Holmstrom Spectral sequence

http://mathoverflow.net/questions/8052/why-are-spectral-sequences-so-ubiquitous

http://mathoverflow.net/questions/22188/introductory-book-on-spectral-sequences

Notes by Murfet

Hatcher Spectral Seqs book draft

http://ncatlab.org/nlab/show/spectral+sequence

References:

• Cartan and Eilenberg.
• Bott and Tu (el).
• Hilton and Stammbach

Probably also other things in the Homological algebra folder, including MacLane: Homology, and Gelfand-Manin: Methods.

Verdier has a notion of “spectral object”. See Deligne: Décompositions dans… (Motives vol) for this. See also Deligne: Theoreme de Lefschetz et criteres de degenerescence de suites spectrales (1968, IHES)

Grayson (page 45 in K-theory handbook) has brief discussion on spectral seqs, for example those omig from a filtration on a spectrum. Read this again.

Gillet in K-theory handbook seems to be a very good starting point for a systematic treatment of spectral sequences

Weibel (p. 19) refers to Grothendieck’s spectral sequence as a special case of the hypercohomology spectral sequence for the composition of two functors.

Some spectral sequences in algebraic geometry: Weight, Leray, local to global, hypercohomology, composition of two functors.

See List of spectral sequences

Deligne, P.: D´eg´en´erescence de suites spectrales et Th´eor`emes de Lefschetz.

Jardine, J.F.:The Leray spectral sequence. J. Pure Appl. Algebra 61, 189–196 (1989)

Segal, G.: Classifying spaces and spectral sequences. Publ. math. IHES 34, 105–112 (1968)

Maunder: The spectral sequence of an extraordinary cohomology theory (1963)

Check MacLane: Homology

Paranjape: Some spectral sequences for filtered complexes and applications (1996)

See also notes under Sheaf cohomology

McCleary book. In homol alg folder

Thomason article

Dwyer: Higher divided squares… “Every cosimplicial space gives rise to a spectral sequence. E.g. the Eilenberg-Moore ss. Study of operations on such spectral seqs.”

Barakat on algorithm for computing with ss of a filtered cplx

LNM0134 treats the Eilenberb-Moore ss and interprets it as a Kunneth ss for a cohomology theory on an overcategory Top/B.

Pirashvili and Redondo: Cohomology of the Grothendieck construction, on arXiv. Treats a very general spectral sequence.

http://mathoverflow.net/questions/76337/exhaustiveness-and-regularness-of-a-filtration-of-a-complex

French on comparison between two ss for unstable homotopy groups (Adams and Goerss-Hopkins)

arXiv:1009.1125 The Goodwillie tower and the EHP sequence from arXiv Front: math.AT by Mark Behrens We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime 2. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the P map, and the Goodwillie differentials to the H map. Furthermore, we study an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. We show that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. We use our theory to re-compute the 2-primary unstable stems through the Toda range (up to the 19-stem). We also study the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching’s operad structure on the derivatives of the identity. These operations act on the mod 2 stable homology of the Goodwillie layers of any functor from spaces to spaces.

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Notes from Tohoku

Section 2.4

Consider an abelian cat $C$. Def of filtered (decreasing) object in $C$, and morphism. Filtered objects form an additive cat. Def of associated graded, a covariant functor to graded objects.

Def of spectral sequence in $C$, as in Tamme. The cat of spectral sequences in $C$ form an additive cat. Spectral functor, cohomological spectral sequence.

Example: Let $K$ be a bicomplex in $C$, with “finite diagonals”. Then have two spectral sequences converging to the cohomology of the associated simple complex. For these the $E_2^{pq}$ terms are defined by taking the $p$-th cohomology in one direction of the $q$-th cohomology in the other direction.

Let $F$ be a covariant functor from one abelian cat $C$ to another $C'$. Suppose the first cat has enough injectives. Let $K$ be a complex in $C$, bounded on the left. Then by Cartan-Eilenberg (Chap XVII) we can construct two spectral sequences $IF$ and $IIF$, bothe computing (what I think is) the derived functors of $F$. This doesn’t make any sense, probably I missed something here. He says that these sequences converge to “the hyperhomology functors of $F$”, and they seem to use starting terms involving the derived functors. We recall the construction: Let $K$ be concentrated in positive degrees. Take a double complex $L$ which in each degree is a resolution of $K$. Also, each node of $L$ should be $F$-acyclic (higher derived functors vanishing). Two ways of constructing such an $L$, including an explicit description of injective objects in the cat of complexes in $C$. More stuff about this hyperhomology, cannot decode the notation.

Apparently the above machinery generates, as a special case, the usual Grothendieck spectral sequence theorem:

Thm: Let $C$, $C'$, $C''$ be three abelian cats, the first two having enough injectives. Let $F:C \to C'$ and $G: C' \to C''$ be covariant functors, such that $G$ is left exact, and $F$ sends injectives to $G$-acyclics. Then there exists a cohomological spectral functor on $C$ with values in $C''$, with abutment the right derived functor of $GF$ (convenablement filtre), with initial term $E_2^{pq}(A) = R^p G (R^qF(A))$.

Several remarks on this thm.

nLab page on Spectral sequence