--- date: 2021-06-06 tags: [ web/quick-notes ] --- # 2021-06-06 ## 12:12 Refs: [A1 Homotopy](A1%20Homotopy.md) - Context: the infinity category of spaces, i.e. [homotopy types](homotopy%20types) - Take smooth manifolds, take the [Yoneda embedding](Yoneda%20embedding) to $\Presh(\smooth\Mfd)$: these satisfy a Mayer-Vietoris gluing property, and homotopy invariance in the sense that \[ \Hom(\wait, X) \cong \Hom(\wait \cross I, X) .\] - Why the first argument: homotopy invariance *as a presheaf* - AG setting: $\Presh(\smooth\Sch_{/k})$, send to presheaves to define [motivic spaces](motivic%20spaces). - Satisfies a Nisnevich gluing condition and $\AA^1$ invariance Similar homotopy invariance: $F(\AA^1 \cross X)\cong F(X)$. - See [Betti realization](Betti%20realization) for $k=\CC$: $\smooth\Sch_{/\CC}\to \Spaces$ where $X\mapsto X(\CC)$. - From topology: identify $\B \U_n(\CC) = \Gr_n(\CC)$ to get \[ \Vect_{/\CC}^{\rank = n}(U) \cong \pi_0 \Maps(U, \B \U_n(\CC)) .\] - Problem in AG: there are two rank 2 [vector bundles](vector%20bundles.md) on $\PP^1 \cross \AA^1$ whose fibers over 0 and 1 are $\OO^2$ and $\OO(1) \oplus \OO(-1)$. - **Theorem**: for $U$ smooth affine $\kSch$, there is an equivalence of rank $n$ vector bundles on $U$ mod equivalence to $\pi_0 \Maps_{\Spaces(k)}(U, \B \GL_n)$ where again $\B\GL_n \cong \Gr_n$. - Would like this for non-smooth non-affine [schemes](schemes)? - [algebraic K theory](algebraic%20K%20theory) : finitely generated projective \(R\dash\)modules mod equivalence with $\oplus$, then take group completion to get $K_0(R)$. - $K_0(k) \cong \NN^\gp \cong \ZZ$. - To get a space: take $K(R) \da \Proj(\rmod)^\gp$ to get a space, set $K_i R \da \pi_i K(R)$ - Prop: the $K$ theory space here is a motivic space, $K: \smooth\Sch_{/k}\op \to \Spaces$. - Interesting fact: $\Loop^\infty \SS \cong (\Finset, \disjoint)^\gp$. - **Note from Yuri Sulyma:** "B is (widespread but) really bad notation for geometric realization. You should think of B as part of an equivalence \[ \B: \ts{\text{monoidal categories}} \to \ts{\text{pointed connected (2-)categories}} \] up to the [Quillen equivalence](Quillen%20equivalence.md) \[ \Kan \mapsvia{\sim} \Spaces \] - [geometric realization](geometric%20realization.md) takes a (quasi-)category (or [simplicial set](simplicial%20set.md)) and inverts all the morphisms. So $M^\gp = \Omega\realize{\B M}$: you take $M$, [deloop](deloop) to turn the objects into morphisms, invert all the morphisms, then take loops to get your objects back." - **Theorem** ([Morel-Voevodsky](Morel-Voevodsky)): $X\in \smooth\kSch$ \[ K(X) \cong \Maps_{\Spaces(k)}(X, \ZZ \cross \Gr_\infty) .\] - Uses [stratification](stratification) of $\Gr$ by affines, thanks [Schubert calculus](Schubert%20calculus)! - There is an $\AA^1$ homotopy equivalence on affines: $K \homotopic \ZZ \cross \Gr_\infty$. Also, \[ \Betti(k) \homotopic \ZZ \cross \BU = \Loopinf \KU .\] - **Theorem**: Can replace $\Gr$ with $\Hilb_\infty(\AA^\infty)$. Very singular! - **Definition**: $\Hilb_d(\AA^n)(T)$ are maps $Z\injects \AA^n\cross T$ over $T$ which are finite [flat morphism](Unsorted/faithfully%20flat.md) of degree $d$ over $T$. Morally: $d\dash$tuples of points in $\AA^n$. - Representable! But $\Hilb_\infty$ is a colimit, thus an [Ind scheme](Unsorted/Ind%20objects.md) - This says either the [Hilbert scheme](Hilbert%20scheme.md) or [K-theory](Unsorted/K-theory.md) is hard. - In fact the theorem defines a map $\Gr_{d-1} \to \Hilb_d(\AA^\infty)$ sending a vector space to the tangent space at 0, and proves this is an $\AA^1\dash$homotopy equivalence on affines. - Sends subspace to thick point at zero. - Thick point: point with a tangent direction. - Burt's proof worked! - [Grassmannian](Grassmannian.md) : parameterizes vector bundles with an embedding into $\infty$? - Cool fact for manifolds: $\Emb(M, \RR^\infty)$ is contractible! - [stacks](Unsorted/stacks%20MOC.md): presheaves of [groupoids](groupoids)? See [a stack is a category fibered in groupoids](Unsorted/a%20stack%20is%20a%20category%20fibered%20in%20groupoids.md). :::{.proof title="?"} First step in proof: forget embedding into $\AA^\infty$, send $\Vect_{d-1}$ to finite flat schemes of degree $d$ $\FFlat_d(R)$ over $\spec R$, which are stacks. - Send $V\to R \oplus V$, a [square zero extension](square%20zero%20extension), 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 $\AA^1$ homotopic to the identity on $\FFlat(R)$. - - Cook up an explicit homotopy: take the [Rees algebra](Rees%20algebra) \[ \Rees(A) \da \ts{ a_0 + a_1 t + \cdots \st a_0\in R } \subseteq A[t] .\] - $\Rees(A) / \gens{ t-1 }\homotopic A$ - $\Rees(A) / \gens{ t }\homotopic R \oplus A/R$. ::: - Some analogs of these theorems for: - [Hermitian K theory](Hermitian%20K%20theory) : - Use orthogonal Grassmannian, take vector bundles with extra data of nondegenerate symmetric [bilinear form](Unsorted/quadratic%20form.md). Need $\characteristic k \neq 2$. Take [Gorenstein](Gorenstein.md) closed subschemes, which is extra data of orientation. - [Unsorted/Twisted K theory](Unsorted/Twisted%20K%20theory.md) (WIP) - Twisted with respect to an [Azumaya algebra](Azumaya%20algebra) or [Brauer class](Brauer%20class). ## Talbot, Mike Hill Tags: #homotopy/stable-homotopy #projects/notes/seminars - Mike was thinking about computing [tmf](tmf) at the prime $p=3$, since for $p>3$ it breaks up as a wedge of copies $\BP \gens{ 2 }$ of [Brown-Peterson spectra](Brown-Peterson%20spectra.md) Roughly twice as hard as computing [K-theory](Unsorted/K-theory.md) with [ku](ku)! (Wilson, Adams, Margalis) - For $p=2$: an [Adams spectral sequence](Archive/0200_Stable%20Homotopy%20Seminar%202021/Adams%20Sseq.md) (Mahowald, Davis-Mahowald) built out of \[ H^*(\tmf, \FF_2) \cong A \tensor_{A(2) } \FF_2 && \text{where } A(2) = \gens{ \Sq^1, \Sq^2, \Sq^4 } \] - Cohomology of $H\FF_2$ is the [Steenrod algebra](Steenrod%20algebra.md)? - Can compute $\Ext$, Brunner did this on a computer - For $p=3$, heuristic: should be like [ko](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](Formal%20group.md) over $R$: a power series $x +_F y \da F(x, y) \in R\powerseries{x, y}$ 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](Formal%20group.md) : $f\in R\powerseries{x}$ with $f(x+_F y)= f(x) +_G f(y)$. - The functor $R\to \FGL_{/R}$ 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): $\MU_*$ is the ring representing the first functor. See [MU](MU). - Milnor showed $\MU_* = \ZZ[x_1, \cdots]$. - How to prove representability: take representing object for power series, check what the conditions translate to. - $\MU_* \MU$ represents the second factor (i.e. the $\MU_*$ homology of $\MU$, given by $\pi_*(\MU \smashprod \MU))$. - Example: if $n\in \NN$, then \[ [n]_F (x) = \overset{F}{\sum_{k\leq n}} x = nx + \cdots \] is an endomorphism of $F$. - If $\characteristic R = p$, then $[p]_F (x) = f(x^{p^n})$, if $f'(0) \in R\units$ then the $\height F=n$ and $f(x) = v_n x + \cdots$. - For $R$ a field, it's a theorem that the [formal group law](Unsorted/Formal%20group.md) is a complete invariant for algebraically closed fields. - Having $\height \leq n$ 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](moduli%20of%20formal%20group%20laws.md) : ![](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](deformation%20theory.md) : can push not only in direction in the space, but also into the tangent space directions. - [deformation](deformation.md) : 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 $\hat{G}$ 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 $\hat{G}$ as a $\colim \spec k[x] / x^n$, i.e. a formal version of the [group scheme](group%20scheme.md) whose group law is given by $F$, so if $F=x+y+xy$ then $\hat{G}$ is the [formal completion](formal%20completion) of $\GG_m$ at the identity. - Examples: - $\ZZ/p^n\to \ZZ/p$ - $\ZZ[\elts{u}{k} / \gens{ p, \elts{u}{k} }^m \to \ZZ/p$ - **Theorem** (Lubin-Tate): there is a universal deformation for $(k, F)$ given by \[ \Wittvectors(k)\powerseries{\elts{u}{n-1}} \da E(k, F)_0 .\] See [Lubin-Tate theory](Lubin-Tate%20theory.md) and [Witt vector](Archive/AWS2019/Witt%20vectors.md). - For $k = \FF_p$, we have $\Wittvectors(k) = \ZZpadic$, and there is an action of $\Aut(F)$ and $\Gal(k)$. - **Theorem** ([Goerss-Hopkins-Miller](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](ring%20spectra.md). - **Theorem** (Devinats-Hopkin?): the map \[ L_{K(n)} S^0 \mapsvia{\simeq} E_n^{h\GG_n} \] is an equivalence where \[ E_n &\da E(\FF_{p^n}, F_{\mathrm{Honda}}) \\ \GG_n &\da \Gal(\FF_{p^n}) \semidirect S_n \\ S_n &\da \Aut(F_{\mathrm{Honda}}) .\] - Define [Hopkins-Miller higher real K theories](Hopkins-Miller%20higher%20real%20K%20theories) : for $G \subseteq S_n$ finite, $\EO_n(G) = E_n^{hG}$. - Example: for $n=1, p=2$ we have $\EO_1(C_2) = \KO\complete{2}$, which is completely understood via the [homotopy fixed point spectral sequence](homotopy%20fixed%20point%20spectral%20sequence). - Example: for $n=2$, we know \[ \EO_2(G) = L_{K(2)} \tmf \] or a summand thereof. - Use Adams indexing, look at homotopy fixed point spectral sequence \[ H^s(G; \pi_t E_n)\abuts \pi_{t-s} \EO_n(G) .\] - We know a lot about the bottom row: $H^0(G; \pi_* E_n) = (\pi_* E_n)^G$, using that [group cohomology](group%20cohomology.md) is the derived functor of $G\dash$invariants. - If $\size G$ 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 \[ \OO_n \da \Wittvectors(\FFpn) \gens{ s } / \gens{ sa = a^{ \varphi} S } \] where \( \varphi \) is the Frobenius on $\FFpn$. - It turns out that $\Aut_(F_{\Honda}) = \OO_n\units$ This is the [Dieudonne module](Dieudonne%20module.md) for $F_\Honda$? - Power series rings have a maximal ideal $\mfm$, so consider \[ \pi_{-2} E_n / \gens{ p, \mfm^2} .\] - There is an $\OO_n\units$ [equivariant](equivariant.md) map from $\OO_n/p$ to this, and the question is how to lift: ![](attachments/image_2021-06-06-15-58-57.png) - Devinats-Hopkins: up to [associated graded](associated%20graded), \[ \pi_* E_n \cong \Sym(\OO_n) \localize{\Delta}\complete{I} .\] - Use that the symmetric algebra is a free thing. - Warning: can't make this $S_n$ [equivariant](equivariant.md), 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](tmf.md). - 2007 Talbot: Mike Hopkins was the faculty mentor. - **Theorem** ([Hill-Hopkins-Ravenel](Hill-Hopkins-Ravenel.md)): there is a $G\dash$equivariant lift $\OO_n \to \pi_{-2} E_n$ for any *finite* $G$. - Real K theories see a lot of $\pi_* \SS$. - **Theorem** (Ravenel): for $p\geq 5$, some element $\beta_{p^i / p^i}$ does not survive the [ANSS](ANSS). - Use a map \[ S^0 &\to \EO_{p-1}^{hC_p} \\ \beta_{p^i / p^i} &\mapsto \text{non-permanent cycles} .\] - Exact same argument for [Kervaire invariant 1](Kervaire%20invariant%201.md) : $p=2$ version of this argument?