Spectral Sequences

Textbooks/Course notes

• Adams Blue Book

• Ravenel's Green Book

• A User's Guide

• Chromatic homotopy theory, Bert Guillou https://www.ms.uky.edu/~guillou/S24/751Notes.pdf

  • Chapters 6 and 7 have some great stuff on the Adams spectral sequence, odd primary and 2 primary, with lots of charts and other visuals

• Introduction to the Adams spectral sequence, John Rognes https://www.mn.uio.no/math/personer/vit/rognes/kurs/mat9580v15/adams-sp-seq.010615.pdf

  • Dense and filled with thrills

• The Adams spectral sequence, John Rognes https://www.mn.uio.no/math/personer/vit/rognes/kurs/mat9580v12/notes.050612.pdf

  • genuinely different than the above entry and also phenomenal

Online accounts

• nLab

• Climbing Mount Bourbaki, Akhil Matthew https://amathew.wordpress.com/tag/adams-spectral-sequence/

  • Many wonderful posts on this blog which set up the mod-p Adams spectral sequence, perform examples, and compute stable stems

Papers

• A user's guide to computing stable homotopy groups, Agnes Beaudry and Jonathan Campbell https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/Beaudry-Campbell.pdf

  • Some classical examples, a great starting point

• Algebraic slice spectral sequences, Dominic Culver, Hana Jia Kong, and J.D. Quigley https://arxiv.org/abs/2007.08682

  • Relates slice and motivic adams spectral sequence

Textbook accounts

• Chapter 4 of the Green Book

Online writings

• Constructing v1 and v2 self-maps, Dexter Chua https://dec41.user.srcf.net/exp/self_maps/self_maps.pdf.

  • Some low filtration yoga with the ANSS.

Papers

• A Novice's Guide to the Adams-Novikov Spectral Sequence, Doug Ravenel https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/Novice.pdf

  • Nice little thing

Textbook accounts

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• The Atiyah--Hirzebruch spectral sequence, Caleb Ji https://www.math.columbia.edu/~calebji/atiyah-hirzebruch-final.pdf

  • Examples with complex K-theory of some spaces

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• The Bar spectral sequence, Kimball Strong https://drive.google.com/file/d/1Ob5oa2T07snZCBPGmoB94W7nmpsr7Iqd/view?usp=sharing

  • Setup and application

• The restricted Lie algebra structure on the bar spectral sequence of an interated loop space, Sushanth Sathish Kumar https://math.mit.edu/research/highschool/primes/materials/2021/Kumar.pdf

  • Relates to Dyer-Lashof operations and shuffle products

Papers

• The Hopf ring for complex cobordisom, Doug Ravenel and W. Stephen Wilson https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/hopfring.pdf

  • Great prototypical example in Hopf ring computations

• The HF2-homology of C2-equivariant Eilenberg-Maclane spaces, Sarah Petersen https://arxiv.org/abs/2206.08165

  • C2-equivariant bar spectral sequence 

The bo-resolution is the bo-based Adams spectral sequence. Here are some places where it has appeared, with additional comments.

• bo-Resolutions - Mark Mahowald (1981) https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-92/issue-2/brm-o-resolutions/pjm/1102736799.full

  • Mahowald studies the bo-resolution for the first time in this paper. He uses the spectral sequence to compute the 2-primary v1-periodic stable homotopy groups of spheres. the results are later used, or suitably interpreted here, to prove the height 1 telescope conjecture at the prime 2.

• The bo-Adams spectral sequence - Wolfgang Lellmann and Mark Mahowald (1987). https://www.jstor.org/stable/2000359?origin=crossref

  • Lellmann-Mahowald revisit the bo-resolution. They reprove Mahowald's prior results and study the E2-term of the spectral sequence by separating the E1-term into its v1-torsion and v1-torsion free components.

• On the E2-term of the bo-Adams spectral sequence - Agnes Beaudry, Mark Behrens, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu (2020). https://arxiv.org/abs/1702.00230

  • BBBCX extend Lellmann-Mahowald's Prior work and compute the E2-term of the bo-resolution even further, using a myriad of spectral sequence arguments. They use their results to compute stable stems up to stem 40.

Textbook accounts

Online writings

• A user's guide to the Bockstein spectral sequence, John Palmieri https://sites.math.washington.edu/~palmieri/Papers/bss.pdf

  • Alternative construction and perspective to the Bockstein spectral sequence

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Textbooks/Course notes

• Hopf algebras, Bert Guillou https://www.ms.uky.edu/~guillou/S17/752Notes.pdf

  • Page 38 has great examples

Online writings

Papers

• Stable comodule deformations and the synthetic Adams--Novikov spectral sequence, Francis Baer, Max Johnson, and Peter Marek https://arxiv.org/pdf/2402.14274

  • Categorifies the Cartan-Eilenberg in Fp-synthetic spectra

Textbook accounts

• Complex Cobordism and the Stable Homotopy Groups of Spheres, Doug Ravenel https://www.sas.rochester.edu/mth/sites/doug-ravenel/mu.html

  • See Chapter 5 as well as Section 4 of Chapter 1

Online writings

• https://math.mit.edu/events/talbot/2013/3-Rasekh-CSS.pdf

  • 2013 Talbot writeup

• Dylan Wilson Pre Talbot notes

  • Section 2 is the chromatic spectral sequence

Papers

• Periodic phenomena in the Adams--Novikov spectral sequence, Haynes Miller, Doug Ravenel, and W. Stephen Wilson https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/mrw.pdf

  • Origin story

• Periodic phenomena in the classical Adams spectral sequence, Mark Mahowald and Paul Schick https://www.ams.org/journals/tran/1987-300-01/S0002-9947-1987-0871672-0/

  • Chromatic spectral sequence for the classical Adams spectral sequence E2 page

• The homotopy groups \pi_*(L_2 S^0) at the prime 3, Katsumi Shimomura and Xiangjunn Wang https://www.sciencedirect.com/science/article/pii/S0040938301000337?via%3Dihub

  • An application!

Textbook accounts

• eric peterson

Online writings

• jay writing mileston :)

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• The EHP spectral sequence, Kimball Strong https://e.math.cornell.edu/people/Kimball_Strong/notes/EHP-Sequence.pdf

  • Great setup and computations!

• EHP spectral sequence, nLab https://ncatlab.org/nlab/show/EHP+spectral+sequence

  • Actually a great nLab writeup

• 
  

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• Examples of Goodwillie calculus, Cameron Krulewski https://math.mit.edu/juvitop/pastseminars/notes_2022_Spring/Cam2.pdf

  • Sets up GSS, determines a vanishing line on E1 page in an example.

• The algebraic transfinite (transfinite algebraic?) Atiyah-Hirzebruch spectral sequence, William Balderrama https://williamb.info/notes/goodwillie-curtis.pdf

  • Relates the work of Behrens on the GSS to a similar story for the mod-p Adams spectral sequence.

Papers

• The Goodwillie tower and the EHP spectral sequence, Mark Behrens https://arxiv.org/abs/1009.1125

  • Relates the Goodwillie tower, the GSS, and the EHP spectral sequence. Computes the 2-primary unstable stems up to stem 19.

• Algebraic Goodwillie spectral sequence, Nikolay Konovalov https://arxiv.org/abs/2303.06240

  • something

• Goodwillie calculus, Gregory Arone and Michael Ching https://mching.people.amherst.edu/Work/handbook-article.pdf

  • Handbook of homotopy theory article. Sets up GSS, not much computation.

Textbook accounts

• Homological Algebra, Charles Weibel https://math.mit.edu/~hrm/palestine/weibel/05-spectral_sequences.pdf

  • Chapter 5 is on spectral sequences

Online writings

• The Grothendieck spectral sequece in algebraic geometry, George Torres https://web.ma.utexas.edu/users/gdavtor/notes/gss.pdf

  • Great exposition, setup, and images. Useful examples in last section for algebraic geometers.

Papers

Textbook accounts

Online writings

• Spectral sequences in (equivariant) stable homotopy theory, Arun Debray https://drive.google.com/file/d/1Wjo-AnplXqHo5SuHD3o8xeI1XskJVwFL/view?usp=sharing

  • Sections 1 and 2 have the HFPSS and the classical example for C2 acting on KU

Papers

• RO(G)-graded homotopy fixed point spectral sequence for height 2 Morava E-theory, Zhipeng Duan, Hana Jia Kong, Guchuan Li, Yunze Lu, and Guozhen Wang https://arxiv.org/abs/2209.01830

  • Plays Slice SS off of HFPSS

The kq-resolution is the motivic analogue of the bo-resolution. It is the kq-based motivic Adams spectral sequence, where kq is the very effective Hermitian K-theory spectrum.

• kq-resolutions I, Dominic Culver and J.D. Quigley (2021). https://arxiv.org/abs/1905.11952

  • Culver-Quigley study the C-motivic kq-resolution. They determine the E1-page in terms of Brown--Gitler comodules, compute v1-periodicity in the C-motivic stable stems, and contemplate motivic analogues of the telescope conjecture.

• On the ring of cooperations for real Hermitian K-theory, Jackson Morris (2025). https://arxiv.org/abs/2506.16672

  • Morris extends the work of Culver-Quigley and computes the E1-page of the kq-resolution in R-motivic homotopy. He also proves a splitting result for symplectic K-theory ksp over all base schemes with 2 invertible 

• Rings of cooperations for Hermitian K-theory, Jackson Morris (2025). https://arxiv.org/abs/2509.02786

  • Morris extends previous work and computes the E1-page of the kq-resolution in Fq-motivic homotopy, where Fq is a finite field of characteristic different from 2.

Textbook accounts

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Textbook accounts

• Cohomology of Groups, Kenneth Brown

  • Chapter VII has a wonderful treatment

Online writings

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Textbooks/Course notes

• Complex Cobordism and the Stable Homotopy Groups of Spheres, Douglas Ravenel https://www.sas.rochester.edu/mth/sites/doug-ravenel/mu.html

  • Section 2 of Chapter 3

• Hopf algebras, Bert Guillou https://www.ms.uky.edu/~guillou/S17/752Notes.pdf

  • Pages 41-45

Online writings

• \pi_*ko and some of \pi_*tmf at the prime 2, Rin Ray https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/ray-a1-2.pdf

  • May spectral sequence to compute the E2 page of the Adams spectral sequence computing \pi_*ko, and partial progress towards the same for tmf

• Spectral sequences in (equivariant) stable homotopy theory, Arun Debray https://drive.google.com/file/d/1Wjo-AnplXqHo5SuHD3o8xeI1XskJVwFL/view?usp=sharing

  • Section 4 has the May spectral sequence

Papers

• Peter May's thesis

  • Origin Story

• The motivic Adams spectral sequence, Dan Dugger and Dan Isaksen https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/dandan1.pdf

  • Section 5 has the motivic May spectral sequence

• 
  

Textbook accounts

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• On the Adams spectral sequence for R-modules, Andrew Baker and Andrey Lazarev  https://arxiv.org/pdf/math/0105079

  • Original formulation of the relative Adams spectral sequence

• On a spectrum-level splitting of the BP<2> cooperations algebra, Liz Tatum https://arxiv.org/abs/2208.11772

  • Extensive use of the relative Adams spectral sequence

• A spectrum-level splitting of the kuR-cooperations algebra, Guchuan Li, Sarah Petersen, and Liz Tatum https://arxiv.org/abs/2503.17149

  • Extensive use of the C2-equivariant relative Adams spectral sequence

• Quotient rings of HF2 ^ HF2, Agnes Beaudry, Mike Hill, Tyler Lawson, Danny Shi, and Mingcong Zeng https://arxiv.org/abs/2103.14707

  • Extensie use of the relative Adams spectral sequence

Textbooks/Course notes

• Homotopy theory, Bert Guillou https://www.ms.uky.edu/~guillou/S15/751Notes.pdf

  • Second half of these notes are excellent

Online writings

• Introduction to spectral sequences, Arun Debray https://adebray.github.io/lecture_notes/u17_introduction_to_spectral_sequences.pdf

  • Great set of examples

Papers

Textbook accounts

Online writings

• The slice filtration ECHT Minicourse, Bert Guillou https://www.ms.uky.edu/~guillou/echtSlices/SlicesMinicourse.html

  • Lectures and problem sets for the equivariant slice spectral sequence

• The motivic slice spectral sequence ECHT Minicourse, Oliver Rondigs https://bpb-us-e1.wpmucdn.com/s.wayne.edu/dist/0/60/files/2019/11/echt-lecture.pdf

  • Lectures for the motivic slice spectral sequence, with computations and setup

Papers

• An Atiyah--Hirzebruch spectral sequence for KR-theory, Dan Dugger https://arxiv.org/abs/math/0304099

  • Origin story for C2-equivariant slices

• The slice spectral sequence for a C4 analog of real K-theory, Mike Hill, Mike Hopkins, Doug Ravenel https://arxiv.org/abs/1502.07611

  • Followup to HHR

• The slice spectral sequence for a motivic analogue of the connective K(1)-local sphere, Hana Jia Kong and J.D. Quigley https://arxiv.org/abs/2209.08603

  • Motivic slice spectral sequence computations over a variety of base fields

• The slice spectral sequence for the cyclic group of order p, Vigleik Angeltveit https://arxiv.org/pdf/2111.06559

  • Odd primary cyclic group equivariant slice spectral sequence formulation

• The homological slice spectral sequence in motivic and Real bordism, Christian Carrick, Mike Hill, and Doug Ravenel https://arxiv.org/abs/2304.01960

  • Generalized slice spectral sequences, equivariant and motivic

• On the slice spectral sequence, John Ullman https://arxiv.org/pdf/1206.0058

  • Introduce the regular slice spectral sequence

• R-motivic v1-periodic homotopy, Eva Belmont, Dan Isaksen, and Hana Jia Kong https://arxiv.org/pdf/2204.05937

  • Slice spectral sequence for image of J, related to Kong-Quigley results

• Stable motivic \pi_1 of low-dimensional fields, Kyle Ormsby and Paul Arne Ostvaer https://arxiv.org/pdf/1310.2970

  • Slice spectral sequence for MGL and BPGL<n>

• Vanishing in stable motivic homotopy sheaves, Kyle Ormsby, Oliver Rondigs and Paul Arne Ostvaer https://arxiv.org/pdf/1704.04744

  • Looks at slices of the motivic sphere

• The homotopy groups of the eta-periodic motivic sphere spectrum, Kyle Ormsby and Oliver Rondigs https://arxiv.org/pdf/1906.11670

  • Looks at the \eta-inverted slice spectral sequence

• Algebraic slice spectral sequences, Dominic Culver, Hana Jia Kong, and J.D. Quigley https://arxiv.org/abs/2007.08682

  • Relates slice and motivic adams spectral sequence

• The C2-effective slice spectral sequence for C2-equivariant connective real K-theory, Hana Jia Kong https://arxiv.org/abs/2004.00806

  • relates R-motivic and C2-equivariant slice spectral sequence for very effective Hermitian K-theory and C2-equivariant connective real K-theory

Textbook accounts

Online writings

Papers

• An introduction to filtered and synthetic spectra, Sven van Nigtevecht https://arxiv.org/abs/2509.21127

  • Wonderful treatise on filtered spectra, associated spectral sequences, the element tau, and synthetic spectra

The tmf-resolution is the tmf-based Adams spectral sequence. Here are some places where it has been studied.

• On the existence of a v_2^{32} self-map on M(1,4) at the prime 2, Mark Behrens, Mike Hill, Mike Hopkins, and Mark Mahowald. https://arxiv.org/abs/0710.5426

• Detecting exotic spheres in low dimension using coker J, Mark Behrens, Mike Hill, Mike Hopkins, and Mark Mahowald. https://arxiv.org/abs/1708.06854

• On the ring of cooperations for 2-primary connective topological modular forms, Mark Behrens, Kyle Ormsby, Nat Stapleton, and Vesna Stojanoska. https://arxiv.org/abs/1501.01050

• The telescope conjecture at height 2 and the tmf-resolution, Agnes Beaudry, Mark Behrens Prasit Bhattacharya, Dominic Culver, and Zhouli Xu. https://arxiv.org/abs/1909.13379

• The structure of the v_2-local algebraic tmf-resolution, Prasit Bhattacharya, Mark Behrens, and Dominic Culver. https://arxiv.org/abs/2301.11230

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