## 12.17 Spectral sequences

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

Definition 12.17.1. Let $\mathcal{A}$ be an abelian category.

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 $r \geq 1$.

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

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

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.

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