Spectral sequence

A spectral sequence or spectral sequence is a calculation method in the mathematical sub-area of homological algebra . According to JF Adams , spectral sequences are like exact sequences , only more complicated. As for exact sequences, the same applies to spectral sequences: they do not offer a guarantee of success, but are nevertheless often an effective tool in the hands of experts.

The basic idea goes back to a research announcement published by Leray in 1946 on the cohomological investigation of a continuous mapping. As early as 1947 Koszul had - with the help of a tip from Cartan - abstracted the spectral sequence calculus in its present form, so that Leray also used Koszul's formalism in the complete version of his work.

Abstract definition

A spectral sequence is a sequence for by bigraduierten Abelian groups and boundary operators with . It must apply that the homology of is with respect to the edge mapping . One calls the -th side or the -th term of the spectral sequence. ${\displaystyle (E_{p,q}^{r},d^{r})}$ ${\displaystyle r\geq 2}$ ${\displaystyle E_{**}^{r}}$ ${\displaystyle d^{r}\colon E_{p,q}^{r}\rightarrow E_{p-r,q+r-1}^{r}}$ ${\displaystyle d^{r}\circ d^{r}=0}$ ${\displaystyle E_{**}^{r+1}}$ ${\displaystyle E_{**}^{r}}$ ${\displaystyle d^{r}}$ ${\displaystyle E_{**}^{r}}$ ${\displaystyle r}$ ${\displaystyle r}$

For fixed , it therefore has an infinite sequence , , , .... If this sequence is convergent - for example because it is constant after finitely many terms - then one writes for the limit value. ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle E_{p,q}^{2}}$ ${\displaystyle E_{p,q}^{3}}$ ${\displaystyle E_{p,q}^{4}}$ ${\displaystyle E_{p,q}^{\infty }}$

Often one encounters statements of the form “there is a spectral sequence ”. This notation means that there is a spectral sequence with ${\displaystyle A_{p,q}\Rightarrow B_{p+q}}$

• ${\displaystyle E_{p,q}^{2}\cong A_{p,q}}$; and
• the group can be filtered in such a way that the groups with the filtration quotients. ${\displaystyle B_{n}}$ ${\displaystyle E_{p,q}^{\infty }}$ ${\displaystyle p+q=n}$

In a typical use, one already knows the groups and hopes to calculate the groups using the spectral sequence calculus . ${\displaystyle A_{p,q}}$ ${\displaystyle B_{n}}$

General construction methods

There are three general approaches to constructing a spectral sequence:

• The spectral sequence of a filtered chain complex;
• The spectral sequence of a double complex ;
• The spectral sequence of an exact triple.

The double complex access is only a particularly important special case of the filtered chain complex access. The first two accesses are even defined for all . ${\displaystyle E_{**}^{r}}$ ${\displaystyle r\geq 0}$

Selected spectral sequences

The list below is by no means complete, for example the Eilenberg - Moore spectral sequence and the Adams spectral sequence, both from Algebraic Topology , are missing .

Leray-Serre spectral sequence

Let be a Serre fiber with a simply connected base space . By developing Leray's original approach, Serre gained a spectral sequence . Serre used his spectral sequence to study the homology of loop spaces. The gysin sequence follows directly from this spectral sequence. ${\displaystyle F\rightarrow E\rightarrow B}$ ${\displaystyle B}$ ${\displaystyle H_{p}(B;H_{q}(F))\Rightarrow H_{p+q}(E)}$

Grothendieck spectral sequence

Grothendieck discovered a spectral sequence that calculates the derived functors of a combination of two functors. Let and be two left exact functors between Abelian categories , where and have enough injective objects . It is also valid: If an injective object is of , then an acyclic object is of . Then there is a spectral sequence for each object of . The corresponding statement for left-derived functors also applies. ${\displaystyle F\colon {\mathcal {A}}\rightarrow {\mathcal {B}}}$ ${\displaystyle G\colon {\mathcal {B}}\rightarrow {\mathcal {C}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle I}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle F(I)}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle R^{p}G(R^{q}F(A))\Rightarrow R^{p+q}(G\circ F)(A)}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}}$

Lyndon-Hochschild-Serre spectral sequence

This spectral sequence in group cohomology was  discovered by Hochschild and Serre in 1953 , after preliminary work by Lyndon . It can be derived as an application example of the Grothendieck spectral sequence. Let be a group with normal divisors , and be a module. Then there is a spectral sequence . ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle \mathbb {Z} G}$ ${\displaystyle H^{p}(G/N;H^{q}(N;M))\Rightarrow H^{p+q}(G,M)}$

Atiyah Hirzebruch spectral sequence

Let be a generalized cohomology theory and a CW complex . Then there is a conditionally converging spectral sequence , whereby the topological space is meant, which consists of exactly one point. Atiyah and Hirzebruch used this spectral sequence in the case of the generalized cohomology theory K theory . Maunder used Postnikov systems to give an alternative construction of the Atiyah-Hirzebruch spectral sequence that allows a better description of the differentials. ${\displaystyle h^{*}}$ ${\displaystyle X}$ ${\displaystyle H^{p}(X;h^{q}(*))\Rightarrow h^{p+q}(X)}$ ${\displaystyle *}$

literature

• John McCleary: A User's Guide to Spectral Sequences (=  Cambridge studies in advanced mathematics . No. 58 ). 2nd Edition. Cambridge University Press , 2001, ISBN 0-521-56759-9 .
• Charles A. Weibel : An introduction to homological algebra (=  Cambridge studies in advanced mathematics . No. 38 ). Cambridge University Press , 1994, ISBN 0-521-43500-5 , chap. 5 .
• Henri Cartan , Samuel Eilenberg : Homological Algebra (=  Princeton Mathematical Series . No. 19 ). Princeton University Press , 1956, LCCN  53-010148 , chap. XV, XVI .
• Tilman Bauer: Homological Algebra and Group Homology. Lecture notes winter semester 2004/05. University of Münster , June 18, 2008, accessed on September 3, 2014 (revised version). 

Individual evidence

1. ↑ Klaus Lamotke: Semisimpliziale Algebraische Topologie (=  The basic teachings of the mathematical sciences . No. 147 ). Springer-Verlag , Berlin 1968, ISBN 978-3-662-12989-0 , heading of VI. Chapter , doi : 10.1007 / 978-3-662-12988-3 .
2. ↑ Andreas Dress : On the spectrum sequence of a fiber . In: Inventiones Mathematicae . tape 3 , no. 2 , 1967, p. 172-178 , doi : 10.1007 / BF01389743 .
3. ↑ Volkeruppe: About the convergence of spectral sequences in the stable homotopy theory . In: manuscripta mathematica . tape 6 , no. 4 , 1972, ISSN  0025-2611 , pp. 327-358 , doi : 10.1007 / BF01303687 .
4. ^ JF Adams : Algebraic Topology . a student's guide (=  London Mathematical Society Lecture Note Series . No. 4 ). Cambridge University Press , 1972, ISBN 0-521-08076-2 , p. 13 : “A spectral sequence is… like an exact sequence, but more complicated. ... Like an exact sequence, it does not provide a guarantee that one can carry out any required calculation effectively, but the experts succeed with it more often than not. "
5. ^ Jean Dieudonné : A history of algebraic and differential topology, 1900-1960 (=  Modern Birkhauser Classics ). Reprint of the 1989 edition. Birkhäuser , Boston 2009, ISBN 978-0-8176-4906-7 , pp. 132-141 , doi : 10.1007 / 978-0-8176-4907-4 .
6. ↑ Engl .: exact couple ; for the German term, see p. 75 in Tilman Bauer: Homologische Algebra und Gruppenkohomologie. Lecture notes winter semester 2004/05. University of Münster , June 18, 2008, accessed on June 22, 2016 (revised version). 
7. ↑ John McCleary: A User's Guide to Spectral Sequences (=  Cambridge studies in advanced mathematics . No. 58 ). 2nd Edition. Cambridge University Press , 2001, ISBN 0-521-56759-9 , chap. 7-8.
8. ↑ John McCleary: A User's Guide to Spectral Sequences (=  Cambridge studies in advanced mathematics . No. 58 ). 2nd Edition. Cambridge University Press , 2001, ISBN 0-521-56759-9 , chap. 9 .
9. ^ Saunders Mac Lane : Homology (=  Classics in Mathematics ). Reprint of the 1975 edition. Springer-Verlag , Berlin 1995, ISBN 3-540-58662-8 , pp. 322 .
10. ^ Charles A. Weibel : An introduction to homological algebra (=  Cambridge studies in advanced mathematics . No. 38 ). Cambridge University Press , 1994, ISBN 0-521-43500-5 , p. 128 .
11. ↑ John McCleary: A User's Guide to Spectral Sequences (=  Cambridge studies in advanced mathematics . No. 58 ). 2nd Edition. Cambridge University Press , 2001, ISBN 0-521-56759-9 , chap. 5-6.
12. ↑ Serge Lang : Algebra (=  Graduate Texts in Mathematics . No. 211 ). Revised 3rd edition. Springer-Verlag , New York 2002, ISBN 0-387-95385-X , p. 821 .
13. ↑ ^{a b} Charles A. Weibel : An introduction to homological algebra (=  Cambridge studies in advanced mathematics . No. 38 ). Cambridge University Press , 1994, ISBN 0-521-43500-5 , pp. 150-151 .
14. ^ Tilman Bauer: Homological Algebra and Group Cohomology. Lecture notes winter semester 2004/05. University of Münster , June 18, 2008, p. 83 , accessed on July 12, 2016 (revised version). 
15. ^ Charles A. Weibel : An introduction to homological algebra (=  Cambridge studies in advanced mathematics . No. 38 ). Cambridge University Press , 1994, ISBN 0-521-43500-5 , p. 195 .
16. ^ Tilman Bauer: Homological Algebra and Group Cohomology. Lecture notes winter semester 2004/05. University of Münster , June 18, 2008, p. 88 , accessed on July 12, 2016 (revised version). 
17. ↑ John McCleary: A User's Guide to Spectral Sequences (=  Cambridge studies in advanced mathematics . No. 58 ). 2nd Edition. Cambridge University Press , 2001, ISBN 0-521-56759-9 , p. 496 .
18. ↑ George W. Whitehead : Elements of Homotopy Theory (=  Graduate Texts in Mathematics . No. 61 ). Springer-Verlag , New York 1978, ISBN 0-387-90336-4 , Section XIII.6 .
19. ^ MF Atiyah, F. Hirzebruch: Vector bundles and homogeneous spaces . In: Carl B. Allendoerfer (Ed.): Differential Geometry (=  Proceedings of Symposia in Pure Mathematics . No. 3 ). American Mathematical Society , 1961, LCCN  62-005289 , pp. 7-38 .
20. ^ CRF Maunder: The spectral sequence of an extraordinary cohomology theory . In: Mathematical Proceedings of the Cambridge Philosophical Society . tape 59 , no. 3 , July 1963, ISSN  0305-0041 , p. 567-574 .
21. ^ Daniel Grady, Hisham Sati: Spectral sequences in smooth generalized cohomology . May 11, 2016, arxiv : 1605.03444v1 .