## 12.20 Spectral sequences

A nice discussion of spectral sequences may be found in [Eisenbud]. See also [McCleary], [Lang], etc.

Definition 12.20.1. Let $\mathcal{A}$ be an abelian category.

1. A spectral sequence in $\mathcal{A}$ is given by a system $(E_ r, d_ r)_{r \geq 1}$ where each $E_ r$ is an object of $\mathcal{A}$, each $d_ r : E_ r \to E_ r$ is a morphism such that $d_ r \circ d_ r = 0$ and $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r)$ for $r \geq 1$.

2. A morphism of spectral sequences $f : (E_ r, d_ r)_{r \geq 1} \to (E'_ r, d'_ r)_{r \geq 1}$ is given by a family of morphisms $f_ r : E_ r \to E'_ r$ such that $f_ r \circ d_ r = d'_ r \circ f_ r$ and such that $f_{r + 1}$ is the morphism induced by $f_ r$ via the identifications $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r)$ and $E'_{r + 1} = \mathop{\mathrm{Ker}}(d'_ r)/\mathop{\mathrm{Im}}(d'_ r)$.

We will sometimes loosen this definition somewhat and allow $E_{r + 1}$ to be an object with a given isomorphism $E_{r + 1} \to \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r)$. In addition we sometimes have a system $(E_ r, d_ r)_{r \geq r_0}$ for some $r_0 \in \mathbf{Z}$ satisfying the properties of the definition above for indices $\geq r_0$. We will also call this a spectral sequence since by a simple renumbering it falls under the definition anyway. In fact, the cases $r_0 = 0$ and $r_0 = -1$ can be found in the literature.

Given a spectral sequence $(E_ r, d_ r)_{r \geq 1}$ we define

$0 = B_1 \subset B_2 \subset \ldots \subset B_ r \subset \ldots \subset Z_ r \subset \ldots \subset Z_2 \subset Z_1 = E_1$

by the following simple procedure. Set $B_2 = \mathop{\mathrm{Im}}(d_1)$ and $Z_2 = \mathop{\mathrm{Ker}}(d_1)$. Then it is clear that $d_2 : Z_2/B_2 \to Z_2/B_2$. Hence we can define $B_3$ as the unique subobject of $E_1$ containing $B_2$ such that $B_3/B_2$ is the image of $d_2$. Similarly we can define $Z_3$ as the unique subobject of $E_1$ containing $B_2$ such that $Z_3/B_2$ is the kernel of $d_2$. And so on and so forth. In particular we have

$E_ r = Z_ r/B_ r$

for all $r \geq 1$. In case the spectral sequence starts at $r = r_0$ then we can similarly construct $B_ i$, $Z_ i$ as subobjects in $E_{r_0}$. In fact, in the literature one sometimes finds the notation

$0 = B_ r(E_ r) \subset B_{r + 1}(E_ r) \subset B_{r + 2}(E_ r) \subset \ldots \subset Z_{r + 2}(E_ r) \subset Z_{r + 1}(E_ r) \subset Z_ r(E_ r) = E_ r$

to denote the filtration described above but starting with $E_ r$.

Definition 12.20.2. Let $\mathcal{A}$ be an abelian category. Let $(E_ r, d_ r)_{r \geq 1}$ be a spectral sequence.

1. If the subobjects $Z_{\infty } = \bigcap Z_ r$ and $B_{\infty } = \bigcup B_ r$ of $E_1$ exist then we define the limit1 of the spectral sequence to be the object $E_{\infty } = Z_{\infty }/B_{\infty }$.

2. We say that the spectral sequence degenerates at $E_ r$ if the differentials $d_ r, d_{r + 1}, \ldots$ are all zero.

Note that if the spectral sequence degenerates at $E_ r$, then we have $E_ r = E_{r + 1} = \ldots = E_{\infty }$ (and the limit exists of course). Also, almost any abelian category we will encounter has countable sums and intersections.

Remark 12.20.3 (Variant). It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. To accommodate this (and to get around certain technical issues) we introduce the following notion. Let $\mathcal{A}$ be an abelian category. Let $(T_ r)_{r \geq 1}$ be a sequence of translation or shift functors, i.e., $T_ r : \mathcal{A} \to \mathcal{A}$ is an isomorphism of categories. In this setting a spectral sequence is given by a system $(E_ r, d_ r)_{r \geq 1}$ where each $E_ r$ is an object of $\mathcal{A}$, each $d_ r : E_ r \to T_ rE_ r$ is a morphism such that $T_ rd_ r \circ d_ r = 0$ so that

$\xymatrix{ \ldots \ar[r] & T_ r^{-1}E_ r \ar[r]^-{T_ r^{-1}d_ r} & E_ r \ar[r]^-{d_ r} & T_ rE_ r \ar[r]^{T_ r d_ r} & T_ r^2E_ r \ar[r] & \ldots }$

is a complex and $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(T_ r^{-1}d_ r)$ for $r \geq 1$. It is clear what a morphism of spectral sequences means in this setting. In this setting we can still define

$0 = B_1 \subset B_2 \subset \ldots \subset B_ r \subset \ldots \subset Z_ r \subset \ldots \subset Z_2 \subset Z_1 = E_1$

and $Z_\infty$ and $B_\infty$ (if they exist) as above.

[1] This notation is not universally accepted. In some references an additional pair of subobjects $Z_\infty$ and $B_\infty$ of $E_1$ such that $0 = B_1 \subset B_2 \subset \ldots \subset B_\infty \subset Z_\infty \subset \ldots \subset Z_2 \subset Z_1 = E_1$ is part of the data comprising a spectral sequence!

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