--- date: 2022-02-23 18:45 modification date: Wednesday 23rd February 2022 18:45:08 title: Formal group aliases: [Formal group, formal group, formal group law, FGL, formal group laws, formal groups] --- --- - Tags - #arithmetic-geometry #homotopy/stable-homotopy - Refs: - *Stable homotopy and generalised homology* #resources/books - *A. Frohlich, Formal groups* #resources/books - #resources/notes - In-depth: Hazewinkel's book *Formal Groups and Applications* #resources/books - #resources/notes - #resources/slides #resources/summaries - "The connections between [chromatic homotopy](chromatic%20homotopy%20theory.md) are key. Read Quillen's paper, J.F. Adams' blue book, Ravenel, etc. Ravenel has some slides on Quillen's work (good entry pt)." - Links: - [Dieudonne module](Dieudonne%20module.md) - [Hodge F-crystal](Hodge%20F-crystal.md) - [complex K theory](complex%20K%20theory.md) - [Lazard ring](Unsorted/Lazard%20ring.md) --- # Formal group ![](attachments/Pasted%20image%2020220508195254.png) ![](attachments/Pasted%20image%2020220508202029.png) ![](attachments/Pasted%20image%2020220508202046.png) # Formal group laws ![](attachments/Pasted%20image%2020220508194954.png) # In chromatic homotopy See [chromatic homotopy](Unsorted/chromatic%20homotopy%20theory.md): - Start with the universal complex line bundle $\mcl \downto \B\GL_1(\CC)\homotopic \CP^\infty$. - Tensoring bundles induces operations: $\mcl_1 \tensor \mcl_1 \leadsto \CP^\infty \times \CP^\infty \to \CP^\infty$. - Note $H_*(\CP^\infty; \ZZ) \cong \ZZ\fps{t}$ for $t= c_1$ the first [Chern class](Chern%20class.md) and $H^*(\CP^\infty\cartpower{2}) \cong \ZZ\fps{x, y}$, so applying homology to the product map yields $H^*(\CP^\infty; \ZZ) \to H^*(\CP^\infty\cartpower 2; \ZZ)$ which is entirely determined by an assignment $t\mapsto F(x, y)$. - Since $c_1(\mcl_1 \tensor \mcl_2) = c_1 \mcl_1 + c_1\mcl_2$, this forces $F(x,y) = x + y$. - Generalizing: a cohomology theory $E$ is **complex oriented** iff $E^*(\CP^\infty) \cong E_*\fps{t} \da E^*(\pt)\fps{t}$. - Tensoring similarly induces maps $E_*\fps{t} \to E_*\fps{x, y}$ where $t\mapsto F_E(x, y)$. - Properties that define a 1-dimensional commutative **formal group law**: - $\mcl \tensor \ul{\CC} \cong \mcl \implies F(x, 0) = x$ - $\mcl_1 \tensor \mcl_2 \cong \mcl_2 \tensor \mcl_1 \implies F(x,y) = F(y, x)$. - Associativity of tensoring $\implies F(F(x,y), z) = F(x, F(y, z))$ - Note that these axioms guarantee that $F_E(x, y) = x + y + \bigo(x^2, xy, y^2)$. - Idea: a formal group is a germ of an algebraic group. E.g. $F(x, y) \da x+y \leadsto \hat{\GG_a}$, the germ of $\GG_a$. - Example: the FGL of [complex K theory](Unsorted/complex%20K%20theory.md) is $\hat{\GG_m}$, the germ of $\GG_m$. - $E \mapsto F_E(x, y)$ yields a functor from complex-oriented cohomology theories to FGLs, whose partial inverse is given by [Landweber exactness](Unsorted/Landweber%20exactness.md). - Defining **heights**: given $F\in R\fps{x, y}$, define an $n\dash$series inductively by $[1]_F(x) \da x$ and $[n]_F(x) \da F(x, [n-1]_F(x))$. - In short: $[n]_F(x) = x +_F x +_F + \cdots +_F x$ where $a+_F b \da F(a, b)$. - This yields $[n]_F(x) = nx + \bigo(x^2, xy, y^2)$. - If $\characteristic R = p$ then $[p]_F(x) = \bigo(x^2, xy, y^2)$. - In general, $[p]_F(x) = ux^{p^n} + \bigo(x^{p^n + 1}, y^{p^n+1}, \cdots)$ for some $u\in R\units$, so define $n$ to be the **height**. - Over $\FF_p$, for every height $n$ define the **Honda FGL** $H_{n, p}$ whose $p\dash$series is $[p](x) = x^{p^n}$. - This corresponds to the $n$th [Morava K theory](Unsorted/Morava%20K%20theory.md) $\K(n, p)$. - Examples: - $H_{1, p} \leadsto \hat{\GG_m}(\FF_p)$ and $\K(1, p) = \KU/p$. - $H_{\infty, p} \leadsto \hat{\GG_a}(\FF_p)$ and $\K(\infty, p) = \mathsf{H}\FF_p$ - Define $H_{0, p}$ to identify with $\mathsf{H}\QQ$. - Deformations: - Let $k$ be a perfect field of characteristic $p$, let $F$ be a formal group law over $k$, and let $(A, \mathfrak{m})$ be a complete local ring with projection $A \stackrel{\pi}{\rightarrow} A / \mathfrak{m}$ to its residue field. A **deformation of $F$ from $k$ to $A$** is a formal group law $\bar{F}$ over $A$ and a map $k \stackrel{i}{\rightarrow} A / \mathfrak{m}$ such that $\pi^{*} \bar{F}=i^{*} F$ over $A / \mathfrak{m}$. - Form a set $\Def_{F\slice k}(A)$ of deformations of $F$ from $k$ to $A$ and a functor $\Def_{F\slice k}(\wait):\CRing\to \Set$. This is representable by the **Lubin-Tate ring** $\LT_{F\slice k}$, so $\Def_{F\slice k}(\wait) \cong \Top\CRing^\loc(\LT_{F\slice k}, \wait)$. - There is a **universal deformation** $\tilde F \downto \LT_{F\slice k}$. - The universal deformations of the Honda formal group laws, $\tilde H_{n, p}$ correspond to [Morava E theory](Unsorted/Morava%20E%20theory.md) $E_{n, p}$, which live over the [Lubin-Tate](Unsorted/Lubin-Tate%20theory.md) rings $\LT_{ H_{n, p} {}\slice{\FF_p} }= \ZZpadic\fps{u_1,\cdots, u_{n-1}}$. # Notes ![](attachments/Pasted%20image%2020220408133756.png) - From Tyler Genao: for $(R, \mfm)$ is a complete [local ring](local%20ring), can plug $\mfm$ into a formal group law to construct a group whose underlying set is $\mfm$ - Makes the power series converge. ![](attachments/Pasted%20image%2020220505161428.png) ![](attachments/Pasted%20image%2020220508195431.png) # Examples ![](attachments/Pasted%20image%2020220317205027.png) ![](attachments/Pasted%20image%2020220408133909.png) ![](attachments/Pasted%20image%2020220408133948.png) ![](attachments/Pasted%20image%2020220408134041.png) ![](attachments/Pasted%20image%2020220408134120.png) ![](attachments/Pasted%20image%2020220508195054.png) ## Applications Constructing the [maximal abelian extension](maximal%20abelian%20extension) of a [local field](Unsorted/global%20field.md): ![](attachments/Pasted%20image%2020220408134241.png) ![](attachments/Pasted%20image%2020220408134249.png) # Height #todo