# Spectral Sequences reference

What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep.

I'm looking for a concise encyclopedic approach, in which theorems are stated with the minimum possible hypotheses and clearly written assumptions, but with sufficient explanations/definitions/facts about the surrounding theory of a SS, so that I won't make a mistake using it by misunderstanding the statement. A book that is intended for interested 'visitors', not 'natives' of the theory.

For example, Weibel's Homological Algebra and Davis&Kirk's Lecture Notes in Algebraic Topology and The Stacks Project are extremely useful and well-written, though I find McCleary's A User's Guide to Spectral Sequences and Rotman's An Introduction to Homological Algebra less user-friendly. On the wiki page, there are mentioned several interesting examples of SS.

Where can I learn about the Kunneth SS? I assume the formulation should be similar to: If and $A_\ast,B_\ast$ are chain complexes of $R$-modules, then there are spectral sequences of modules $E^2_{p,q}=\bigoplus_{q'+q''=q}Tor_p^R(H_{q'}A_\ast,H_{q''}B_\ast)\Rightarrow H_{p+q}(A_\ast\otimes B_\ast)$ and $E_2^{p,q}=\prod_{q'+q''=q}Ext^p_R(H_{q'}A_\ast,H_{q''}B_\ast)\Rightarrow H_{p+q}(Hom(A_\ast, B_\ast))$. But what are the minimal assumptions for this to hold? I'd like to have a formulation, such that when $R$ is hereditary, we get the Kunneth shhort exact sequences. I can't find this in any of the mentioned books. Also, if $A_\ast$ and $B_\ast$ are the singular chain complexes, is there a similar SS of algebras for cohomology?

Where can I learn about the Mayer-Vietoris SS of an open covering, Cartan–Leray SS of a quotient space, van Kampen SS of a wedge space, etc.? I feel there's a huge potential with SS, but they are somewhat inaccessible to people from other areas of mathematics. I'd just like to have systematic statements in the form $E_{p,q}^2\ldots\Rightarrow\ldots$ and $E^{p,q}_2\ldots\Rightarrow\ldots$ and explanations of what each of the objects in the formula is, so I can get my hands real dirty with fun computations, appropriate for a novice. Davis&Kirk is perfect for that, I wish the book were ten times longer, so I could devour it all.

• It is not a complete answer, but have you tried reading Boardman's paper "Conditionally convergent spectral sequences"? I find it extremely readable and it should give you all you need about convergence (and construction) of spectral sequences May 8 '14 at 0:30
• I agree with the poster that it would be great to have some sort of encyclopedic repository of "all the spectral sequences". Every time I need a different spectral sequence I find myself having to hunt through a pile of books for the right one that applies. I also agree it's frustrating that there don't seem to be many clear treatments of the general Kunneth and Mayer-Vietoris spectral sequences. Unfortunately, beyond expressing my support for the existence of a good answer, I don't have one! May 8 '14 at 0:53
• The "Kunneth spectral sequence" is, in fact, in Weibel. Theorem 5.6.4. May 8 '14 at 1:34
• @LeonLampret abuttment is the thing it's converging to. Derived $\otimes$ and derived Hom are functors taking in complexes and spitting out a complex. If you stick in complexes concentrated in degree 0, you get a complex whose homology is exactly Tor or Ext as it's usually defined. May 8 '14 at 3:48
• Sorry, I don't know the second edition. I'm taking about the (second printing of) the first edition (I insist in the printing, or the Errata, because I seem to remember that in the first printing there was a convergent change of rings spectral sequence that actually did not converge). May 8 '14 at 9:37

A very readable introduction to spectral sequences is Chapter III of

In particular you can find details about both Mayer-Vietoris and Leray Spectral sequences.

Another good reference:

I just would like to remark that many important spectral sequences are particular cases of the Grothendieck spectral sequence for derived functor of the composition of two functor. For instance the Leray spectral sequence and the exact sequence of low degrees.

Let $$E_{2}^{h,k} \Longrightarrow H^{n}(A)$$ be a spectral sequence whose terms are non trivial only for $h,k \geq 0$. Then we have $$0\mapsto E^{1,0}_{2}\rightarrow H^{1}(A) \rightarrow E^{0,1}_{2} \rightarrow E^{2,0}_{2} \rightarrow H^{2}(A).$$ Let $\mathcal{F}:\mathcal{C}_{1}\rightarrow \mathcal{C}_{2}$ and $\mathcal{G}:\mathcal{C}_{2}\rightarrow \mathcal{C}_{3}$ be two additive covariant functors between abelian categories. Suppose that $\mathcal{G}$ is left exact and that $\mathcal{F}$ takes injective objects of $\mathcal{C}_{1}$ in $\mathcal{G}$-acyclic objects of $\mathcal{C}_{2}$. Then there exists a spectral sequence (Grothendieck spectral sequence) for any object $A$ of $\mathcal{C}_{1}$ $$E_{2}^{h,k} = (R^{h}\mathcal{G}\circ R^{k}\mathcal{F})(A) \Longrightarrow R^{h+k}(\mathcal{G}\circ \mathcal{F})(A).$$ The corresponding exact sequence of low degrees is the following $$0\mapsto R^{1}\mathcal{G}(\mathcal{F}(A)) \rightarrow R^{1}(\mathcal{G}\mathcal{F}(A)) \rightarrow \mathcal{G}(R^{1}\mathcal{F}(A)) \rightarrow R^{2}\mathcal{G}(\mathcal{F}(A)) \rightarrow R^{2}(\mathcal{G}\mathcal{F})(A).$$ As a special case of the Grothendieck spectral sequence we get the Leray spectral sequence. Let $f:X\rightarrow Y$ be a continuous map between topological spaces. We take $\mathcal{C}_{1} = \mathfrak{Ab}(X)$ and $\mathcal{C}_{2} = \mathfrak{Ab}(Y)$ to be the categories of sheaves of abelian groups over $X$ and $Y$ respectively. Then we take $\mathcal{F}$ to be the direct image functor $f_{*}:\mathfrak{Ab}(X)\rightarrow \mathfrak{Ab}(Y)$ and $\mathcal{G} = \Gamma_{Y}:\mathfrak{Ab}(Y) \rightarrow \mathfrak{Ab}$ to be the global section functor, where $\mathfrak{Ab}$ is the category of abelian groups. Note that $$\Gamma_{Y}\circ f_{*} = \Gamma_{X}:\mathfrak{Ab}(X) \rightarrow \mathfrak{Ab}$$ is the global section functor on $X$. By Grothendieck's spectral sequence we know that $(R^{h}\Gamma_{Y}\circ R^{k}f_{*})(\mathcal{E}) \Longrightarrow R^{h+k}(\Gamma_{Y}\circ f_{*})(\mathcal{E}) = R^{h+k}\Gamma_{X}(\mathcal{E})$ for any $\mathcal{E} \in \mathfrak{Ab}(X)$, that is $$H^{h}(Y,R^{k}f_{*}\mathcal{E}) \Longrightarrow H^{h+k}(X,\mathcal{E}).$$ The exact sequence of low degrees looks like $$0\mapsto H^{1}(Y,f_{*}\mathcal{E})\rightarrow H^{1}(X,\mathcal{E})\rightarrow H^{0}(Y,R^{1}f_{*}\mathcal{E}) \rightarrow H^{2}(Y,f_{*}\mathcal{E})\rightarrow H^{2}(X,\mathcal{E}).$$ Finally we can work out the local to global spectral sequence of Ext functors. Let $\mathcal{E} \in \mathfrak{Coh}(X)$ be a coherent sheaf on a scheme $X$. Consider the functor $$\mathcal{H}om(\mathcal{E},-):\mathfrak{Coh}(X) \rightarrow \mathfrak{Coh}(X), \: \mathcal{Q}\mapsto \mathcal{H}om(\mathcal{E},\mathcal{Q}),$$ and the global section functor $$\Gamma_{X}:\mathfrak{Coh}(X) \rightarrow \mathfrak{Ab}, \: \mathcal{Q}\mapsto \Gamma_{X}(\mathcal{Q}).$$ Note that $\Gamma_{X}\circ \mathcal{H}om(\mathcal{E},-) = Hom(\mathcal{E},-)$. By Grothendieck spectral sequence we have $(R^{h}\Gamma_{X}\circ R^{k}\mathcal{H}om(\mathcal{E},-))(\mathcal{Q}) \Longrightarrow R^{h+k}(Hom(\mathcal{E},-)(\mathcal{Q})$ for any $\mathcal{Q}\in \mathfrak{Coh}(X)$, that is $$H^{h}(X,\mathcal{E}xt^{k}(\mathcal{E},\mathcal{Q})) \Longrightarrow Ext^{h+k}(\mathcal{E},\mathcal{Q}).$$ The corresponding sequence of low degrees is $$0\mapsto H^{1}(X,\mathcal{H}om(\mathcal{E},\mathcal{Q})) \rightarrow Ext^{1}(\mathcal{E},\mathcal{Q}) \rightarrow H^{0}(X,\mathcal{E}xt^{1}(\mathcal{E},\mathcal{Q}))\rightarrow H^{2}(X,\mathcal{H}om(\mathcal{E},\mathcal{Q})) \rightarrow Ext^{2}(\mathcal{E},\mathcal{Q}).$$

• This is useful information but seems to not at all be what the question is asking for. May 8 '14 at 0:22
• Hmm, sorry, but where is the Mayer-Vietoris SS in Bott-Tu? What is $E^2_{p,q}$ and what does it converge to? Can't find the statement.
– Leo
May 8 '14 at 0:33
• Pag 185 in Bott-Tu. For the link between Grothendieck spectral sequence and Mayer-Vietoris sequence you may look at this mathoverflow.net/questions/30557/…. The notation $E_2^{h,k}\Rightarrow H^n(A)$ is just a spectral sequence and the short exact sequence is the five-term exact sequence associated to it.
– F_L
May 8 '14 at 0:53

What about the following short note? https://neil-strickland.staff.shef.ac.uk/courses/bestiary/ss.pdf