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Context
Homological algebra
Higher algebra
Contents
Idea
A spectral sequence is called multiplicative or a spectral ring if there is a bi-graded algebra structure on each page such that the differentials act as graded derivations of total degree 1.
For example the Serre-Atiyah-Hirzebruch spectral sequence with coefficients in a ring spectrum.
Constructions
From spectral products on Cartan-Eilenberg systems
The following gives sufficient conditions for a Cartan-Eilenberg spectral sequence to be multiplicative. This is due to (Douady 58). The following is taken from (Goette 15a).
Definition
Let , und be Cartan-Eilenberg systems. A spectral product is a sequence of homomorphisms
such that for all , , , the following two diagrams commute:
and
Write for the Cartan-Eilenberg spectral sequence induced from the Cartan-Eilenberg system .
Proposition
A spectral product as in def. 1 induces products
such that
-
-
,
-
is induced by .
(Goette 15a, following Douady 58, theorem II).
Proof
Assume by induction that is induced by . In particular,
This is clear for if we put because and .
Let , be represented by , with , . Using the first diagram and the construction of , we conclude that
From the second diagram, we get
This proves the Leibniz rule (2).
From the Leibniz rule and the facts that and , we conclude that induces a product on , which proves (3). Because is induced by , so is , and we can continue the induction.
Examples
We discuss that the multiplicative structure on the cohomology Serre-Atiyah-Hirzebruch spectral sequence for multiplicative generalized cohomology. This is taken from (Goette 15b).
Definition
The spectral product , def. 1, on the Cartan-Eilenberg system of def. 2 is that given by the following morphism
Together with the diagonal map , for , we define
Proposition
With def. 3, then for all , , , the following diagram commutes
Hence by prop. 1 the spectral product of def. 3 defines a mutliplicative structure on the Serre-WhiteheadAtiyah-Hirzebruch spectral sequence for multiplicative generalizted cohomology.
Proof
The upper square commutes because the maps are natural transformations. For the lower square, we consider the boundary morphism of the triple
The following diagram commutes:
By extend this diagram to the right using the maps once concludes that the lower square above also commutes.
References
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Adrien Douady, La suite spectrale d’Adams : structure multiplicative Séminaire Henri Cartan, 11 no. 2 (1958-1959), Exp. No. 19, 13 p (Numdam)
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Brayton Gray, Products in the Atiyah-Hirzebruch spectral sequence and the calculation of , Trans. Amer. Math. Soc. 260 (1980), 475-483 (web)
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Stanley Kochmann, prop. 4.2.9 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
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John McCleary, section 2.3 in A User’s Guide to Spectral Sequences, Cambridge University Press (2000)
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Daniel Dugger, Multiplicative structures on homotopy spectral sequences I (arXiv:math/0305173)
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Daniel Dugger, Multiplicative structures on homotopy spectral sequences II (arXiv:math/0305187)
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Sebastian Goette, MO comment a, MO comment b Feb 15, 2015