--- date: 2022-02-23 18:45 modification date: Friday 18th March 2022 15:19:48 title: chromatic homotopy theory aliases: [chromatic homotopy theory, "chromatic homotopy", "chromatic"] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #homotopy/stable-homotopy - Refs: - Jacob Lurie. [Chromatic homotopy theory](https://www.google.com/url?q=https%3A%2F%2Fpeople.math.harvard.edu%2F~lurie%2F252x.html&sa=D&sntz=1&usg=AOvVaw3tcvIFbapx5o2faAuOarsa). #resources/notes/lectures - An overview of the lecture titles can be found [here](https://drive.google.com/file/d/1LXzy9Vb1XiYkPPMLPvlXpUVQ5-KwnN5G/view?usp=sharing) and possible elucidations of the material can be found in the [lecture notes](https://7df51482-a-62cb3a1a-s-sites.googlegroups.com/site/chrisschommerpriesmath/Home/course-notes-and-materials/Chromatic.pdf?attachauth=ANoY7crJ272qNkzcGC1gzHir0usuNZJLDKc-RRkY74_UC_1yiTa-kcJBw3-Qa8hIWkLN79yE_C8TOKLeO2VBcD_silPCCm10IjdriZ89IdLSiS3DO8A7Rwl4DP-13D7FZou5m27QRIrFff1QtqkzPNcldksUC9Ssu2U7ZqBsxggG0mH-I_rGnd1MontKUjJLH3LvGYClmd9cdwGX2Fb6PIrSy41iHaPJn3W1KuJdVljuWRHljvWdA_83RyBilVTE5mMVh_1PppxvDWGj-vaAz09VUDW9k8x9ww%3D%3D&attredirects=0) - [ ] [Lecture 1.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture1.pdf) #projects/to-read - [ ] [Lecture 2.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture2.pdf) - [ ] [Lecture 3.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture3.pdf) - [ ] [Lecture 4.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf) - [ ] [Lecture 5.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf) - [ ] [Lecture 6.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf) - [ ] [Lecture 7.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf) - [ ] [Lecture 8.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf) - [ ] [Lecture 9.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture9.pdf) - [ ] [Lecture 10.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture10.pdf) - [ ] [Lecture 11.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf) - [ ] [Lecture 12.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture12.pdf) - [ ] [Lecture 13.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture13.pdf) - [ ] [Lecture 14.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture14.pdf) - [ ] [Lecture 15.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture15.pdf) - [ ] [Lecture 16.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture16.pdf) - [ ] [Lecture 17.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture17.pdf) - [ ] [Lecture 18.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture18.pdf) - [ ] [Lecture 19.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture19.pdf) - [ ] [Lecture 20.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf) - [ ] [Lecture 21.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf) - [ ] [Lecture 22.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf) - [ ] [Lecture 23.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf) - [ ] [Lecture 24.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf) - [ ] [Lecture 25.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf) - [ ] [Lecture 26.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture26.pdf) - [ ] [Lecture 27.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture27.pdf) - [ ] [Lecture 28.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture28.pdf) - [ ] [Lecture 29.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture29.pdf) - [ ] [Lecture 30.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf) - [ ] [Lecture 31.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture31.pdf) - [ ] [Lecture 32.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture32.pdf) - [ ] [Lecture 33.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture33.pdf) - [ ] [Lecture 34.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture34.pdf) - [ ] [Lecture 35.](https://people.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf) - Piotr Pstrągowski. [Finite height chromatic homotopy theory](https://www.google.com/url?q=https%3A%2F%2Fpeople.math.harvard.edu%2F~piotr%2F252y_notes.pdf&sa=D&sntz=1&usg=AOvVaw28yulKMtqhPUjwMgns_AFR). #resources/notes/lectures - Lennart Meier. [Elliptic homology and topological modular forms](https://www.google.com/url?q=https%3A%2F%2Fwebspace.science.uu.nl%2F~meier007%2FTMF-Lecture.pdf&sa=D&sntz=1&usg=AOvVaw3uTZCN-Jql01pKX_bBJA1E). #resources/notes/lectures - Michael Hopkins. [Complex oriented cohomology theories and the language of stacks](https://www.google.com/url?q=https%3A%2F%2Fpeople.math.rochester.edu%2Ffaculty%2Fdoug%2Fotherpapers%2Fcoctalos.pdf&sa=D&sntz=1&usg=AOvVaw0vGGV3D7ErRwaOE6xi5ISQ). #resources/notes/lectures - Sanath Devalapurkar. [Chromatic homotopy theory](https://www.google.com/url?q=https%3A%2F%2Fsanathdevalapurkar.github.io%2Ffiles%2Fiap-2018.pdf&sa=D&sntz=1&usg=AOvVaw0R7AliDJr6CsVIDNzgDT3w). #resources/notes/lectures - [Behrens Hopkins Hill](https://www3.nd.edu/~mbehren1/papers/exotic2.pdf) #resources/papers - A lot of references in the bibliography here: #resources/summaries - Links: - [stable homotopy](Unsorted/stable%20homotopy.md) - [phantom maps](phantom%20maps.md) - [chromatic homotopy theory](chromatic%20homotopy%20theory.md) - [Fracture theorem](Fracture%20theorem) - The chromatic spectral sequence - [Nishida's Theorem](Nishida's%20Theorem.md) - [Bousfield localization](Bousfield%20localization.md) - [Morava stabilizer group](Morava%20stabilizer%20group) - [Morava E theory](Morava%20E%20theory.md) - [Formal group](Formal%20group.md) - [Formal group](Formal%20group.md) - [Lubin-Tate space](Lubin-Tate%20space) - [Lubin-Tate theory](Lubin-Tate%20theory.md) - [Kervaire invariant 1](Kervaire%20invariant%201.md) - [topological modular forms](topological%20modular%20forms) - [Dieudonne module](Dieudonne%20module.md) - [Redshift](Redshift) - [K-theory](Unsorted/K-theory.md) - [p-divisible group](p-divisible%20group) - [Balmer spectrum](Unsorted/tensor%20triangulated%20category.md) --- # chromatic homotopy theory See [tensor triangular geometry](Unsorted/tensor%20triangulated%20category.md). ![](attachments/Pasted%20image%2020220508204755.png) ![](attachments/Pasted%20image%2020220419150128.png) The chromatic view-point, which studies stable homotopy theory via its relationships to the moduli of [formal groups](Unsorted/Formal%20group.md), and related topics such as topological modular forms, use a sizable amount of (fairly abstract) algebraic geometry. And Lurie's work on [derived algebraic geometry](derived%20algebraic%20geometry.md) was motivated in part by establishing foundations adequate to the task of defining equivariant forms of [topological modular forms](topological%20modular%20forms). Kervaire and Milnor defined $\Theta_n$ to be the group of [homotopy spheres](homotopy%20spheres) up to [h-cobordism](h-cobordism.md) (where the group operation is given by connect sum). By the [h-cobordism theorem](h-cobordism%20theorem) ($n > 4$) and Perelman’s proof of the [Unsorted/Poincare conjectures](Unsorted/Poincare%20conjectures.md) ($n = 3$). For $n \neq 4$, $\Theta_n = 0$ if and only if $S^n$ has a unique [smooth structure](smooth%20structure.md) (i.e. there are no [exotic spheres](exotic%20spheres) of dimension $n$). We wish to consider the following question: For which $n$ is $\Theta_n = 0$? The general belief is that there should be finitely many such $n$, and these n should be concentrated in relatively low dimensions. Relation to [complex cobordism](Unsorted/complex%20bordism.md), [complex oriented cohomology theories](Unsorted/complex%20oriented%20cohomology%20theory.md), and [FGLs](Unsorted/Formal%20group.md) via the [Landweber exact functor theorem](Unsorted/Landweber%20exactness.md). ![](attachments/Pasted%20image%2020220508194119.png) See [Morava E theory](Unsorted/Morava%20E%20theory.md), [Morava K theory](Unsorted/Morava%20K%20theory.md), [thick subcategory](Unsorted/thick%20subcategory.md), [tmf](Unsorted/tmf.md). Relation to [stacks](Unsorted/stacks%20MOC.md): ![](attachments/Pasted%20image%2020220508194333.png) ## The chromatic tower Fix a prime $p$ the **chromatic tower** of a [Thom spectrum](Unsorted/Thom%20space.md) $X$ is the tower of [Bousfield localization](Unsorted/Bousfield%20localization.md): $$ X \rightarrow \cdots \rightarrow X _ { E ( n ) } \rightarrow X _ { E ( n - 1 ) } \rightarrow \cdots \rightarrow X _ { E ( 0 ) } $$ where $E(n)$ is the $n$th [Johnson-Wilson spectrum](Johnson-Wilson%20spectrum) $(E(0) = \mH \QQ$, by convention) with $$ E ( n ) _ { * } = \mathbb { Z } _ { ( p ) } \left[ v _ { 1 } , \dots , v _ { n - 1 } , v _ { n } ^ { \pm } \right] $$ The fibers of the chromatic tower $$ M _ { n } X \rightarrow X _ { E ( n ) } \rightarrow X _ { E ( n - 1 ) } $$ are called the monochromatic layers. The [spectral sequence](spectral%20sequence.md) associated to the chromatic tower is the **chromatic spectral sequence** $$ E _ { 1 } ^ { n , * } = \pi _ { * } M _ { n } X \Rightarrow \pi _ { * } X\plocal $$ Let $M_\ell$ denote the [DM moduli stack of elliptic curves](Unsorted/moduli%20stack%20of%20elliptic%20curves.md) over $\spec(\ZZ)$. For a commutative ring $R$, the [groupoid](groupoid.md) of $R$-points of $M_\ell$ is the groupoid of elliptic curves over $R$. This [Unsorted/stacks MOC](Unsorted/stacks%20MOC.md) carries a [line bundle](line%20bundle.md) $\omega$ where for an elliptic curve $C$, the fiber over $C$ is given by $\omega C = T^∗_e C,$ the tangent space of $C$ at its basepoint $e$. The stack $M_{\ell}$ admits a compactification $\Mell$ whose $R$ points are generalized [elliptic curves](Projects/2022%20Advanced%20Qual%20Projects/Elliptic%20Curves/Elliptic%20Curves.md). The space of integral [modular forms](Unsorted/modular%20form.md) of [weight](weight%20of%20a%20modular%20form) $k$ is defined to be the space of sections $$ H ^ { 0 } \left( \bar{\Mell}; \omega\tensorpowerk{n} \right) $$ Motivated by the definition of integral modular forms and the [descent spectral sequence](descent%20spectral%20sequence) in the case of $U = M_\ell$ , the spectrum $\TMF$ (see [TMF](TMF)) is defined to be its global sections: \[ \TMF \da \OO ^ { \Top } \left( \bar{\Mell} \right) \] # Chromatic convergence ![](attachments/Pasted%20image%2020220508204709.png) ![](attachments/Pasted%20image%2020220508204726.png) ![](attachments/Pasted%20image%2020220209191526.png) ![](attachments/Pasted%20image%2020220209191600.png) ![](attachments/Pasted%20image%2020220508190047.png) Can be done on a [thick subcategory](Unsorted/thick%20subcategory.md): ![](attachments/Pasted%20image%2020220508190152.png) ## Chromatic filtration An application to [K-theory](Unsorted/K-theory.md): ![](attachments/Pasted%20image%2020220209211652.png) # The thick subcategory and nilpotent theorems ![](attachments/Pasted%20image%2020220508204517.png) # Relation to tensor triangular geometry and tmf See [tensor triangular geometry](Unsorted/tensor%20triangulated%20category.md) and [tmf](Unsorted/tmf.md). Explaining the infamous chromatic stratification picture: ![](attachments/Pasted%20image%2020220508205207.png) ![](attachments/Pasted%20image%2020220508205136.png) ![](attachments/Pasted%20image%2020220508205057.png) # Applications ![](attachments/Pasted%20image%2020220508204613.png) ![](attachments/Pasted%20image%2020220508204632.png)