$$, send to presheaves to define motivic spaces.

• Satisfies a Nisnevich gluing condition and $${\mathbb{A}}^1$$ invariance Similar homotopy invariance: $$F({\mathbb{A}}^1 \times X)\cong F(X)$$.
• See Betti realization for $$k={\mathbb{C}}$$: $${\mathsf{sm}}{\mathsf{Sch}}_{/{\mathbb{C}}}\to {\mathsf{Spaces}}$$ where $$X\mapsto X({\mathbb{C}})$$.

• From topology: identify $${\mathbf{B}}{\operatorname{U}}_n({\mathbb{C}}) = {\operatorname{Gr}}_n({\mathbb{C}})$$ to get \begin{align*} { \mathsf{Vect} }_{/{\mathbb{C}}}^{\operatorname{rank}= n}(U) \cong \pi_0 \mathop{\mathrm{Maps}}(U, {\mathbf{B}}{\operatorname{U}}_n({\mathbb{C}})) .\end{align*}

• Problem in AG: there are two rank 2 vector bundles on $${\mathbb{P}}^1 \times{\mathbb{A}}^1$$ whose fibers over 0 and 1 are $${\mathcal{O}}^2$$ and $${\mathcal{O}}(1) \oplus {\mathcal{O}}(-1)$$.

• Theorem: for $$U$$ smooth affine $${\mathsf{Sch}_{/k}}$$, there is an equivalence of rank $$n$$ vector bundles on $$U$$ mod equivalence to $$\pi_0 \mathop{\mathrm{Maps}}_{{\mathsf{Spaces}}(k)}(U, {\mathbf{B}}\operatorname{GL}_n)$$ where again $${\mathbf{B}}\operatorname{GL}_n \cong {\operatorname{Gr}}_n$$.

• Would like this for non-smooth non-affine schemes?
• algebraic K theory : finitely generated projective $$R{\hbox{-}}$$modules mod equivalence with $$\oplus$$, then take group completion to get $$K_0(R)$$.

• $$K_0(k) \cong {\mathbb{N}}^ {\operatorname{gp} } \cong {\mathbb{Z}}$$.
• To get a space: take $K(R) \coloneqq\mathop{\mathrm{Proj}}({\mathsf{R}{\hbox{-}}\mathsf{Mod}})^ {\operatorname{gp} }$ to get a space, set $$K_i R \coloneqq\pi_i K(R)$$

• Prop: the $$K$$ theory space here is a motivic space, $$K: {\mathsf{sm}}{\mathsf{Sch}}_{/k}^{\operatorname{op}}\to {\mathsf{Spaces}}$$.

• Interesting fact: ${\Omega}^\infty {\mathbb{S}}\cong ({\mathsf{FinSet}}, {\textstyle\coprod})^ {\operatorname{gp} }$.

• Note from Yuri Sulyma: "B is (widespread but) really bad notation for geometric realization. You should think of B as part of an equivalence \begin{align*} {\mathbf{B}}: \left\{{\text{monoidal categories}}\right\} \to \left\{{\text{pointed connected (2-)categories}}\right\} \end{align*}

up to the Quillen equivalence

\begin{align*} {\mathsf{Kan}}\xrightarrow{\sim} {\mathsf{Spaces}} \end{align*}

• geometric realization takes a (quasi-)category (or simplicial set) and inverts all the morphisms.

So $$M^ {\operatorname{gp} } = \Omega{ {\left\lvert {{\mathbf{B}}M} \right\rvert} }$$: you take $$M$$, deloop to turn the objects into morphisms, invert all the morphisms, then take loops to get your objects back."

• Theorem (Morel-Voevodsky): $$X\in {\mathsf{sm}}{\mathsf{Sch}_{/k}}$$ \begin{align*} K(X) \cong \mathop{\mathrm{Maps}}_{{\mathsf{Spaces}}(k)}(X, {\mathbb{Z}}\times{\operatorname{Gr}}_\infty) .\end{align*}

• Uses stratification of $${\operatorname{Gr}}$$ by affines, thanks Schubert calculus!

• There is an $${\mathbb{A}}^1$$ homotopy equivalence on affines: $$K \simeq{\mathbb{Z}}\times{\operatorname{Gr}}_\infty$$. Also, \begin{align*} {\operatorname{Betti}}(k) \simeq{\mathbb{Z}}\times{\mathbf{B}}{\operatorname{U}}= {\Omega}^\infty{\operatorname{KU}} .\end{align*}

• Theorem: Can replace $${\operatorname{Gr}}$$ with $$\operatorname{Hilb}_\infty({\mathbb{A}}^\infty)$$. Very singular!

• Definition: $$\operatorname{Hilb}_d({\mathbb{A}}^n)(T)$$ are maps $$Z\hookrightarrow{\mathbb{A}}^n\times T$$ over $$T$$ which are finite flat morphism of degree $$d$$ over $$T$$.

Morally: $$d{\hbox{-}}$$tuples of points in $${\mathbb{A}}^n$$.

• Representable! But $$\operatorname{Hilb}_\infty$$ is a colimit, thus an Ind scheme

• This says either the Hilbert scheme or K-theory is hard.

• In fact the theorem defines a map $${\operatorname{Gr}}_{d-1} \to \operatorname{Hilb}_d({\mathbb{A}}^\infty)$$ sending a vector space to the tangent space at 0, and proves this is an $${\mathbb{A}}^1{\hbox{-}}$$homotopy equivalence on affines.

• Sends subspace to thick point at zero.

• Thick point: point with a tangent direction.
• Burt’s proof worked!

• Grassmannian : parameterizes vector bundles with an embedding into $$\infty$$?

First step in proof: forget embedding into $${\mathbb{A}}^\infty$$, send $${ \mathsf{Vect} }_{d-1}$$ to finite flat schemes of degree $$d$$ $${\mathsf{FFlat}}_d(R)$$ over $$\operatorname{Spec}R$$, which are stacks.

• Send $$V\to R \oplus V$$, a square zero extension, add trivial multiplication.

• Inverse: take an algebra $$A\to A/R$$ by killing the unit.

• Not an equivalence of stacks! Since $$A\not\cong A/R \oplus R$$, but the surprising fact is $$A\to A/R \oplus R$$ is $${\mathbb{A}}^1$$ homotopic to the identity on $${\mathsf{FFlat}}(R)$$.

• Cook up an explicit homotopy: take the Rees algebra \begin{align*} {\operatorname{Rees}}(A) \coloneqq\left\{{ a_0 + a_1 t + \cdots {~\mathrel{\Big\vert}~}a_0\in R }\right\} \subseteq A[t] .\end{align*}

• $${\operatorname{Rees}}(A) / \left\langle{ t-1 }\right\rangle\simeq A$$

• $${\operatorname{Rees}}(A) / \left\langle{ t }\right\rangle\simeq R \oplus A/R$$.

• Some analogs of these theorems for:
• Hermitian K theory :
• Use orthogonal Grassmannian, take vector bundles with extra data of nondegenerate symmetric bilinear form. Need $$\operatorname{ch}k \neq 2$$. Take Gorenstein closed subschemes, which is extra data of orientation.
• Unsorted/Twisted K theory (WIP)
• Twisted with respect to an Azumaya algebra or Brauer class.

## Talbot, Mike Hill

• Mike was thinking about computing tmf at the prime $$p=3$$, since for $$p>3$$ it breaks up as a wedge of copies $${\operatorname{BP}}\left\langle{ 2 }\right\rangle$$ of Brown-Peterson spectra

Roughly twice as hard as computing K-theory with ku! (Wilson, Adams, Margalis)

• For $$p=2$$: an Adams spectral sequence (Mahowald, Davis-Mahowald) built out of \begin{align*} H^*( \mathrm{tmf} , {\mathbb{F}}_2) \cong A \otimes_{A(2) } {\mathbb{F}}_2 && \text{where } A(2) = \left\langle{ \operatorname{Sq}^1, \operatorname{Sq}^2, \operatorname{Sq}^4 }\right\rangle \end{align*}

• Cohomology of $$H{\mathbb{F}}_2$$ is the Steenrod algebra?
• Can compute $\operatorname{Ext}$, Brunner did this on a computer
• For $$p=3$$, heuristic: should be like ko at $$p=2$$ in terms of complexity.

• Also thinking about Hopkins-Miller higher real K theories.

• First Talbot: huge efforts by Norrah!!!

• Important for Talbot to be a safe space to not necessarily be an expert
• formal group laws over $$R$$: a power series $x +_F y \coloneqq F(x, y) \in R { \left[ {x, y} \right] }$ such that

• $$x +_F 0 = x$$
• $$x +_F y = y +_F x$$
• $$x+_F (y +_F z) = (x +_F y) +_F z$$.
• A morphism of formal group laws : $f\in R { \left[ {x} \right] }$ with $$f(x+_F y)= f(x) +_G f(y)$$.

• The functor $$R\to \mathsf{FGL}_{/R}$$ is representable, as is the functor sending $$R$$ to formal group laws over $$R$$ along with an isomorphism $$f$$ such that $$f'(0) = 1$$.

• Theorem (Quillen): $${\operatorname{MU}}_*$$ is the ring representing the first functor. See MU.

• Milnor showed $${\operatorname{MU}}_* = {\mathbb{Z}}[x_1, \cdots]$$.

• How to prove representability: take representing object for power series, check what the conditions translate to.

• $${\operatorname{MU}}_* {\operatorname{MU}}$$ represents the second factor (i.e. the $${\operatorname{MU}}_*$$ homology of $${\operatorname{MU}}$$, given by $$\pi_*({\operatorname{MU}}\wedge{\operatorname{MU}}))$$.

• Example: if $$n\in {\mathbb{N}}$$, then \begin{align*} [n]_F (x) = \overset{F}{\sum_{k\leq n}} x = nx + \cdots \end{align*} is an endomorphism of $$F$$.

• If $$\operatorname{ch}R = p$$, then $$[p]_F (x) = f(x^{p^n})$$, if $$f'(0) \in R^{\times}$$ then the $$\operatorname{ht}F=n$$ and $$f(x) = v_n x + \cdots$$.

• For $$R$$ a field, it’s a theorem that the formal group law is a complete invariant for algebraically closed fields.

• Having $$\operatorname{ht}\leq n$$ is a closed condition, since asking for $$v_{\leq n}$$ to vanish is a Zariski closed condition.
• Picture of the moduli of formal group laws : • How to glue: sheaf condition on opens? Extensions on closed sets? But how do you talk about gluing an open to a closed set?

• Explained by deformation theory : can push not only in direction in the space, but also into the tangent space directions.

• deformation : a ring map $$A\to k$$ with a nilpotent kernel:  • Here $$\widehat{G}$$ sends a ring $$R$$ to the set of nilpotent elements of $$R$$, and $$F$$ gives that a group structure – the algebraic geometry gadget corresponding to the formal group law $$F$$.

• Can obtain $$\widehat{G}$$ as a $$\colim \operatorname{Spec}k[x] / x^n$$, i.e. a formal version of the group scheme whose group law is given by $$F$$, so if $$F=x+y+xy$$ then $$\widehat{G}$$ is the formal completion of $${\mathbb{G}}_m$$ at the identity.
• Examples:

• $${\mathbb{Z}}/p^n\to {\mathbb{Z}}/p$$
• $${\mathbb{Z}}[{ {u}_1, {u}_2, \cdots, {u}_{k}} / \left\langle{ p, { {u}_1, {u}_2, \cdots, {u}_{k}} }\right\rangle^m \to {\mathbb{Z}}/p$$
• Theorem (Lubin-Tate): there is a universal deformation for $$(k, F)$$ given by \begin{align*} {\mathbb{W}}(k) { \left[ {{ {u}_1, {u}_2, \cdots, {u}_{n-1}}} \right] } \coloneqq E(k, F)_0 .\end{align*}

See Lubin-Tate theory and Witt vector.

• For $$k = {\mathbb{F}}_p$$, we have $${\mathbb{W}}(k) = { {\mathbb{Z}}_{\widehat{p}} }$$, and there is an action of $$\mathop{\mathrm{Aut}}(F)$$ and $${ \mathsf{Gal}} (k)$$.

• Theorem (Goerss-Hopkins-Miller): there is a canonical functor $$(k, F) \to E(k, F)$$ such that

• Even periodicity: $$\pi_{2m+1} E(k, F) = 0$$ and $$\pi_{2m+2} E(k, F) \cong \pi_{2m} E(k, F)$$
• $$\pi_0 E(k, F) = E(k, F)_0$$.
• This lifts the AG problem to a problem in commutative ring spectra.

• Theorem (Devinats-Hopkin?): the map \begin{align*} L_{K(n)} S^0 \xrightarrow{\simeq} E_n^{h{\mathbb{G}}_n} \end{align*} is an equivalence where \begin{align*} E_n &\coloneqq E({\mathbb{F}}_{p^n}, F_{\mathrm{Honda}}) \\ {\mathbb{G}}_n &\coloneqq{ \mathsf{Gal}} ({\mathbb{F}}_{p^n}) \rtimes S_n \\ S_n &\coloneqq\mathop{\mathrm{Aut}}(F_{\mathrm{Honda}}) .\end{align*}

• Define Hopkins-Miller higher real K theories : for $$G \subseteq S_n$$ finite, $${\mathsf{E} {\operatorname{O}}}_n(G) = E_n^{hG}$$.

• Example: for $$n=1, p=2$$ we have $${\mathsf{E} {\operatorname{O}}}_1(C_2) = {\operatorname{KO}}{ {}_{ \widehat{2} } }$$, which is completely understood via the homotopy fixed point spectral sequence.

• Example: for $$n=2$$, we know \begin{align*} {\mathsf{E} {\operatorname{O}}}_2(G) = L_{K(2)} \mathrm{tmf} \end{align*} or a summand thereof.

• Use Adams indexing, look at homotopy fixed point spectral sequence \begin{align*} H^s(G; \pi_t E_n)\Rightarrow\pi_{t-s} {\mathsf{E} {\operatorname{O}}}_n(G) .\end{align*}

• We know a lot about the bottom row: $$H^0(G; \pi_* E_n) = (\pi_* E_n)^G$$, using that group cohomology is the derived functor of $$G{\hbox{-}}$$invariants.

• If $${\sharp}G$$ is prime to $$p$$ then $$H^{> 0}(G, \pi* E_n) = 0$$.

• We don’t know much else about anything in this spectral sequence!

• Consider \begin{align*} {\mathcal{O}}_n \coloneqq{\mathbb{W}}({\mathbb{F}_{p^n}}) \left\langle{ s }\right\rangle / \left\langle{ sa = a^{ \varphi} S }\right\rangle \end{align*} where $$\varphi$$ is the Frobenius on $${\mathbb{F}_{p^n}}$$.

• It turns out that $$\mathop{\mathrm{Aut}}_(F_{\mathrm{\operatorname{Honda}}}) = {\mathcal{O}}_n^{\times}$$

This is the Dieudonne module for $$F_\mathrm{\operatorname{Honda}}$$?

• Power series rings have a maximal ideal $${\mathfrak{m}}$$, so consider \begin{align*} \pi_{-2} E_n / \left\langle{ p, {\mathfrak{m}}^2}\right\rangle .\end{align*}

• There is an $${\mathcal{O}}_n^{\times}$$ equivariant map from $${\mathcal{O}}_n/p$$ to this, and the question is how to lift: • Devinats-Hopkins: up to associated graded, \begin{align*} \pi_* E_n \cong \operatorname{Sym}({\mathcal{O}}_n) \left[ { \scriptstyle { {\Delta}^{-1}} } \right]{ {}_{ \widehat{I} } } .\end{align*}

• Use that the symmetric algebra is a free thing.

• Warning: can’t make this $$S_n$$ equivariant, so can’t compute the whole thing using this approach.

• “Up to associated graded” means there’s a spectral sequence relating.

• Mike thought about this a lot in grad school! But wound up doing his these on the first topic about $$H^*$$ of tmf.

• 2007 Talbot: Mike Hopkins was the faculty mentor.

• Theorem (Hill-Hopkins-Ravenel): there is a $$G{\hbox{-}}$$equivariant lift $${\mathcal{O}}_n \to \pi_{-2} E_n$$ for any finite $$G$$.

• Real K theories see a lot of $$\pi_* {\mathbb{S}}$$.

• Theorem (Ravenel): for $$p\geq 5$$, some element $$\beta_{p^i / p^i}$$ does not survive the ANSS.

• Use a map \begin{align*} S^0 &\to {\mathsf{E} {\operatorname{O}}}_{p-1}^{hC_p} \\ \beta_{p^i / p^i} &\mapsto \text{non-permanent cycles} .\end{align*}
• Exact same argument for Kervaire invariant 1 : $$p=2$$ version of this argument?