- Tags
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Refs:
- Stable homotopy and generalised homology #resources/books
- A. Frohlich, Formal groups #resources/books
- https://etale.site/xkcd/dieudonne-modules.pdf #resources/notes
- In-depth: Hazewinkel’s book Formal Groups and Applications #resources/books
- http://www-personal.umich.edu/~alimil/223anotes.pdf#page=82 #resources/notes
- https://etale.site/writing/ustars-tmf-beamer.pdf#page=103 #resources/summaries
- “The connections between chromatic homotopy are key. Read Quillen’s paper, J.F. Adams’ blue book, Ravenel, etc. Ravenel has some slides on Quillen’s work (good entry pt).”
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Links:
- Dieudonne module
- Hodge F-crystal
- complex K theory
- Lazard ring
Formal group
Formal group laws
In chromatic homotopy
See chromatic homotopy:
- Start with the universal complex line bundle \({\mathcal{L}}\downto {\mathbf{B}}\operatorname{GL}_1({\mathbf{C}})\simeq{\mathbf{CP}}^\infty\).
- Tensoring bundles induces operations: \({\mathcal{L}}_1 \otimes{\mathcal{L}}_1 \leadsto {\mathbf{CP}}^\infty \times {\mathbf{CP}}^\infty \to {\mathbf{CP}}^\infty\).
- Note \(H_*({\mathbf{CP}}^\infty; {\mathbf{Z}}) \cong {\mathbf{Z}}{\left[\left[ t \right]\right] }\) for \(t= c_1\) the first Chern class and \(H^*({\mathbf{CP}}^\infty{ {}^{ \scriptscriptstyle\times^{2} } }) \cong {\mathbf{Z}}{\left[\left[ x, y \right]\right] }\), so applying homology to the product map yields \(H^*({\mathbf{CP}}^\infty; {\mathbf{Z}}) \to H^*({\mathbf{CP}}^\infty{ {}^{ \scriptscriptstyle\times^{2} } }; {\mathbf{Z}})\) which is entirely determined by an assignment \(t\mapsto F(x, y)\).
- Since \(c_1({\mathcal{L}}_1 \otimes{\mathcal{L}}_2) = c_1 {\mathcal{L}}_1 + c_1{\mathcal{L}}_2\), this forces \(F(x,y) = x + y\).
- Generalizing: a cohomology theory \(E\) is complex oriented iff \(E^*({\mathbf{CP}}^\infty) \cong E_*{\left[\left[ t \right]\right] } \coloneqq E^*({\operatorname{pt}}){\left[\left[ t \right]\right] }\).
- Tensoring similarly induces maps \(E_*{\left[\left[ t \right]\right] } \to E_*{\left[\left[ x, y \right]\right] }\) where \(t\mapsto F_E(x, y)\).
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Properties that define a 1-dimensional commutative formal group law:
- \({\mathcal{L}}\otimes\underline{{\mathbf{C}}} \cong {\mathcal{L}}\implies F(x, 0) = x\)
- \({\mathcal{L}}_1 \otimes{\mathcal{L}}_2 \cong {\mathcal{L}}_2 \otimes{\mathcal{L}}_1 \implies F(x,y) = F(y, x)\).
- Associativity of tensoring \(\implies F(F(x,y), z) = F(x, F(y, z))\)
- Note that these axioms guarantee that \(F_E(x, y) = x + y + { \mathsf{O}} (x^2, xy, y^2)\).
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Idea: a formal group is a germ of an algebraic group. E.g. \(F(x, y) \coloneqq x+y \leadsto \widehat{{\mathbf{G}}_a}\), the germ of \({\mathbf{G}}_a\).
- Example: the FGL of complex K theory is \(\widehat{{\mathbf{G}}_m}\), the germ of \({\mathbf{G}}_m\).
- \(E \mapsto F_E(x, y)\) yields a functor from complex-oriented cohomology theories to FGLs, whose partial inverse is given by Landweber exactness.
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Defining heights: given \(F\in R{\left[\left[ x, y \right]\right] }\), define an \(n{\hbox{-}}\)series inductively by \([1]_F(x) \coloneqq x\) and \([n]_F(x) \coloneqq F(x, [n-1]_F(x))\).
- In short: \([n]_F(x) = x +_F x +_F + \cdots +_F x\) where \(a+_F b \coloneqq F(a, b)\).
- This yields \([n]_F(x) = nx + { \mathsf{O}} (x^2, xy, y^2)\).
- If \(\operatorname{ch}R = p\) then \([p]_F(x) = { \mathsf{O}} (x^2, xy, y^2)\).
- In general, \([p]_F(x) = ux^{p^n} + { \mathsf{O}} (x^{p^n + 1}, y^{p^n+1}, \cdots)\) for some \(u\in R^{\times}\), so define \(n\) to be the height.
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Over \({ \mathbf{F} }_p\), for every height \(n\) define the Honda FGL \(H_{n, p}\) whose \(p{\hbox{-}}\)series is \([p](x) = x^{p^n}\).
- This corresponds to the \(n\)th Morava K theory \({\mathsf{K}}(n, p)\).
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Examples:
- \(H_{1, p} \leadsto \widehat{{\mathbf{G}}_m}({ \mathbf{F} }_p)\) and \({\mathsf{K}}(1, p) = {\operatorname{KU}}/p\).
- \(H_{\infty, p} \leadsto \widehat{{\mathbf{G}}_a}({ \mathbf{F} }_p)\) and \({\mathsf{K}}(\infty, p) = \mathsf{H}{ \mathbf{F} }_p\)
- Define \(H_{0, p}\) to identify with \(\mathsf{H}{\mathbf{Q}}\).
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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 \(\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu\) over \(A\) and a map \(k \stackrel{i}{\rightarrow} A / \mathfrak{m}\) such that \(\pi^{*} \mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu=i^{*} F\) over \(A / \mathfrak{m}\).
- Form a set \(\operatorname{Def} _{F_{/ {k}} }(A)\) of deformations of \(F\) from \(k\) to \(A\) and a functor \(\operatorname{Def} _{F_{/ {k}} }({-}):\mathsf{CRing}\to {\mathsf{Set}}\). This is representable by the Lubin-Tate ring \(\LT_{F_{/ {k}} }\), so \(\operatorname{Def} _{F_{/ {k}} }({-}) \cong {\mathsf{Top}}\mathsf{CRing}^{\mathsf{loc}}(\LT_{F_{/ {k}} }, {-})\).
- There is a universal deformation \(\tilde F \downto \LT_{F_{/ {k}} }\).
- The universal deformations of the Honda formal group laws, \(\tilde H_{n, p}\) correspond to Morava E theory \(E_{n, p}\), which live over the Lubin-Tate rings \(\LT_{ H_{n, p} {}_{/ {{ \mathbf{F} }_p}} }= { {\mathbf{Z}}_{\widehat{p}} }{\left[\left[ u_1,\cdots, u_{n-1} \right]\right] }\).
Notes
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From Tyler Genao: for \((R, {\mathfrak{m}})\) is a complete local ring, can plug \({\mathfrak{m}}\) into a formal group law to construct a group whose underlying set is \({\mathfrak{m}}\)
- Makes the power series converge.
Examples
Applications
Constructing the maximal abelian extension of a local field: 
