This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
In the previous post, we used filtrations and induction to prove the balancing of \(\Ext\) and \(\Tor\). This can be thought of as a primitive form of the spectral sequence that we will discuss in this post. A spectral sequence is a systematic method for approximating the cohomology of a cochain complex equipped with a filtration, passing through pages step by step. Let us formally define this data.
Spectral Sequences
Definition 1 A spectral sequence is a collection of the following data.
- Bigraded object \(E_r=(E_r^{p,q})_{p,q}\),
- A differential \(d_r\) of bidegree \((r,1-r)\) on \(E_r\), i.e., \(d_r:E_r^{p,q}\rightarrow E_r^{p+r, q-r+1}\) (so that \(d_r^2=0\))
For each \(r\), the bigraded complex \((E_r^{p,q}, d_r)\) is called the \(r\)-th page. These two data are connected by the formula
\[E_{r+1}^{p,q}\cong \frac{\ker(d_r^{p,q}: E_r^{p,q}\rightarrow E_r^{p+r,q-r+1})}{\im(d_r^{p-r,q+r-1}: E_r^{p-r, q+r-1}\rightarrow E_r^{p,q})}\]If we visualize the elements of the \(E_r\) page as points \((p,q)\) in the plane, then \(d_r^{p,q}\) goes from the point \((p,q)\) to \((p+r, q-r+1)\), and these points form a cochain complex. In particular, from this viewpoint \(E_{r+1}^{p,q}\) can be thought of as the cohomology at the point \((p,q)\) of the cochain complex passing through \((p,q)\).
Based on our experience, if we think of a spectral sequence as that of some double complex’s total complex, then the differentials on each of these pages can be thought of as a careful analysis of each component of the differential in the total complex, namely
\[d^n:\bigoplus_{p+q=n}C^{p,q}\rightarrow \bigoplus_{p+q=n+1}C^{p+q}\]We analyzed this, and our main goal was ultimately to compute the homology of this total complex. For this purpose, in the proof of §Ext and Tor, ⁋Proposition 3, we defined a filtration using the horizontal/vertical degrees of the total complex \(A^\bullet=\Tot(K)^\bullet\). Therefore, we need to introduce the notion of a filtered complex more generally.
Filtrations
As mentioned above, the following Definition 2 can be thought of as a much more general version, but the philosophy of breaking down a complex more finely is essentially the same.
Definition 2 A decreasing filtration \(F\) on a cochain complex \(A^\bullet\) is a sequence of subcomplexes \((F^p A^\bullet)_p\) satisfying the condition
\[\cdots \supset F^{p-1}A^\bullet \supset F^pA^\bullet \supset F^{p+1}A^\bullet \supset \cdots\]A cochain complex equipped with a (decreasing) filtration is called a filtered complex, denoted \((A^\bullet, F)\).
In particular, since \(F^p A^\bullet\) is assumed to be a subcomplex of \(A^\bullet\), the differential of \(A^\bullet\) restricts well to \(F^pA^\bullet\) and the cohomology with respect to this differential is also well-defined. Intuitively, as \(p\) increases, \(F^p A^\bullet\) becomes smaller, and one can understand that new information is added at each step. In the proof of §Ext and Tor, ⁋Proposition 3 above, we thought of \(F^{p+1}A^\bullet/F^pA^\bullet\) to apply induction and regarded this as the original double complex \(K^{p, \bullet-p}\); in the general case too, this information is important in that it exactly contains the \(p\)-th filtration. The cochain complex obtained in this way,
\[\gr^p A^\bullet = F^p A^\bullet / F^{p+1} A^\bullet\]is called the associated graded complex with respect to \(F\). Of course, the differential of this complex comes from the differential of the original cochain complex \(A^\bullet\).
The most important thing about a filtration is that when a filtered complex \(A^\bullet\) is given, the filtration is naturally defined at the cohomology level as well. We will use this filtration to define the convergence of the spectral sequence induced from a filtered complex.
Definition 3 Let \((A^\bullet, F)\) be a filtered complex. Then the image at the cohomology level of the inclusion \(F^pA^\bullet\rightarrow A^\bullet\) is defined as
\[F^p H^n = \operatorname{im}\bigl(H^n(F^p A^\bullet) \to H^n(A^\bullet)\bigr)\]This filtration consists of the cohomology classes induced by cocycles contained in \(F^p A^\bullet\). As \(p\) increases, \(F^p A^\bullet\) becomes smaller, so \(F^p H^n\) also becomes smaller.
Convergence of Spectral Sequences
Now we explain the convergence of spectral sequences. Intuitively, since each page \(E_r^{p,q}\) of a spectral sequence is an object that is progressively refined as \(r\) increases, we must examine what this approximation ultimately converges to. Therefore, we first define the following.
Definition 4 A spectral sequence \(\{E_r^{p,q}, d_r\}\) is regular if for each \((p,q)\), \(E_r^{p,q} = E_{r+1}^{p,q}\) holds for sufficiently large \(r\). The stabilized value is then defined as \(E_\infty^{p,q}\).
Since regularity means that the pages of the spectral sequence no longer change at each bidegree, it enables the following definition.
Definition 5 A spectral sequence \(\{E_r^{p,q}, d_r\}\) converges to a filtered graded object \((H^n, F)\) if for each \((p,q)\),
\[E_\infty^{p,q} \cong F^p H^{p+q} / F^{p+1} H^{p+q} = \gr^p H^{p+q}\]holds. We write this as \(E_r^{p,q} \Rightarrow H^{p+q}\).
Again, thinking of the familiar example of the total complex of a double complex, this can be thought of as a generalization of the fact that when we put the horizontal filtration on the double complex and sent \(p\) to \(0\), the homology of the total complex came out.
If for each \(n\), \(\bigcap_p F^p H^n = 0\) (Hausdorff condition), \(\bigcup_p F^p H^n = H^n\) (exhaustive condition), and the spectral sequence is regular, then one can uniquely reconstruct \(H^n\) by collecting the information of each \(\gr^p H^{p+q}\). When these three conditions are satisfied, the spectral sequence is said to strongly converge to \((H^n, F)\). On the other hand, if any one of these three conditions is not met, it is said to weakly converge, and in this case \(E_\infty^{p,q}\) alone cannot uniquely determine \(H^n\).
In general, regularity of a spectral sequence does not always hold. However, if for all \(r\), the \(E_r^{p,q}\) exist only in the first quadrant, that is, if \(E_r^{p,q}\neq 0\) is possible only when \(p,q\geq 0\), then taking \(d_r\) with sufficiently large \(r\) causes the point to escape into the second or fourth quadrant and become \(0\), so it is always regular. That is, the following holds.
Proposition 6 A first quadrant spectral sequence, i.e., a spectral sequence such that \(E_r^{p,q} = 0\) for \((p,q)\) with \(p < 0\) or \(q < 0\), is always regular. Moreover, when such a spectral sequence converges as \(E_r^{p,q} \Rightarrow H^{p+q}\), for each \((p,q)\) we have \(E_r^{p,q} = E_\infty^{p,q}\) for sufficiently large \(r\).
Filtrations and Spectral Sequences
Until now, we have satisfied our intuition by thinking of the total complex of a double complex and the spectral sequence associated to it, but the connection is somewhat unclear in that we still do not know what spectral sequence this total complex defines. In this section, we explain the concrete method of constructing a spectral sequence from an arbitrary filtered complex. In particular, in the proof of §Ext and Tor, ⁋Proposition 3, the objects on both sides correspond to the spectral sequences given by the filtrations taken in the vertical and horizontal directions, respectively, and the basic idea of the proof is that these converge to the same object.
Let \((A^\bullet, F)\) be a filtered complex. Then we can construct the \(E_0\) page directly by the formula
\[E_0^{p,q} = \gr^p A^{p+q} = F^p A^{p+q} / F^{p+1} A^{p+q}\]Now let us define the differential on this precisely. First, since the filtration preserves the differential, from this we obtain
\[F^p A^{p+q}\rightarrow F^p A^{p+q+1}\]Let us write this as \(F^p d\) for convenience. Then the desired differential is obtained by composing \(F^p d\) with the quotient
\[F^pA^{p+q+1}\rightarrow F^pA^{p+q+1}/F^{p+1}A^{p+q+1}\]and then using the first isomorphism theorem to factor this through as
\[F^p A^{p+q}/F^{p+1}A^{p+q}\rightarrow F^pA^{p+q+1}/F^{p+1}A^{p+q+1}\]At this point, the first isomorphism theorem applies well: if \(a \in F^{p+1}A^{p+q}\), then \(d(a) \in F^{p+1}A^{p+q+1}\), so it goes to \(0\) in the codomain quotient \(F^p A^{p+q+1}/F^{p+1}A^{p+q+1}\).
More generally, the differential \(d_r\) on the \(E_r\) page is also defined in a similar way. Essentially, since \(E_r^{p,q}\) is constructed by taking quotients of \(F^pC^{p+q}\) in several stages, an element of \(E_r^{p,q}\) can be thought of as a suitable equivalence class \([x]\) of some element \(x\in F^pC^{p+q}\). Now \(d_r^{p,q}: E_r^{p,q}\rightarrow E_r^{p+r, q-r+1}\) is given by the formula
\[d_r^{p,q}([x])=[dx]\in E_r^{p+r, q-r+1}\tag{$\ast$}\]Of course, showing that this correspondence is well-defined and defines a differential requires somewhat complicated calculations (link), but what is important is that the elements of \(E_r^{p,q}\) can be defined by the following two conditions:
- \(dx\in F^{p+r}C^{p+q+1}\), and
- If \(x,y\in F^{p+r}C^{p+q}\) differ only by a boundary of the \((r-1)\)-th stage, then \(x,y\) are regarded as the same.
In particular, what (\(\ast\)) tells us is that all the \(d_r\) are essentially the same as \(d\), and the index \(r\) is used only to measure the degree to which the filtration is skipped. That is, the following holds.
Proposition 7 The \(E_r^{p,q}\) and \(d_r\) constructed from a filtered complex \((A^\bullet, F)\) as above satisfy the conditions of a spectral sequence in Definition 1. Namely,
\[d_r \circ d_r = 0\]holds, and \(E_{r+1}^{p,q} \cong H(E_r, d_r)\).
Moreover, the spectral sequence defined in this way also possesses a kind of functoriality.
Proposition 8 Let \(f : (A^\bullet, F) \to (B^\bullet, G)\) be a chain map between filtered complexes. That is, for each \(p\),
\[f(F^p A^\bullet) \subset G^p B^\bullet\]holds. Then \(f\) induces a well-defined map \(f_r : E_r(A) \to E_r(B)\) for each \(r\).
Proof
Since \(f\) is a chain map, it sends cocycles to cocycles and boundaries to boundaries. Also, since \(f(F^p) \subset G^p\), we have \(f(Z_r^{p,q}(A)) \subset Z_r^{p,q}(B)\) and \(f(B_r^{p,q}(A)) \subset B_r^{p,q}(B)\). Therefore \(f\) induces a well-defined map on each \(E_r\).
Then our core result is that this spectral sequence actually reaches the cohomology of the original complex. Let us first define the following.
Definition 9 A filtered complex \((A^\bullet, F)\) is bounded if for each \(n\) there exists a sufficiently large \(p\) satisfying \(F^pA^n=0\), and a sufficiently small \(p\) satisfying \(F^pA^n=A^n\).
That is, this means that if we fix Definition 2 to degree \(n\) and start with the filtration
\[\cdots\supset F^{p-1}A^n\supset \cdots F^p A^n \supset \cdots F^{p+1}A^n\supset\cdots\]there, this filtration is bounded. Then for each \((p,q)\) there exist \((m, n)\) such that \(F^nA^{p+q}=A^{p+q}\) and \(F^mA^{p+q}=0\), so for a fixed \((p,q)\), if we take \(r\) large enough that the differential \(d_r\) escapes this interval, one can check that a bounded filtered complex is regular.
Moreover, from the above description of the elements of \(E_r^{p,q}\) we can show that
\[E_\infty^{p,q}\cong F^p H^{p+q}(A^\bullet)/F^{p+1}H^{p+q}(A^\bullet)\]and from this we obtain the following result.
Proposition 10 Let \((A^\bullet, F)\) be a bounded filtered complex, and let \((E_r^{p,q})\) be the spectral sequence it defines. Then \((E_r^{p,q})\) converges to the filtered graded object \((H^\bullet, F)\) of Definition 3. That is, \(E_r^{p,q}\Rightarrow H^{p+q}(A^\bullet)\).
Spectral Sequences of Double Complexes
We have taken the proof of the balancing of \(\Ext\) and \(\Tor\) in §Ext and Tor, ⁋Proposition 3 as the motivation for our theory. We finish this post by examining the spectral sequence defined from a double complex.
Example 11 Consider the total complex \(\Tot(K)^\bullet\) of an arbitrary double complex \(K^{p,q}\). We can put a filtration on this total complex in two ways.
First, define the vertical filtration as
\[F^p_v \Tot(K)^n = \bigoplus_{j \geq p} K^{n-j, j}\]Then the associated graded object for this filtration is \(\mathrm{gr}^p_v \Tot(K)^{p+q} = K^{p,q}\), so the \(E_0\) page is
\[E_0^{p,q} = K^{p,q}\]and \(d_0\) is a derivation of degree \((0,1)\) given by the vertical differential \(d_v\). Therefore the \(E_1\) page is
\[E_1^{p,q} = H^q_v(K^{p,\bullet})\]and the \(d_1\) on the \(E_1\) page is a derivation of degree \((1,0)\) induced by the horizontal differential \(d_h\). On the other hand, if we now define the horizontal filtration as
\[F^p_h \Tot(K)^n = \bigoplus_{i \geq p} K^{i, n-i}\]then similarly the \(E_0\) page is
\[E_0^{p,q} = K^{p,q}\]and \(d_0\) is given by the horizontal differential \(d_h\). Therefore the \(E_1\) page is
\[E_1^{p,q} = H^p_h(K^{\bullet, q})\]and \(d_1\) is induced by the vertical differential \(d_v\).
In particular, let \(K^{p,q}\) be a first quadrant double complex. Then both filtrations define bounded filtered complexes, so by Proposition 10 each spectral sequence converges to \(H^\bullet(\Tot(K))\). From this we can reconstruct the proof of §Ext and Tor, ⁋Proposition 3 in fancier language.
References
[GM] S. I. Gelfand, Y. I. Manin, Methods of homological algebra, Springer, 2003. [Wei] C. A. Weibel, An introduction to homological algebra, Cambridge University Press, 1994. [God] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, 1958.
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