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

Tags: #stable-homotopy #projects/notes/seminars

  • 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 :

attachments/image_2021-06-06-15-32-17.png

  • 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:

    attachments/image_2021-06-06-15-35-33.png

    attachments/image_2021-06-06-15-37-51.png

  • 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:

attachments/image_2021-06-06-15-58-57.png

  • 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?

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