
## 12.22 Spectral sequences: double complexes

Definition 12.22.1. Let $\mathcal{A}$ be an additive category. A double complex in $\mathcal{A}$ is given by a system $(\{ A^{p, q}, d_1^{p, q}, d_2^{p, q}\} _{p, q\in \mathbf{Z}})$, where each $A^{p, q}$ is an object of $\mathcal{A}$ and $d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ and $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ are morphisms of $\mathcal{A}$ such that the following rules hold:

1. $d_1^{p + 1, q} \circ d_1^{p, q} = 0$

2. $d_2^{p, q + 1} \circ d_2^{p, q} = 0$

3. $d_1^{p, q + 1} \circ d_2^{p, q} = d_2^{p + 1, q} \circ d_1^{p, q}$

for all $p, q \in \mathbf{Z}$.

This is just the cochain version of the definition. It says that each $A^{p, \bullet }$ is a cochain complex and that each $d_1^{p, \bullet }$ is a morphism of complexes $A^{p, \bullet } \to A^{p + 1, \bullet }$ such that $d_1^{p + 1, \bullet } \circ d_1^{p, \bullet } = 0$ as morphisms of complexes. In other words a double complex can be seen as a complex of complexes. So in the diagram

$\xymatrix{ \ldots & \ldots & \ldots & \ldots \\ \ldots \ar[r] & A^{p, q + 1} \ar[r]^{d_1^{p, q + 1}} \ar[u] & A^{p + 1, q + 1} \ar[r] \ar[u] & \ldots \\ \ldots \ar[r] & A^{p, q} \ar[r]^{d_1^{p, q}} \ar[u]^{d_2^{p, q}} & A^{p + 1, q} \ar[r] \ar[u]_{d_2^{p + 1, q}} & \ldots \\ \ldots & \ldots \ar[u] & \ldots \ar[u] & \ldots }$

any square commutes. Warning: In the literature one encounters a different definition where a “bicomplex” or a “double complex” has the property that the squares in the diagram anti-commute.

Example 12.22.2. Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be abelian categories. Suppose that

$\otimes : \mathcal{A} \times \mathcal{B} \longrightarrow \mathcal{C}, \quad (X, Y) \longmapsto X \otimes Y$

is a functor which is bilinear on morphisms, see Categories, Definition 4.2.20 for the definition of $\mathcal{A} \times \mathcal{B}$. Given complexes $X^\bullet$ of $\mathcal{A}$ and $Y^\bullet$ of $\mathcal{B}$ we obtain a double complex

$K^{\bullet , \bullet } = X^\bullet \otimes Y^\bullet$

in $\mathcal{C}$. Here the first differential $K^{p, q} \to K^{p + 1, q}$ is the morphism $X^ p \otimes Y^ q \to X^{p + 1} \otimes Y^ q$ induced by the morphism $X^ p \to X^{p + 1}$ and the identity on $Y^ q$. Similarly for the second differential.

Let $A^{\bullet , \bullet }$ be a double complex. It is customary to denote $H^ p_ I(A^{\bullet , \bullet })$ the complex with terms $\mathop{\mathrm{Ker}}(d_1^{p, q})/\mathop{\mathrm{Im}}(d_1^{p - 1, q})$ (varying $q$) and differential induced by $d_2$. Then $H^ q_{II}(H^ p_ I(A^{\bullet , \bullet }))$ denotes its cohomology in degree $q$. It is also customary to denote $H^ q_{II}(A^{\bullet , \bullet })$ the complex with terms $\mathop{\mathrm{Ker}}(d_2^{p, q})/\mathop{\mathrm{Im}}(d_2^{p, q - 1})$ (varying $p$) and differential induced by $d_1$. Then $H^ p_ I(H^ q_{II}(A^{\bullet , \bullet }))$ denotes its cohomology in degree $p$. It will turn out that these cohomology groups show up as the terms in the spectral sequence for a filtration on the associated total complex.

Definition 12.22.3. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet , \bullet }$ be a double complex. The associated simple complex $sA^\bullet$, also sometimes called the associated total complex is given by

$sA^ n = \bigoplus \nolimits _{n = p + q} A^{p, q}$

(if it exists) with differential

$d_{sA}^ n = \sum \nolimits _{n = p + q} (d_1^{p, q} + (-1)^ p d_2^{p, q})$

Alternatively, we sometimes write $\text{Tot}(A^{\bullet , \bullet })$ to denote this complex.

If countable direct sums exist in $\mathcal{A}$ or if for each $n$ at most finitely many $A^{p, n - p}$ are nonzero, then $sA^\bullet$ exists. Note that the definition is not symmetric in the indices $(p, q)$.

There are two natural filtrations on the simple complex $sA^\bullet$ associated to the double complex $A^{\bullet , \bullet }$. Namely, we define

$F_ I^ p(sA^ n) = \bigoplus \nolimits _{i + j = n, \ i \geq p} A^{i, j} \quad \text{and} \quad F_{II}^ p(sA^ n) = \bigoplus \nolimits _{i + j = n, \ j \geq p} A^{i, j}.$

It is immediately verified that $(sA^\bullet , F_ I)$ and $(sA^\bullet , F_{II})$ are filtered complexes. By Section 12.21 we obtain two spectral sequences. It is customary to denote $({}'E_ r, {}'d_ r)_{r \geq 0}$ the spectral sequence associated to the filtration $F_ I$ and to denote $({}''E_ r, {}''d_ r)_{r \geq 0}$ the spectral sequence associated to the filtration $F_{II}$. Here is a description of these spectral sequences.

Lemma 12.22.4. Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet , \bullet }$ be a double complex. The spectral sequences associated to $K^{\bullet , \bullet }$ have the following terms:

1. ${}'E_0^{p, q} = K^{p, q}$ with ${}'d_0^{p, q} = (-1)^ p d_2^{p, q} : K^{p, q} \to K^{p, q + 1}$,

2. ${}''E_0^{p, q} = K^{q, p}$ with ${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \to K^{q + 1, p}$,

3. ${}'E_1^{p, q} = H^ q(K^{p, \bullet })$ with ${}'d_1^{p, q} = H^ q(d_1^{p, \bullet })$,

4. ${}''E_1^{p, q} = H^ q(K^{\bullet , p})$ with ${}''d_1^{p, q} = (-1)^ q H^ q(d_2^{\bullet , p})$,

5. ${}'E_2^{p, q} = H^ p_ I(H^ q_{II}(K^{\bullet , \bullet }))$,

6. ${}''E_2^{p, q} = H^ p_{II}(H^ q_ I(K^{\bullet , \bullet }))$.

Proof. Omitted. $\square$

These spectral sequences define two filtrations on $H^ n(sK^\bullet )$. We will denote these $F_ I$ and $F_{II}$.

Definition 12.22.5. Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet , \bullet }$ be a double complex. We say the spectral sequence $({}'E_ r, {}'d_ r)_{r \geq 0}$ weakly converges to $H^ n(sK^\bullet )$, abuts to $H^ n(sK^\bullet )$, or converges to $H^ n(sK^\bullet )$ if Definition 12.21.9 applies. Similarly we say the spectral sequence $({}''E_ r, {}''d_ r)_{r \geq 0}$ weakly converges to $H^ n(sK^\bullet )$, abuts to $H^ n(sK^\bullet )$, or converges to $H^ n(sK^\bullet )$ if Definition 12.21.9 applies.

As mentioned above there is no consistent terminology regarding these notions in the literature. In the situation of the definition, we have weak convergence of the first spectral sequence if for all $n$

$\text{gr}_{F_ I}(H^ n(sK^\bullet )) = \oplus _{p + q = n} {}'E_\infty ^{p, q}$

via the canonical comparison of Lemma 12.21.6. Similarly the second spectral sequence $({}''E_ r, {}''d_ r)_{r \geq 0}$ weakly converges if for all $n$

$\text{gr}_{F_{II}}(H^ n(sK^\bullet )) = \oplus _{p + q = n} {}''E_\infty ^{p, q}$

via the canonical comparison of Lemma 12.21.6.

Lemma 12.22.6. Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet , \bullet }$ be a double complex. Assume that for every $n \in \mathbf{Z}$ there are only finitely many nonzero $K^{p, q}$ with $p + q = n$. Then

1. the two spectral sequences associated to $K^{\bullet , \bullet }$ are bounded,

2. the filtrations $F_ I$, $F_{II}$ on each $H^ n(K^\bullet )$ are finite,

3. the spectral sequences $({}'E_ r, {}'d_ r)_{r \geq 0}$ and $({}''E_ r, {}''d_ r)_{r \geq 0}$ converge to $H^*(sK^\bullet )$,

4. if $\mathcal{C} \subset \mathcal{A}$ is a weak Serre subcategory and for some $r$ we have ${}'E_ r^{p, q} \in \mathcal{C}$ for all $p, q \in \mathbf{Z}$, then $H^ n(sK^\bullet )$ is in $\mathcal{C}$. Similarly for $({}''E_ r, {}''d_ r)_{r \geq 0}$.

Proof. Follows immediately from Lemma 12.21.11. $\square$

Here is our first application of spectral sequences.

Lemma 12.22.7. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex. Let $A^{\bullet , \bullet }$ be a double complex. Let $\alpha ^ p : K^ p \to A^{p, 0}$ be morphisms. Assume that

1. For every $n \in \mathbf{Z}$ there are only finitely many nonzero $A^{p, q}$ with $p + q = n$.

2. We have $A^{p, q} = 0$ if $q < 0$.

3. The morphisms $\alpha ^ p$ give rise to a morphism of complexes $\alpha : K^\bullet \to A^{\bullet , 0}$.

4. The complex $A^{p, \bullet }$ is exact in all degrees $q \not= 0$ and the morphism $K^ p \to A^{p, 0}$ induces an isomorphism $K^ p \to \mathop{\mathrm{Ker}}(d_2^{p, 0})$.

Then $\alpha$ induces a quasi-isomorphism

$K^\bullet \longrightarrow sA^\bullet$

of complexes. Moreover, there is a variant of this lemma involving the second variable $q$ instead of $p$.

Proof. The map is simply the map given by the morphisms $K^ n \to A^{n, 0} \to sA^ n$, which are easily seen to define a morphism of complexes. Consider the spectral sequence $({}'E_ r, {}'d_ r)_{r \geq 0}$ associated to the double complex $A^{\bullet , \bullet }$. By Lemma 12.22.6 this spectral sequence converges and the induced filtration on $H^ n(sA^\bullet )$ is finite for each $n$. By Lemma 12.22.4 and assumption (4) we have ${}'E_1^{p, q} = 0$ unless $q = 0$ and $'E_1^{p, 0} = K^ p$ with differential ${}'d_1^{p, 0}$ identified with $d_ K^ p$. Hence ${}'E_2^{p, 0} = H^ p(K^\bullet )$ and zero otherwise. This clearly implies $d_2^{p, q} = d_3^{p, q} = \ldots = 0$ for degree reasons. Hence we conclude that $H^ n(sA^\bullet ) = H^ n(K^\bullet )$. We omit the verification that this identification is given by the morphism of complexes $K^\bullet \to sA^\bullet$ introduced above. $\square$

Remark 12.22.8. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet , \bullet , \bullet }$ be a triple complex. The associated total complex is the complex with terms

$\text{Tot}^ n(A^{\bullet , \bullet , \bullet }) = \bigoplus \nolimits _{p + q + r = n} A^{p, q, r}$

and differential

$d^ n_{\text{Tot}(A^{\bullet , \bullet , \bullet })} = \sum \nolimits _{p + q + r = n} d_1^{p, q, r} + (-1)^ pd_2^{p, q, r} + (-1)^{p + q}d_3^{p, q, r}$

With this definition a simple calculation shows that the associated total complex is equal to

$\text{Tot}(A^{\bullet , \bullet , \bullet }) = \text{Tot}(\text{Tot}_{12}(A^{\bullet , \bullet , \bullet })) = \text{Tot}(\text{Tot}_{23}(A^{\bullet , \bullet , \bullet }))$

In other words, we can either first combine the first two of the variables and then combine sum of those with the last, or we can first combine the last two variables and then combine the first with the sum of the last two.

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